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The Many Roles For Modeling In Ecosystem Science and
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presented by

Eli P. Fenichel

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THE MANY ROLES FOR MODELING IN ECOSYSTEM SCIENCE AND
MANAGEMENT

By

Eli P. Fenichel

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Fisheries and Wildlife

2007

ABSTRACT
THE MANY ROLES FOR MODELING IN ECOSYSTEM SCIENCE AND
MANAGEMENT
By

Eli P. Fenichel

Sustainability can be defined as maintaining or increasing society’s “genuine
wealth” or the pool of society’s capital. This pool of capital includes financial,
physical, natural, and human capital. Sustainable natural resource management science
is therefore fundamentally a capital management science. Natural resource
management involves making tradeoffs across potential investments to maintain or
enhance the pool of capital, with a focus on natural capital. Making such tradeoffs,
however, is complicated by the complexities of ecological and economic systems.
Models are useful tools for exposing, analyzing, and aiding in making tradeoffs among
potential natural resource management strategies or investments. This dissertation uses
a variety modeling approaches to expose, analyze, and aid in making tradeoffs
associated with thee fishery management problems. A common theme among these
three problems is they all involve aquatic species introduction; the first two address
aquatic infectious pathogens and the third one addresses fish stocking and control of an
invasive Species. All chapters are motivated by Great Lakes issues involving fish
stocking, but they all have broader implications.

Making tradeoffs is a fundamental part of natural resource management. In
order to make tradeoffs, information about the ecological and economic systems is
needed. Collecting this information involves tradeoffs as well (Chapters 2 and 3). The

complex nature of ecological and economic interactions complicates making tradeoffs,

and models can help organize thinking and facilitate analysis. Increasing awareness of
the joint production of ecosystem services, multiple equilibria and non-convexities, and
the imperfectly targeted nature of management actions (Chapter 4) should be viewed in
light of advancing knowledge about ecological and economic systems. It is not that
tradeoffs are more complex than in the absence of this knowledge, but rather that there
is a growing appreciation and desire to account for potential, unintended effects of

management actions.

Copyright by
Eli P. Fenichel
2007

TABLE OF CONTENTS

LIST OF
TABLES .......................................................................................... viii
LIST OF FIGURES .............................................................................. ix
CHAPTER 1
INTRODUCTION ................................................................................ 1
CHAPTER 2
A MODEL OF RENIBA C TERI UM SALMONINAR UM DYNAMICS IN GREAT
LAKES FISH POPULATIONS .................................................................. 5
Introduction ............................................................................... 5
System Background ...................................................................... 8
Model Development ...................................................................... 9
Simulations .............................................................................. 1 7
Results ................................................................................... 19
Model sensitivity for baseline conditions and the effects of
vertical transmission .......................................................... 19
The effects of management .................................................... 21
Impact of a reservoir and Whitefish .......................................... 26
Discussion ................................................................................ 28
Conclusion and Research Recommendations ........................................ 31
Appendix ................................................................................. 36
Tables ..................................................................................... 38
Figures .................................................................................... 42
CHAPTER 3

FISH PATHOGEN SCREENING AND ITS INFLUENCE ON THE
LIKELIHOOD OF ACCIDENTAL PATHOGEN INTRODUCTION

DURING FISH TRANSLOCATIONS ...................................................... 49
Introduction .............................................................................. 49
How Many Fish to Screen for Unwanted Pathogens ................................ 51
How Long to Continue Screening for Pathogens .................................. 56
A case study......' ........................................................................ 57
Screening effort for Lake Ontario sea lamprey and the likelihood
of pathogen translocation ...................................................... 58
Determining the duration of a Lake Ontario sea lamprey pathogen
Screeningprogram.....................' ....................................... 61
Discussion and Conclusion ............................................................ 62
Tables ..................................................................................... 67
Figures .................................................................................... 68
CHAPTER 4

vi

A BIOEC'ONOMIC MODEL OF CHINOOK SALMON STOCKING
AND ALEWIFE CONTROL TO MAXIMIZE ECOSYSTEM BENEFITS

THAT ACCOUNTS FOR INSTITUTIONAL CONSTRAINTS ............. - ........... 74
Introduction ............................................................................. 74
A Bioeconomic model of salmon stocking .......................................... 78

A model of angler behavior .................................................. 79
Fish population models ........................................................ 81
Optimizing social welfare through stocking .............. ' ........................... 8 3
Interpreting the optimality conditions ....................................... 86
Solving for the nonlinear feedback rule ..................... ‘ ................ 89
Functions specification for simulation ................................................ 90
Economic functions ............................................................ 90
Ecological functions ............................................................ 91
Optimal management ................................................................... 92
The case of no alewife damages ............................................. 92
Dynamics when w = wmax ............................................. 93
Dynamics when w = 0 ............................................... 94
Dynamics when w = w*(s, a) ........................................ 94
Optimal stocking strategies .......................................... 96
Optimal management with alewife-induced damages ..................... 99
Describing the phase plane ........................................... 99
Optimal stocking strategies ......................................... 100
Explaining the change in stocking strategy by accounting
for ecosystem disservices .......................................... 101
Sensitivity Analysis .................................................................... 102
Discussion ............................................................................... 105
The interaction of imperfectly targeted controls and
non-convexities ................................................................ l 05
Implication for salmon stocking policy .................................... 107
Appendix ............................................................................... 109
Figures .................................................................................. 112
REFERENCES .................................................................................. 117

vii

LIST OF TABLES

Table 2.1 Hatchery practice scenarios used to describe the potential output
of infected fish by hatcheries .................................................................. 38

Table 2.2 Elasticity results for wild Chinook salmon without vertical
transmission by age and health class ......................................................... 39

Table 2.3 Elasticity results for wild Chinook salmon with vertical transmission
by age and health class ......................................................................... 40

Table 2.4. The equilibrium number of wild + hatchery Chinook salmon,
by age, resulting from 15 combinations of between stocking levels and
hatchery practices ......................................................................................................... 41

Table 3.1 The probability that a given pathogen would be present in a

sample of a given Size (or detected in a subsample with

specificity = sensitivity = 1), for different prevalence levels among

source fish .................................................................................................................... 67

viii

LIST OF FIGURES

Figure 2.1. A conceptual model of the SEIE system ....................................... 42

Figure 2.2. The sequence of events throughout a simulated year ......................... 43

Figure 2.3. The percent of the wild Chinook population in exposed and

infected health classes, by age, for two different stocking levels at equilibrium.

Figure 2.4. A ZOO-year simulation of wild Chinook exposed and infected

..44

prevalence at age-4 .............................................................................. 45

Figure 2.5. The percent of the population in the exposed and infected health

classes by hatchery practice at equilibrium .................................................. 46

Figure 2.6. A ZOO-year simulation of wild Chinook exposed and infected

prevalence at age 4 .............................................................................. 47

Figure 2.7. The percent of the wild Chinook population in exposed and
infected health classes, among ages, and the effects of adding an

environmental reservoir and Whitefish to the model at equilibrium ..................... 48

Figure 3.1. Decision tree giving the probability of transferring a pathogen

given screening .................................................................................. 68

Figure 3.2. Number of fish that need to be tested to achieve 95% confidence
that a pathogen is absent from the transfer batch at a given prevalence level

(assuming sensitivity = Specificity = 1) ...................................................... 69

Figure 3.3. Probability that an infected fish would be transferred given
different levels of source prevalence and screening effort, assuming

translocation of 600 fish ....................................................................... 70

Figure 3.4. Probability (labeled on the contour lines) of translocating at

least one infected fish, given the (screening and transfer batch numbers ............... 71

Figure 3.5. Probability of translocating at least one infected fish, given the
screening sample size (x-axis) and the probability of a false negative

screening result (labeled contour lines) ..................................................... 72

Figure 3.6. Effects of screening costs, the cost of a disease outbreak, and
the probability of pathogen introduction on the number of years that a
screening program should continue, given that no infected individuals are

detected ......................................................................................... 73

ix

Figure 4.1. Phase plane illustrating the dynamics of stocking salmon at
the maximum rate, w = wma" .................................................................................... 112

Figure 4.2. Phase plane illustrating the dynamics of stocking zero salmon,
w = 0 ........................................................................................... 113

Figure 4.3. Phase plane illustrating the dynamics of the optimal solution in
the case with no alewife damages ......................................................... 114

Figure 4.4. Phase plane illustrating the dynamics of the optimal solution in
the case with alewife damages ............................................................. 115

Figure 4.5. Phase plane illustrating the dynamics of the optimal solution in
the case with no alewife damages and increased dependence on alewife for
recruitment to the fishery; 4) increased from 0.1 to 0.2 ................................. 116

CHAPTER 1

INTRODUCTION

Sustainability can be defined as maintaining or increasing society’s “genuine wealth”
(Arrow et al. 2004) or the pool of society’s capital. This pool of capital includes
financial, physical, natural, and human capital. Sustainable natural resource
management science is therefore fundamentally a capital management science. Natural
resource management involves making tradeoffs across potential investments to
maintain or enhance the pool of capital, with a focus on natural capital. Making such
tradeoffs, however, is complicated by the complexities of ecological and economic
systems. Models are useful tools for exposing, analyzing, and aiding in making
tradeoffs among potential natural resource management strategies or investments. This
dissertation uses a variety modeling approaches to expose, analyze, and aid in making
tradeoffs associated with three fishery management problems. A common theme
among these three problems is they all involve aquatic invasive species, two address
aquatic infectious pathogens and one addresses fish stocking and control of an invasive
species. All chapters are motivated by Great Lakes issues involving fish stocking, but
they all have broader implications as well.

In a number of management situations it can be difficult to know where to begin
investing resources in management or what tradeoffs exist. Increased research is one
potential solution, but this raises the question of how beSt to invest resources in
research. This is a common problem in the management of aquatic disease systems.

There is often little data at the population or system level. The second chapter of this

dissertation develops a multiple-host pathogen model for Lake Michigan focusing on
interactions between a pathogen, Renibacterium salmom'narum (the etiological agent of
bacterial kidney disease) and its hosts. This model is used to organize existing
information about the system. The model is analyzed to determine what kinds of
information are most likely to have the largest impact on indicators of concern. This
can help research funding agencies make tradeoffs among different research proposals.
Moreover, the model exposes certain management tradeoffs. For example, the model
indicates that there is likely a tradeoff in terms of adult stock sizes between increasing
the number of smolts stocked and increasing bio-security measures.

Once managers have an idea of what information is needed to assist in making
management decisions, there is a need to design data collection methods that are
effective, efficient, and connected to the management objective. Sampling is the result
of the need to make tradeoffs. There are never enough resources to collect or analyze
all the potential data. Yet, sampling design is often only based on statistical
considerations. Chapter 3 presents necessary concepts for developing aquatic pathogen
screening programs and connects probability theory to management objectives and the
need to make tradeoffs. Chapter 3 emphasizes that estimating only the chance of
pathogen detection is not enough; managers also should consider the likelihood and
consequences of facilitating pathogen introduction when designing screening programs.
Finally, Chapter 3 gives conditions for when a screening program should terminate
after a number of successive screening events, during which no infected individuals
have been detected. This result is based on the tradeoffs between the costs of screening

and the cost of a pathogen introduction.

Mathematical bioeconomic investment models (sensu Clark 1990) can aid
managers in exploring management tradeoffs. These models explicitly treat natural
resources as capital. Chapter 4 presents this type of model for the Chinook salmon
(Oncorhynchus tschawytscha) stocking program in Lake Michigan. Linear control
theory is used to solve for a stocking program that maximizes net present social
benefits as a function of angler welfare, averted alewife (Alosa pseudoharengus)
damages, and stocking costs. The model system includes an open access recreation
fishery dependent on salmon. Anglers respond to incentives provided by the salmon
stock. Alewives are an invasive species that provide ecosystem disservices by
damaging infrastructure, natural amenities, and preying on the eggs of valued native
fish. Salmon provide benefits to anglers and control the alewife population, but are
also dependent on the alewife population for survival. The model reveals that these
interactions result in complex tradeoffs that result in multiple equilibria and non-
convexities. These are related to the imperfect nature of controls for targeting alewife
management. Policy findings include that i) the optimal level of stocking is a function
of the current salmon and alewife stock and therefore stocking should be based on
“feedback” rules; ii) eradication (conservation) of invasive alewife may not be optimal
and will depend on initial conditions; iii) consideration of the ecosystem disservices
caused by alewives alters the nature of the problem in a qualitative way; and iv) the
maximum catch does not occur at the optimal equilibrium, but rather as part of an
approach path as alewife are depleted.

Making tradeoffs is a fundamental part of natural resource management. In

order to make tradeoffs, information about the ecological and economic systems is

needed. Collecting this information involves tradeoffs as well (Chapters 2 and 3). The
complex nature of ecological and economic interactions complicates making tradeoffs,
and models can help organize thinking and facilitate analysis. Increasing awareness of
the joint production of ecosystem services, multiple equilibria and non-convexities, and
the imperfectly targeted nature of management actions (Chapter 4) should be viewed in
light of advancing knowledge about ecological and economic systems. It is not that
tradeoffs are more complex than in the absence of this knowledge, but rather that there
is a growing appreciation and desire to account for potential, unintended effects of
management actions.

This dissertation presents three chapters represented three ways models can be

useful in making natural resource management decisions.

CHAPTER 2
A MODEL OF RENIBA CTERI UM SALMONINARUM DYNAMICS IN
GREAT LAKES FISH POPULATIONS:
Implications for bacterial kidney disease management and research in Lake

Michigan

2.1 Introduction
Fishery managers are increasingly concerned about fish health, but little is known about
the dynamics and effects of pathogens in fish populations outside of the laboratory,
aquaculture, and hatchery facilities. Quantitative modeling has provided a useful
approach to organize and improve understanding of host-pathogen dynamics in wild
populations, but such efforts have been largely applied to terrestrial ecosystems
(McCallum et al. 2004), with a few exceptions from marine systems (Murray et al.
2001; Ogut et al. 2005), and have not been applied to managed freshwater ecosystems.
The presence of Renibacterium salmoninarum (Rs), the causative agent for bacterial
kidney disease (BKD) is well established in the Great Lakes, a large managed
freshwater ecosystem. Benjamin and Bence (2003b) estimated a substantial increase in
salmonid mortality in Lake Michigan associated with increased prevalence of Rs
(Holey et al. 1998). Rs is also believed to be “one of the most important bacterial
diseases affecting wild and propagated anadromous salmonid stocks [in the Pacific
Northwest]” (Wiens and Kaattari 1999), but little is known about the pathogen’s
dynamics in the wild.

Here we develop a model of fish-Rs dynamics based on free-swimming fish
populations to aid fishery managers in two ways. First, a model can help synthesize

and interpret complex datasets to draw inferences about the possible consequences of

management strategies. Patterson (1996) uses a complex dataset to estimate disease
induced mortality caused by the phycomycete fungal parasite Ichthyophonus hoferi in
North Sea herring (Clupea harengus) from catch-at-age data combined with prevalence
data, but does not evaluate how alternate management strategies may affect this
parameter. Patterson (1996) notes that failing to account for disease effects can lead to
on overestimation of fishing mortality andthis may have implications for harvest
policy. Barlow (1991; 1996) and Smith and Cheeseman (2002) employ host-pathogen
models to evaluate alternative control strategies for bovine tuberculosis
(Mycobacterium bovis) in New Zealand possums (Trichosurus vulpecula) and English
badgers (Meles meles) respectively.

Second, host-pathogen models also can be used to develop and evaluate hypotheses
about system dynamics, thereby exposing critical uncertainties and identifying
opportunities for efficient and insightful data collection. A lack of data on disease
transmission, however, does not pose a barrier for fulfilling these objectives (Caley and
Hone 2004). For example, Murray et al. (2001) developed a model of herpesvirus
transmission in pilchards to examine assumptions about transmission processes. Ogut
et al. (2005) demonstrated how host-pathogen models can be used to gain insight into
the disease dynamics, explore uncertainties, and estimate parameters for a captive
Chinook salmon (Oncorhynchus tschawytscha) populatidn infected with Aeromonas
salmonicida (the etiologic agent of furunculosis).

Following terrestrial investigations, existing fish-pathogen modeling efforts have
focused on single fish populations and ignore the influence of management activities on

the system. However, the ability of pathogens to influence ecological relationships

among Species is receiving increased attention (Hatcher et al. 2006), and it is often
necessary to consider pathogen dynamics in the context of a broader host community to
understand the effects of a pathogen on a particular host species (Dobson 2004, Tsao
2004). Moreover, many freshwater systems, such as Lake Michigan, are highly
influenced by human activities (e.g. stocking of salmonids) that affect host community
dynamics.

We incorporate management features into a multiple-host disease model of Rs in
Lake Michigan to qualitatively explore the implications of modeling assumptions,
model sensitivity, and management actions on host-pathogen dynamics. Three host
stocks were included in the model: two types of non-native Chinook salmon - those that
result from spawning in the wild (wild Chinook), and those that result from collection
of gametes from feral adults and hatchery production (hatchery Chinook) - and native,
wild spawning lake Whitefish, Coregonus clupeaformis. The Chinook salmon
population was divided into wild and hatchery spawning components to enable analysis
of management effects and to help design future experiments. Chinook salmon and
lake Whitefish usually, but not always, occupy different habitats, do not consume the
same prey, and do not eat each other, therefore we have assumed that direct pathogen
transmission between these two species is negligible. However, pathogen transmission
between these two species may be facilitated by an environmental reservoir. A
reservoir captures the ability of Rs to survive in water or sediment and may also
represent, to some extent, the presence of infectious pathogen occurring within other

currently unknown host species that are not explicitly’modeled.

2.2 System Background

In 1987, Chinook salmon in Lake Michigan suffered a large increase in
mortality (Benjamin and Bence 2003a) associated with a widely observed outbreak of
BKD and a high prevalence of Rs, first documented in the Spring of 1988 (Holey et al.
1998). Researchers subsequently documented Rs andclinical signs of BKD in lake
Whitefish populations (Jonas et al. 2002). Furthermore, Rs has been documented in
other Lake Michigan salmonines and in sea lampreys (Eissa et al. 2006) and is likely
present in other Lake Michigan fishes (M. Faisal, Michigan State University, personal
communication).

Knowledge of Rs/BKD dynamics in Lake Michigan is extremely limited,
despite evidence of the importance of this disease for salmonid dynamics (Holey et al.
1998; Benjamin and Bence 2003a, b). Most knowledge about fish-bome disease comes
from observations and studies conducted in hatcheries or fish production facilities
(Stephen and Thorbum 2002), and the dynamics of BKD are no exception (Wiens and
Kaattari 1999). From such experiments it is known that Rs can be transmitted
horizontally (i.e., directly) among juvenile and adult salmonids (Mitchum and Sherman
1981; Balfry et al. 1996), as well as vertically from adult females to eggs (summarized
in Wiens and Kaattari 1999). Further research has shown that fish may become
infected when the pathogen is present in sufficiently high levels in the surrounding
water (summarized in Hamel 2005). However, it is unclear which transmission
processes drive pathogen dynamics among free-swimming fish populations or are

important for pathogen maintenance.

2.3 Model DeveIOpment

Host-pathogen models are population models where the host population is distributed
among health classes, and pathogen dynamics are tracked based on the proportion of
the population in each health class. We use a susceptible-exposed-infected-exposed
(SEIE) host-pathogen framework (Fig 1).

The SEIE model is a modification of the standard susceptible-infected-recovered
(SIR) framework for modeling host-pathogen relationships (Anderson and May 1991).
The health states S, E, I, and R represent four subpopulations - susceptible, exposed,
infected, and recovered, respectively. Susceptible individuals do not host the pathogen
currently, but may do so in the future. Infected individuals are ones that are infectious
and subject to disease induced mortality. Recovered individuals have acquired complete
immunity (Anderson and May 1991). With respect to BKD, we assume that any
immune population is negligible, and thus, we drop the R class, as is common for
chronic diseases (Barlow 1991). Furthermore, the Slow development of BKD motivates
the inclusion of a latent/exposed (E) stage (Heesterbeek and Roberts 1995 and Murray
et al., 2001). Exposed fish in health class E harbor Rs, and host the pathogen; however,
they are not yet infectious, nor are they subject to disease induced mortality. In our
model, infection may abate, and individuals in class I may return to class E. This can
be thought of as an infection in remission.

The number of individuals in a. given health class depends on rates of transition
among the different classes, including the rates at which individuals become exposed to
the pathogen, become infectious, die due to disease, or go into remission. Taking host

population dynamics into account is important for a chronic disease like BKD because

pathogen dynamics occur on a comparable timescale with that of hosts. Thus, the
number of individuals among different health classes depends on birth and death rates,
which are affected by abiotic factors and density-dependence. In these systems, the
pathogen can persist in the population for multiple generations (i.e., is endemic, see
Barlow 1991 and 1996 for examples). In contrast, for highly transmissible and ofien
virulent pathogens, because pathogen dynamics occur on much shorter timescales than
that of hosts, recruitment of new susceptibles through. birth is usually ignored. In
Murray et al.’s‘ (2001) pilchard example, the pathogen infects a. large number of fish
and then dies out within one generation (i.e., is epidemic).

Our host-pathogen model is a system of population models for three fish host types
- lake Whitefish (w), wild Chinook, (c), and hatchery Chinook, (h). In addition to fish
population hosts, we consider two pathogen reservoir types, lake (I), and river (r). At
any moment the model tracks five sources of pathogen — wild spawned chinook,
hatchery chinook, lake Whitefish, the river reservoir, and the lake reservoir.
Furthermore, we used an age-structured model for Chinook salmon and a stage-
structured (pre-recruit, post-recruit) model for lake Whitefish (over much of their life,
lake whitefish population dynamics are affected by similar factors in similar ways).
Thus there are five age classes for each Chinook type and two stage classes of
Whitefish. Each age (stage) of each host type is essentially treated as an individual host
type. Chinook salmon shift from riverine habitats at age-0 to lake habitats during
juvenile life stages, and then back to riverine habitats to reproduce. Only Chinook

salmon at age-0 are exposed to the river reservoir. Lake Whitefish spend their entire

10

life in the lake proper and typically do not recruit to the commercial fishery until age-3
or 4.

Given this general framework, we now explain our model in detail. The model
operates on an annual time step, and within each year there are five sequential steps
(Fig. 2): (l) recruitment and aging, (2) pre-transmission mortality, (3) changes to the
pathogen reservoirs, (4) disease transmission (movement of fish among the S, E, and I
states), and (5) post-transmission mortality. In the following section, these steps are
discussed in detail.

Step 1. At the beginning of each time step, the age composition of each host group
is updated. For the age—structured Chinook salmon populations this means that all fish
of age a at time t that survive the year become age a+1 at time t+1. For the stage-
structured lake Whitefish, however, only a portion of the surviving stage-0 fish become
stage-l at the start of the next time step.

Recruitment of both wild Chinook salmon and lake Whitefish is calculated from a

I

Ricker (1954) stock-recruitment relationship, I‘ = aAe-yA , where A is the mature
host abundance (with A' =A in the standard model), r is recruitment and (1,7 are
parameters. For stocked Chinook salmon, recruitment is Simply the number of fish
stocked. The mature Chinook salmon producing wild recruitment is assumed to include
both wild and hatchery fish of all spawning age classes. Lake Whitefish recruitment
(progeny of fish in stage-l) are added to those that remain in stage 0 from the previous
year.

The Ricker function includes a. density-independent term (a) related to the

- A'
maximum production of offspring per adult, and a density-dependent term (e y ), that

11

accounts for limitations on recruitment due to nest crowding a lack of oxygen or similar
processes. We assume that the magnitude of density-dependence is determined by the

entire spawning population for both wild Chinook and lake Whitefish. For lake

Whitefish A' =77”, 2' N w,l , where NW, [is the population of stage 1 Whitefish. For the

wild Chinook salmon this A, = 77¢- ; ZZWi,a,Ni,a , where l// is a measure of the
i a

relative fecundity and proportion mature for each age class, and where NM is the

population of host type i (i=c,h) at age a.1 In contrast, we assume that the density-
independent term iS scaled only by the spawning adults in the specified cohort. Thus,

in our model A at A'. This gives the following system of equations:

Nh,0, :St
N _22 t. N —7CnCt ~_{ ’7}
c,0t “ az,a z,a,te 1" c,
(2.1) i a

’7 77
NW,0t :((l'p)Nw,0t +Ointhe w Wt)

The parameter p is the proportion of Whitefish recruited from stage-0 to stage- 1 during
the previous year (see below) and s, is the number of hatchery salmon that are stocked.
In addition to determining how many progeny are produced, we also must index
progeny by infectious status, because their distribution across health classes affects the
total mortality rate they experience during the next step of the model. Rs can be
transmitted vertically, i.e., from mother to egg through reproduction (summarized in

Wiens and Kaattari 1999). To model vertical transmission we modify the density-

 

' We assume that #4, =aa/a4, where age-4 has the largest value of 0:, though more general specifications
are possible.

12

independent component of recruitment in equation (2.1) to distinguish the contribution
of infected and uninfected adults to infected and susceptible offspring. We assume that
spawners in class E do not transmit pathogen to their progeny, and that any offspring
that contract the pathogen from vertical transmission are placed in class I. The
parameter v is introduced to account for the proportion of infected progeny produced
from class I spawners (i.e., the vertical transmission rate). We assumed a 10% rate of
vertical transmission, which is within the observed range (l%-14%) found in laboratory
experiments for coho salmon (Oncorhynchus kisutch) (summarized in Hamel 2005).

For wild Chinook recruitment the resulting equations are:
_. '7’ '77“
51,0 - ZZai,a(N,-,a,, " 1i,a,, + (1 ._ Vlli,a,,)9 ' '
i a

Ii,0 = zzai,a(VIi,a,t k-i’ini
i a

(2.2)

where 5130 and 11,0 represent recruits of susceptible fish from uninfected and infected

parents, and infected fish from infected parents, respectively. For lake Whitefish the

term (1 -p) S50 or (1 -p) 1,30 is added as in equation (2.1) to account for susceptible or

infected fish remaining in stage 0.

Step 2 (and 5). Instantaneous mortality is applied twice during the year (Figure
2.2), before and after disease transmission. Let the proportion of the year represented
by the solid line on the left side of Figure 2.2 be ‘t (in our simulations we assume I =

0.5). The standard equation for applying instantaneous mortality is

(2-3) Ni,a,t+r " Ni,a,te ,

13

which assumes that the total mortality rate, zm (for host type i and age (stage) a) is
constant. In a host-pathogen model even if the component instantaneous mortality rates
(i.e., natural, non-BKD mortality (m), fishing mortality, (f), and BKD-induced mortality
(d)) are constant, the value of z in this model could not be constant throughout the year
because the proportion of N that experiences disease mortality (d) changes as a result of
disease transmission (step 3) during the year. Therefore, we model the mortality of

each health class separately.

(m+f )1

Si,a,t+r : Si,a,te

E- = . e
(2.4) t,a,t+r t,a,t

Ii,a,t+r = Ii,a,te
For the second application of mortality (right solid line in Figure 2.2), r is replaced with
l-r and t is replaced with 1:. This approach implicitly assumes that disease transmission
occurs over a relatively short period of time during the year. A shorter time step would
be required to model a transmission process that occurred throughout the year, but this
is only numerically Significant to the disease and population dynamics if the changes
among classes of host are fairly large and disease mortality is comparable to magnitude
of the other mortality sources.
We assumed that the instantaneous rate of BKD-induced mortality, d, is 0.8 of the
rate of natural mortality, m. This value was derived empirically to allow for pathogen
persistence at and interior level in the model — substantially higher values led to

pathogen extinction, and substantially lower values led to 100% infection.

14

Step 3. The next event in the model is updating the environmental reservoirs.
Reservoir size is defined in terms of infectious units. Environmental reservoirs do not
have population growth relationships, as Rs does not reproduce outside of the host. The
source of Rs to reservoirs is assumed to be from the carcasses of infected individuals
that died over the past year. Pathogen loading of the reservoir is modeled as
proportional to the number of individuals dying due to natural (non-BKD) or BKD-
induced mortality that are in health class I. We assume the rate of shedding by live fish
is negligible (or directly proportionally to dead fish). In the absence of replenishment,
the abundance of pathogen in the reservoir is expected to decline and is modeled by an
exponential decay function.

Step 4. Once they are established in an age (stage) class, susceptible fish move into
the exposed class via contact with infected fish through random mixing. During each
time step fish move from health class S to E at time 1: according to horizontal
transmission function Hm, which is specific to host type and age. Horizontal

transmission generally follows Dobson (2004)
Sr+A = _Hi,a(Sr a It ) : Sr — Si,a,r Z X :B(i,a),(j,k)1j,k,r
j k

(2.5) ET-l-A 2' Hi,a(Sz"It' ) = IT + Si’a,72§fl(i,a),(j,k)1j,k
J

’2'

where B is the per capita rate of pathogen transmission from host j cohort k to host i
cohort a (Dobson 2004), where i, j index all potential host types and a, k index their age
or stage, and A represents a small change in time. .

Immediately following the recruitment of susceptibles to the exposed class, the

advancement process between the exposed and infected classes occurs. Traditionally,

15

exposed individuals are included to model a fixed latency period, and advancement is
simply determined by a rate parameter that represents the inverse of the average latency
period (Heesterbeek and Roberts 1995). However, researchers do not know what an
average latency period is for Rs/BKD, and furthermore, the BKD epidemic in Lake
Michigan coincided with a period of reduced Chinook growth rates, suggesting that
nutritional stress may play an important role in triggering clinical disease (Holey et al.
1998). Here, we make advancement a function of nutritional stress. We assume that
nutritional stress is a function of intraspecific competition and that increased host

densities approximate reduced food resources per individualz, thus making

advancement a function of ZZNi,a,T+A . As the density of the host population
i 0

increases, ceteris paribus, so does the rate at which exposed individuals become
infected. Conversely, low host densities result in increased food resources per
individual and thus a net recovery from the infected to the exposed class. That is,
infected fish return to the exposed class at a greater rate than exposed fish become
infected. We define the advancement function Q aS the process by which exposed fish

advance to the infected class (negative advancement implies remission to the exposed),

AET+2A = Er+A — Q(Y(A)’ Ei,a,r+A91i,a,r+A)
(2.6) AIr+2A : IT+A + Q(Y(A)aEi,a,r+AaIi,a,r+A)

where A = ZZNi,a,r+A and where Y is a function that defines the rate of
i a

movement between classes E and I and must have two qualities. First, IY] .<_ 1, and

 

2 This implicitly assumes that the prey resource is fixed.

16

second, there must be some A =/\7 where Y (A7 )=O, i.e., where there is no net

movement from exposed to infected. Function Y takes the form

lA—qeq)
YA =‘ ‘
(2'7) ( ) V(A—qeq)+ qeq +q50%

 

where (150% is the population density at which 50% of the exposed (infected)

individuals become infected (exposed) and qe = A7 . The parameter qe also
. q q

determines the direction of movement:

Y(A)Ei,a,r+A : ifA > qeq

Y A ,E' a]. =
.(2.8) Q( ( ) "W” WHA) -Y(A)Ii,a,r+A’ifA<qeq

Simulations

Two modeling workshops and numerous correspondences with fish health
specialists and Great Lakes fishery biologists were conducted to solicit expert opinion
on model processes and parameter values for calibration of the model. There are no
estimates for important parameters in the model such as transmission rates. However,
using realistic estimates for recruitment, survival, and mortality parameters, it was
possible to bound the range of feasible transmission parameters (see Appendix for
parameter values used in the simulations). Transmission parameters where chosen to
create realistic scenarios, so that meaningful cases could be examined. For most cases,
transmission parameters were chosen so that prevalence rates could stabilize at interior

levels (i.e., the pathogen persisted but did not result in 100% infection). In other cases,

17

transmission parameters were set so that infection could not be maintained and the
pathogen vanished from the system.

We conducted a series of simulations to identify emergent properties resulting
from the combination of population and host-pathogen theories. The model was
simulated for a sufficiently long time period so that transient dynamics subsided and the
system appeared to arrive at equilibrium (where changes in the values essentially were
zero). Most simulations ran for 100 years, but 200 year simulations were used when
the system was deliberately perturbed during the first 100 years. The initial source of
Rs is assumed to be hatchery fish, though this is assumption is explored in the results
section.

We investigated how various management activities may influence model
outcomes. First, we investigated scenarios that only included hatchery and wild
Chinook salmon. This allowed us to explore the model’s sensitivity to demographic
parameters, the effects of stocking rate and hatchery practices (see Table 2.1 for
descriptions of hatchery practice scenarios), and the interactions between stocking rates
and hatchery practices. We also characterized conditions that could lead to an outbreak
of BKD like that observed on Lake Michigan without the complication of alternative
hosts. Next, we included an environmental reservoir in the model to understand how
this may impact the results of Chinook salmon only simulations. Finally, we included
Whitefish and examined interactions with Chinook salmon. We implemented the model

in Microsoft Visual Basic interfaced through a Mic-roset‘t Excel Workbook.

l8

2.4 Results
Model sensitivity for baseline conditions and the effect of vertical transmission

We began by exploring the sensitivity of the model to parameter assumptions.
The model, even when restricted to Chinook salmon, has a large number of parameters
and analysis and presentation of model sensitivity of each one is not feasible.
Moreover, in the absence of data, a detailed sensitivity analysis for each parameter may
be less illuminating than other sorts of analysis (discussed later in the Results sections).
We focus our sensitivity analyses on parameters for which there are the least amount of
data in the literature and on parameters that ecological and epidemiological theory
predict would have the greatest effect on model outcomes. The first model ignores
vertical transmission and disease induced mortality to focus on the impact of horizontal
transmission.

To examine sensitivity of a given parameter, we adjusted the value of the
parameter by 5% and then compared simulation results to that of the default value. To
assess sensitivity, we employed the concept of elasticity (Pindyck and Rubinfeld 2001
p. 30; Coulson and Godfray 2007) and calculated the percent change in the density of
wild Chinook salmon at equilibrium in each health class divided by a 5 percent change
in the model parameter. If the percent change in the density at equilibrium was > 5%
(yielding an elasticity > Hi), then the model was deemed to be elastic and sensitive to
changes in the parameter; likewise, if the percent change was < 5% (elasticity < Ill),
then the model was deemed to be insensitive to changes to the parameter. Parameters
explored included the Ricker density-dependence parameter (y), the transmission

parameter (B), and the two parameters in the advancement function (qeq and q50%,). The

19

model showed greatest sensitivity to the Ricker density-dependence parameter but was
also sensitive to the transmission parameter and q50%, with elasticities >> |1| (Table
2.2). These results were consistent across all ages of Chinook salmon and for both the
exposed and infected health classes. Wild Chinooksalmon prevalence rates varied by
health class and age between being inelastic and elastic with respect to changes in the
qeq parameter (Table 2.2).

The sensitivity of wild Chinook salmon prevalence to changes in parameters
generally declined when vertical transmission was added to the model (Table 2.3). For
all age and health classes except age-4 health class E, elasticities were largest for y
followed by 0. Many elasticities, however, were < |l| when vertical transmission was
added to model, including all age-3 and age-4 exposed elasticities as well as all
elasticities with respect to q50%. Finally, all elasticities with respect to v were < |l|.
When results were inelastic (< |l|) this implies that v does not have a strong direct effect
on the outcome.

The presence of vertical transmission does seem to have large effect on the
model even though small changes in vertical transmission may not have
disproportionately large effects. The inclusion of vertical transmission had two effects
that need to be considered. First, vertical transmission increased the prevalence of both
exposed and infected individuals of all ages at equilibriUm. The second effect was that
increased vertical transmission increased the number of age-O exposed fish. This
occurs because increased vertical transmission increases the number of age-0 infected
fish with which susceptible age-0 fish make contact, i.e., additional infected fish

provide an additional source for secondary exposures. Disease-induced mortality

20

reduces the prevalence of disease, ceteris paribus. This is because disease induced
mortality selects against infected individuals, thereby reducing the probability that these

individuals transmit pathogen to susceptible individuals prior to death.

The effects of management

The sensitivity analyses indicated that parameters that directly influence
density-dependent processes had a strong influence on model results. Therefore, it is
likely that manipulations to host density due to management practices will substantially
influence model outcomes. Managers primarily influence Chinook salmon density in
Lake Michigan through stocking. There are two components to a stocking program that
could influence Rs prevalence in the lake: the total number and the infection prevalence
of fish stocked. We simulated three different stocking levels: 5, 8, and 10 million fish
per year, which correspond to recent levels of Chinook stocking into Lake Michigan (5
million) and 1.6x and 2x increases in numbers stocked. We also examined five
hatchery pathogen management scenarios (Table 2.1) that span a plausible range of
prevalence of infected, stocked fish. At each stocking level we ran simulations for each
of the hatchery practice scenarios (Table 2.4).

When hatchery practices are effective at limiting the prevalence of infection to
5% of that found in the parents that supply the hatcheries with gametes (scenario 1:
good, stock-dependent), the resulting prevalence of exposed fish exceeds that of
infected fish at all ages (Figure 2.3) when 5 million fish are stocked. A 60% increase in
stocking more than doubles the prevalence of fish in 'both exposed and infected health

classes, and at the higher stocking rate, the preValence of infected fish approaches that

21

of exposed fish by age 4. This is true for both hatchery and wild Chinook salmon. The
prevalence at equilibrial densities varies between hatchery and wild Chinook salmon by
< 2%.

A temporary increase in stocking (with hatchery scenario 1 still in place) led to
an increase in prevalence for an extended period (Figure 2.4). When stocking was
increased from 5 million to 8 million fish in years 85 and 86 of a simulation and
subsequently returned to 5 million, the prevalence of infection continued to increase
and peaked in year 94 (Figure 2.4). Increased host density had a temporary positive
feedback effect, as evidenced by the fact that by year 91 all fish stocked in year 86 have
been removed from the system so that there was no longer a direct effect from stocking;
yet prevalence continued to increase. However, the overall increase in prevalence of
infected fish was small. Initially, increasing stocking slightly decreases the prevalence
of exposed individuals (Figure 2.4). At a higher density, fish move from the exposed to
the infected class at a greater rate, but prevalence of infected fish has not yet increased
enough to recruit new fish to the exposed class. More susceptible fish become exposed
once the prevalence of infected fish increases.

The other four hatchery management scenarios all led to substantially greater
prevalence of exposed and infected Chinook salmon in both wild and hatchery sub-
populations (Figure 2.5). The differences are greatest for age-0 fish, particularly for
satisfactory or poor practices, and diminish as fish age. More than two-thirds of fish
are exposed or infected by age four, except for simulations conducted with good stock-
dependent hatchery practices. As hatchery practices deteriorate (i.e., a higher

propbrtion of stocked fish. are infected), there is a greater difference in prevalence

22

between wild and hatchery raised fish, especially for the “poor” scenario, where 95% of
hatchery fish were infected while only 45% of wild age 4 fish were infected. Hatchery
practices that mimicked what would be expected in wild spawned fish (natural
replication) produced fewer exposed and infected fish than hatchery practices that
limited infection prevalence to 5%. This is due to our assumption about vertical
transmission, rates and may not hold if the rate were increased. For example, a 10%
vertical transmission rate implies that if <50% of wild spawning fish are infected, then
natural replication (scenario 3) will release fewer than 5% infected fish.

A simulated short-term deterioration of hatchery practices (Figure 2.6) had a
similar, but much greater effect than increased stocking (Figure 2.4) on prevalence
patterns (note the difference in the y—axis between Figures 2.4 and 2.6). A change in
hatchery practices in years 85 and 86 from good stock-dependent to poor led to a
sudden change in the distribution of fish within health classes. By year 86 there was a
large increase in the density of exposed fish, but by year 88, prevalence of exposed fish
began to decline. The prevalence of infected fish continued to rise until year 91, a year
after fish stocked in year 86 would have died. As with increased stocking, the increase
in prevalence of infected fish persisted beyond the lifespan of the directly affected
(hatchery) fish, but not for as long as with the stocking rate change. This Shorter lasting
effect of poorer hatchery practices may be due to the high prevalence rates that are
achieved — the system cannot maintain Such high prevalence rates without a continued
influx of infected fish. Moreover, fish density — the other factor affecting prevalence —

declines as more fish experience disease-induced mortality.

23

Finally, we examined how the interaction of stocking, hatchery practices, and
disease dynamics affect fish density (Table 2.4). From these simulations, we can make
three important observations about the interaction between population size, stocking
level, hatchery practices, and disease dynamics. First, increasing stocking by 100%
only increases the population of age-0 fish by 28% - 32%, depending on the hatchery
practice scenario. This is a consequence of density dependence in the stock-recruitment
relationship - increased stocking in a system where survival from spawning to age-0
decreases with increasing spawning stock. The largest percent gain in age-0 fish was
made with good stock-dependent hatchery practices because fewer of these fish are
exposed to disease-induced mortality.

Second, the percent increases in age-4 fish with increases in stocking above 5
million fish are also proportionally much less than the commensurate increase in
stocking. In contrast to age—0 fish, however, the greatest percent gain realized at age-4
from stocking was achieved when hatchery practices were satisfactory or poor. This
counterintuitive result was due to the combined effects of density-dependent
transmission and disease-induced mortality. When hatchery practices were relatively
good, an increase in density due to stocking led to increased transfer of fish from
susceptible to exposed and from exposed to infected states and consequently more fish
died due to disease induced mortality. When hatchery practices were poor, most fish
start life already infected so increased stocking rates cannot increase the rate of
infection much further — the effect of increased density on the rate of infection matters

less once most of the fish are infected.

24

Finally, improving hatchery practices resulted in the greatest number of age-4
fish overall, as would be expected. When hatchery practices were good and stock
dependent, and 5 million fish were stocked, the model predicted just fewer than 2
million age-4 fish (Table 2.4). In contrast, when twice as many fish were stocked under
poor hatchery practices the model only predicted 1.6 million age-4 fish. Of course this
difference will ultimately depend on the actual rate of disease induced mortality,
transmission, and other parameters.

We have assumed that infected Chinook salmon experience a disease-induced
mortality rate that is only 0.8 of the natural mortality rate. This assumption has allowed
Rs to persist in the system even under the best hatchery practices and low stocking
levels. If the disease induced mortality rate were substantially higher, then Rs would
die-out under the stock-dependent good hatchery practices scenario because, on
average, infected fish die before they infect at least one additional fish. As hatchery
practices deteriorate, Rs could persist in the system, but prevalence rates would remain
low. Furthermore, such low disease induced mortality rates are inadequate to create a
major fish-kill as was observed in the late 19805 in Lake Michigan.

In our model increases in stocking alone, and thus increases in Chinook salmon
density, cannot create a fish kill. Instead, a temporary lapse in hatchery practices is
required. If we were to assume that hatchery practices normally ranged between the
satisfactory and good stock-independent scenarios, then 'a lapse to the poor hatchery
practices scenario could create a substantial fish-kill, provided disease-induced
mortality rates are sufficiently high. For example, increasing the disease-induced

mortality rate to 3.6 times the natural mortality rate (with no change in the fishing

25

mortality), and assuming satisfactory hatchery practices as the status quo, would allow
two years of poor hatchery practices to reduce the total Chinook salmon stock to 82%,
73%, 67%, 63%, 59% of the equilibrium stock siZe for age 0 to 4 fish respectively if 5
million fish were stocked, and 76%, 63%, 57%, 51%, 47% if 8 million fish were
stocked. Assuming that hatchery practices were good as opposed to satisfactory to start
with results in larger reductions in the stock, but would also result in low equilibrium
prevalence of Rs. Larger reductions stem fromthe fact that better hatchery practices
produce a greater equilibrium density of fish that can be subject to a fish kill. Stocking
more fish also sets up a situation for larger reductions, both due to the disease dynamics

and the simple fact that stocking more fish allows more fish to die during a die-off.

Impact of a reservoir and Whitefish

The effect of including a reservoir was to increase the number of newly exposed
host individuals per unit time. By definition a reservoir provides a higher force of
infection (Caley and Hone 2005) on the target Species than the target Species directs
upon itself at a given point in time (Horan and Fenichel unpublished). The
environmental reservoir was initially calibrated so that it had a small but positive effect
on the prevalence of exposed individuals at equilibrium (Figure 2.7). We then analyzed
the sensitivity, using elasticity calculations, of changes in prevalence with respect to
reservoir parameters for each wild Chinook salmon health class. Equilibrium
prevalence was relatively insensitive (elasticities < III) to changes in the reservoir
parameters. However, the prevalence in the wild Chinook salmon population at

equilibrium was slightly more sensitive to increases in the lake reservoir than the river

26

reservoir. Prevalence at equilibrium was also more sensitive to the parameter
associated with pathogen release than to the parameter associated with pathogen decay.

The addition of a second host population, Whitefish, further increased the
prevalence of exposed and infected salmon (Figure 2.7). The Whitefish population,
which was assumed to interact with the Chinook salmon populations only through
additions of Rs to the lake reservoir, had a large effect on prevalence in Chinook
salmon at equilibrium. When Whitefish are included, the prevalence of exposed and
infected salmon approximately doubles (Figure 2.7), due to dying infected Whitefish
boosting the Size of the lake reservoir.

The source of Rs in the Great Lakes is unknown. When Rs was discovered in
the Great Lakes in 1955, it was assumed that Rs was introduced with imported hatchery
fish eggs, though this view has come under question (Hnath and Faisal 2004). We used
the model to explore the hypothesis that Rs historically existed in Lake MiChigan at low
levels, but that hatcheries and salmon introductions amplified Rs. There are insufficient
data to formally test this hypothesis, but our model enabled us to attempt to recreate
conditions necessary for it to be possible. We began by assuming a minimal,
undetectable, and stable prevalence of Rs in Whitefish (prevalence << 0.1%) with no
Chinook salmon in the system. This led to a stable environmental reservoir of Rs.
When uninfected Chinook salmon were introduced into the system two events would
occur simultaneously. The stocked fish would be exposed to the existing reservoir and
some would become infected. As well, they would begin to establish wild spawning
populations. Subsequently, the infected Chinook salmon would begin contributing to

the environmental reservoir, resulting in increased prevalence of exposed Whitefish.

27

Under these conditions we were able to develop simulations that resulted in equilibrium
levels of infection in salmon which were comparable to levels that occurred in the
model when the original source of Rs was the hatchery. However, for Rs to persist in
the Whitefish population at undetectably low levels prior to the introduction of Chinook
salmon, the Whitefish to Whitefish transmission rates would have to be .quite small.
This would result in a low prevalence, of Rs in Whitefish at equilibrium, even in the
presence of salmon, which is in contrast to recent findings (M. L. Jones, unpublished
data). The density of Whitefish in the model is much greater than the density of
Chinook salmon, therefore even when a significant number of salmon become infected
they have a small impact on the reservoir relative to the Whitefish population.3 It is
possible that Rs has always been part of the Great Lakes ecosystem, but the scenario we
describe above (lake Whitefish as a native, historic source of Rs) seems unlikely. There
are number of alternative hypotheses, however. For example, Rs may have been
present in Whitefish at higher, detectable rates, but they were never tested for the
bacterium and clinical signs of BKD were generally absent. Alternatively, Rs may have
indeed been introduced with hatchery fish or eggs, but at a time prior to the start of

modern Chinook stocking program.

2.5 Discussion
The results of this modeling exercise are best interpreted qualitatively due to the

general lack of quantitative knowledge about parameter eStimates. Nevertheless, we

 

3 There are certainly more Whitefish than Chinook salmon in Lake Michigan in terms of numbers, but the
difference in biomass may not be as great. It is possible that biomass may be more important for
contribution to a reservoir. However, our model operates on numbers in a fixed volume of water
(density). Whether the amount of Rs shed is a function of numbers or biomass is an empirical question
that has yet to be answered.

28

feel that a number of useful inferences can be drawn about relationships among
different components of the Lake Michigan Rs system. These inferences can help
prioritize future research and recommend management experiments. Our modeling
results suggest, not surprisingly, that hatcheries may play. an important role in Lake
Michigan Rs dynamics, and that density management of Chinook salmon can be
important to pathogen dynamics. The role of hatcheries, however, is not independent
of natural recruitment. An understanding of the interaction between wild recruitment
and stocking is needed to better understand how the stocking of Chinook salmon affects
disease dynamics. Finally, while a multiple species approach will be necessary to fully
understand Rs dynamics in Lake Michigan, simpler models focusing on Chinook
salmon populations alone can provide valuable insights for management.

Both Holey et al. (1998) and Hnath and Faisal (2004) have expressed concern
that hatchery practices can affect Rs dynamics and outbreaks of BKD in Lake
Michigan. Hnath and Faisal (2004) state that practices such as using Heath—type
stacking incubators can contribute the spread of Rs between eggs. This effectively
increases the rate of vertical transmission in hatcheries. The role of vertical
transmission (or horizontal transmission ameng eggs) may not have been fully
appreciated. Our results Show that there are two effects of increased rates of infection
in eggs. The first effect is clear; if there are mere infected eggs, this will lead to more
infected fiSh. The lesslobvious second effect is that age-0 that are born infected (either
due to vertical transmission or hatchery induced egg-toiegg transmission), represent an
additional source of exposure for susceptible wild and hatchery age-0 fish. Our model

also suggests that even short-term failures in hatchery procedures that result in releases

29

of a high proportion of infected fish can lead to a drastic and prolonged rise in infection
of both wild and hatchery salmon. Finally, increasing stocking rates are not likely to
overcome poor hatchery practices with respect to density of older Chinook salmon, due
to disease induced mortality and density-dependent compensation in wild recruitment
(Table 2.4). -

Increasing natural prOduction of Chinook salmon in Lake Michigan complicates
management of Rs. Managers would benefit from a better understanding of the
recruitment dynamics of Lake Michigan wild Chinook salmon. Specifically, it would
be useful to know if stocking of Chinook salmon leads to compensation in wild
production at current abundance levels — as we assumed in our models. The density of
Chinook salmon also affects Rs-host dynamics. Thus, managers need to account for the
response of wild Chinook salmon when making stocking decisions both for disease and
production objectives. Because of compensatory processes, it is possible that a
reduction in stocking may not reduce the number of fish in the lake and may free
resources to improve remaining hatchery conditions. Finally, wild spawning fish may
be less likely to create infected eggs than hatcheries. Hamel (2005) reports that it is
common for Pacific hatcheries to produce eggs with a 90% prevalence of Rs.

A variety of fish species in Lake Michigan may carry Rs. Thus, to understand
the ecology of pathogens alternative hosts often need to be considered (Dobson 2004).
Dividing Chinook salmon into wild and hatchery is the'first step in a multiple host
approach. The concept of “pathogen spillover” from a reservoir to a host of interest has
begun to receive attention (Daszak et al. 2000; 'Power and Mitchell 2004). The

inclusion of an environmental reservoir is used in part to capture potential Rs dynamics

30

in other species, but is only a first approximation of how Rs dynamics in other species
might affect Chinook salmon. To fully understand how Chinook salmon, Rs, and other
species interact, the dynamics of those other Species will need to be considered
explicitly.” C '

It would be valuable for managers to understand how stocking contributes to the
spread and maintenance of Rs in Chinook salmon. Our modeling has highlighted an
indicator that may be useful for determining the relative role of density and hatchery
practices. If there are significant differences in prevalence between hatchery and wild-
reared Chinook salmon, hatchery practices likely have a large affect on Rs dynamics,
whereas similar prevalence would imply that density has a greater impact on Rs

dynamics

2.6 Conclusion and Research Recommendations

Models that integrate population and epidemiological theory seldom have been
used to improve understanding and management of fish populations or fish health, and
pathogen and disease models have not been explicitly considered in the leading fish
population dynamics texts (Hilbom and Walters 1992; Quinn and Deriso 1999). These
texts acknowledge the importance of disease in Shaping mortality, but simply lump
disease induced mortality into unmanageable natural mortality. Moreover, such natural
mortality estimates are applied to the entire population (combining health classes) and
are often assumed constant for older fish. This implies that the proportion of fish in a
specific health class does not change. For some populations mortality may change

substantially as a result of host-pathogen dynamics. This is especially a concern in

31

systems where a substantial number of fish are stocked, or where invading species
might be bringing with them new pathogens. More careful consideration of the role of
host—pathogen dynamics is merited in these cases, and the development of models is
critical to “building an ecological approach to fish health management,” (Stephen and
Thorbum 2002). In this paper we have introduced a mechanistic approach for
modeling the role played by pathogens in shaping fish population dynamics.

Models enable the organization of information that allows formulation of
meaningful hypotheses and identification of priority research. In the absence of good
empirical data upon which to base parameter estimates, it is best to emphasize the
qualitative behavior of the models and their sensitivity to a range of plausible parameter
values. Such qualitative conclusions nevertheless provide a strong basis for determining
critical areas of uncertainty and thus priorities for future research. Accordingly, we
recommend seven priorities for improving understanding of Rs dynamics in Lake
Michigan. We suspect that these recommendations would be applicable to other
fishery-pathogen systems as well, particularly where hatcheries are used to supplement
fishing opportunities.

First, from our results we hypothesize that if hatchery practices, i.e. the
proportion of infected fish being released from hatcheries, play an important role in
pathogen dynamics, then prevalence 'in hatchery reared fish should be substantially
higher than prevalence in wild reared fish. A method of discriminating hatchery from
wild fish, such as tagging or marking hatchery fish or the use of natural chemical

markers, would prove valuable for such comparisons.

32

 

Second, it is imperative that researchers measure the stock-recruitment
relationship of wild spawning Chinook salmon in Lake Michigan. The prevalence for
all but one health-age class was most sensitive to the density-dependence parameter in
the Ricker function. It is important for managers to have better idea about the three-
way interaction between wild recruitment, stocking, and disease transmiSSion, how the
level of stocking influences wild recruitment. Moreover, this is a critical element to
understanding vertical transmission — the next research recommendation.

Third, the role of vertical transmission should be further examined. Assessing
size of runs of wild spawning Chinook salmon and testing resulting eggs would provide
information both about wild recruitment and vertical transmission in the wild.
Alternatively, 5- month old wild-spawned smolts could be collected and examined for
Rs, and prevalence rates compared to those being released from hatcheries. Ultimately,
prevalence goals for hatcheries will need to be connected to vertical transmission in the
wild.

Fourth, the model we used to represent the hypothesized link between
nutritional stress and the advancement (function Q) of Chinook salmon hosts from
exposed to infected states (Holey et al. 1998), needs to be further investigated. Rs
dynamics can be sensitive to parameters in the Q function, and little is known about this
relationship other than the general observation of increased prevalence of BKD when
Chinook salmon grth rates were low in the late 19803. The predator-prey dynamics
of Lake Michigan salmonines have been the subject of Considerable study (Szalai 2003,
Madenjian et al. 2005, also see chapter 4), but this needs to be coupled with a better

mechanistic understanding of how nutritional stress. affects Rs dynamics.

33

Fifth, we urge a more structured approach to assessing fish health at the
population level. It is imperative that sampling be designed to estimate epidemiological
parameters, which cannot rely on sampling opportunistically nor testing only those fish
contributing gametes to hatchery facilities. Stocks that are regularly assessed may have
sampling designs adequate for fish health assessment already in place. Additionally,
integrating stock assessment models with epidemiologiCal models could facilitate
estimation of transmission parameters. Statistical approaches that integrate stock and
health assessment should be the subject of future research, but improved health
assessment sampling designs can yield a wealth of information prior to the
development of methods for full integration of stock and health assessment.

Sixth, the feasibility of testing museum specimens of ' Whitefish and other
potential hosts such as lake trout and other Coregonid Species for presence of Rs
(nucleic material) should be investigated. The detection of Rs in Whitefish that pro-date
modern salmon introductions (or even the BKD outbreak in the 19805) would provide
insight into the ecological role of Rs. Our model, along with recent data (unpublished)
indicates that if Rs historically had been present in Whitefish, prevalence rates would
have been reasonably high.

Finally, although much can be learned from studies focused on Chinook salmon,
our model suggests that other species could play an important role in the Rs dynamics
in Lake Michigan, even if they do not directly interact with Chinook salmon as
predators, competitors or prey. We found that RS dynamics in Chinook salmon can
depend on the contribution of environmental reservoirs as a source of the pathogen, and

other abundant species that harbor the pathogen could play an important role in

34

determining the dynamics of this reservoir. At a minimum it would be valuable to
determine how widespread Rs is in aquatic species that are either abundant or interact
strongly with Chinook salmon, or both. Two obvious candidates would be sea lampreys
and alewives.

Alone, models cannot resolve all fishery management questions. However,
models are a necessary component of fishery management and fishery research.
Models play a major role is stock assessment and harvest policy (Hilbom and Walters
1992; Quinn and Deriso 1999), but have played virtually no role in fish health
management. In the future, we hope that the same analytical rigor that has been applied
to other areas of fisheries management is applied to fish health. In this paper, we have
developed such a model to address the affects of a pathogen on a fishery. We hope that
this model can serve as an example for addressing other emerging and chronic fish

health issues.

35

2.7 Appendix. Parameters and values used in the model; the same values were used for
both wild and hatchery Chinook salmon.

Parameter values
With disease

 

 

. Increase With .
induced . . Altematrve
Base . disease Whitefish .
mortality . whrtefish
case . induced and the .
and vertical . . hypothesrs
. . mortality reservorr

transmrssron

Chinook Salmon Parameters

Ricker (1, age 0 0 0 0 0 0

Ricker or, age 1 0 0 O 0 0

Ricker or, age 2 l l I l l

Ricker or, age 3 5 5 5 5 5

Ricker or, age 4 10 10 10 IO 10

Rickery 0.39 0.39 0.39 0.39 0.39

Instantaneous fishing

mortality, age 0 O 0 O 0 0

Instantaneous fishing

mortality, age I O O 0 O 0

Instantaneous fishing

mortality, age 2 0.1 0.1 0.1 0.1 0.1

Instantaneous fishing

mortality, age3 0.15 0.15 0.15 0.15 0.15

Instantaneous fishing

mortality, age 4 0.2 0.2 0.2 0.2 0.2

Instantaneous natural

mortality, age 0 0.4 0.4 0.4 0.4 0.4

'“S‘anti‘neous natural 0.25 0.25 0.25 0.25 0.25

mortality, age 1

Instantaneous natural

mortality, age 2 0.25 0.25 0.25 0.25 0.25

'“S‘amfneous "3“"3' 0.25 0.25 0.25 0.25 0.25

mortality, age 3

Instantaneous natural

mortality, age4 0.25 0.25 0.25 0.25 0.25

Instantaneous disease

induced mortality 0 0 2 0'9 O 2 O 2

Transmission

coefficient, [3, between 0.11 0.11 0.11 0.11 0.11

Chinook '

Vertical transmrssron 0 O l 0.1 0 1 O I

rate

q“, 2.5 2.5 2.5 2.5 2.5

950% I40 I 40 I40 I40 140

Probability of

infection from 0 0.1 l 0.1 l 0.1 l 0.1 l

reservoir

 

36

2.7 Appendix continued. Parameters and values used in the model; the same values
were used for both wild and hatchery Chinook salmon.

With disease

Increase

With

 

 

 

 

 

Base induced disease Whitefish Alterrtr‘atrve
case morta rty and induced and the White 18h.
vertical . . hypothesrs
. . mortality reservorr
transmrssron
Whitefish
Parameters
Ricker a, stage 0 0 0 0 0 0 I
Ricker a, stage 1 0 0 0 7.25 7.25 ;
Ricker y 0 0 0 0.015 0.015 .
Instantaneous 1
fishing mortality, 0 0 0 O 0
age 0
Instantaneous
fishing mortality, 0 0 0 0.34 0.34
age 1
Instantaneous
natural mortality, 0 0 O 0.4 0.4
age 0
Instantaneous
natural mortality, 0 0 0 0.25 0.25
age 1
Instantaneous
disease induced 0 0 0 0.2 0.2
mortality
Transmission
coefficient, B, 0 0 0 0.01 0.00425
between Whitefish
vemca.‘ . 0 0 0.1 0.1
transmrssron rate
qeq O 0 0 200 200
(150% 0 0 0 800 800
Probability of
infection from 0 0 0 0.0035 0.0035
reservoir
Reservoir
Parameters
Loading rate (river) 0 0 0 0.05 0.05
Exponential lose rate 0 O 0 0.75 0.75
(I'IVCI')
Loading rate (lake) 0 0 0 0.05 0.05
Exponential lose rate 0 0 0 0.75 0.75

(lake)

 

37

2.8 Tables

Table 2.1. Hatchery practice scenarios used to describe the potential output of infected

 

 

fish by hatcheries.

Name Description

1. Good — stock The prevalence of Rs in stocked fish is 5% of prevalence
dependent seen in wild spawned fish. Chosen as an ideal outcome.

2. Good — stock If there is at least one infected fish in the wild population,
independent then the prevalence of Rs in stocked fish is 5%. Chosen

based on standard hatchery protocol.

3. Natural Replication The prevalence of Rs in stocked fish is equal to the
prevalence seen in wild spawned fish.

4. Satisfactory If there is at least one infected fish in the wild population,
then the prevalence of Rs in stocked fish is 19%. Chosen
based on what is believed to be the historical maximal
tolerance for hatchery infection in Michigan.

5. Poor If there is at least one infected fish in the wild population,
then the prevalence of Rs in stocked fish is 95%. Chosen
to represent hatcheries with no biosecurity (based on
Hamel 2005).

 

 

 

 

38

Table 2.2. Elasticity results for wild Chinook salmon without vertical transmission by
age and health class. Elasticity is the percent change in the prevalence of a given health
class at age divided by 5% (the amount by which the parameter value was increased). If
the elasticity > Ill, then the model is deemed to be elastic and sensitive to changes in the
parameter; likewise, if elasticity < Ill, then the model is deemed to be insensitive to

changes to the parameter.

Health

 

 

Parameter Class Age Class
O l 2 3 4

R‘Cker dens‘ty' E -12.68 -1224 -11.77 -11.27 -10.71
dependence parameter ,y

1 -12.95 -12.74 -12.43 -l2.14 -11.85
D‘sease transm‘ssm“ E 10.24 8.88 7.61 6.39 5.23
parameter, 0

I 10.24 9.44 8.59 7.84 7.16
Advancement E -7.55 -7.10 -6.61 -6.09 554
parameter, (150%

I -8.14 -7.89 -7.59 -7.30 -7.02
Advancement
(equilibrium) parameter, E -1.02 -0.90 -0.82 -0.73 -0.63
qeq

I -1.11 -1.17 -1.04 -0.96 -0.90

 

39

Table 2.3. Elasticity results for wild Chinook salmon with vertical transmission by age
and health class. Elasticity is the percent change in the prevalence of a given health
class at age divided by 5% (the amount by which the parameter value was increased). If
the elasticity > Ill, then the model is deemed to be elastic and sensitive to changes in the
parameter; likewise, if elasticity < lll, then the model is deemed to be insensitive to

changes to the parameter.

 

 

Health

Parameter Class Age Class
0 1 2 3 4

RM“ dens‘ty' E -3.77 -277 -l.81 -0.88 0.03
dependence parameter ,y

1 -3.73 -3.64 -324 -2.83 -245
”56356 transm‘ss‘on E 3.03 2.01 1.17 0.48 -0.09
parameter,B

1 2.42 2.24 1.87 1.53 1.23
Advancement E -020 -0.10 -004 0.01 0.07
parameter, (150%

1 -0.23 -O.28 -022 -0.18 -015
Advancement
(equilibrium) parameter, E -1.78 -l.18 -0.58 -0.01 0.54
qeq

1 -225 -2.18 -1.94 -170 -1.47
vemca' transm‘ss‘o" E 0.48 0.32 0.18 0.06 -004
parameter,v

1 0.80 0.59 0.45 0.35 0.28

 

40

Table 2.4. The equilibrium number of wild + hatchery Chinook salmon, by age,

resulting from 15 combinations of between stocking levels and hatchery practices. See

Table 2.1 for definitions of each hatchery practice.

 

 

 

 

 

 

 

 

 

 

Stocking Age % gain at age 4

level above stocking
0 1 2 3 4 5 million fish

1 - good stock dependent

5 million 8.926 6.907 4.812 3.171 1.976

8 million 10.662 8.066 5.421 3.405 2.004 1.4%

10 million 11.765 8.785 5.803 3.578 2.067 4.6%

2 - good stock-independent

5 million 9.042 6.897 4.705 3.013 1.814

8 million 10.698 8.011 5.323 3.305 1.925 6.1%

10 million 11.771 8.711 5.707 3.495 2.008 10.7%

3 - natural replication

5 million 9.007 6.910 4.751 3.073 1.871

8 million 10.696 8.016 5.330 3.312 1.930 3.2%

10 million 11.771 8.705 5.701 3.489 2.004 7.1%

4 - satisfactory

5 million 9.078 6.803 4.544 2.846 1.674

8 million 10.686 7.851 5.128 3.139 1.806 7.9%

10 million 11.722 8.547 5.537 3.361 1.918 14.6%

5 - poor

5 million 8.853 6.315 4.047 2.448 1.400

8 million 10.362 7.242 4.550 2.701 1.517 8.3%

10 million 11.347 7.847 4.879 2.868 1.595 13.9%

 

41

2.9 Figures

Deaths (m+f)
__ Susceptible (S)

 

  

H 1 Contact pathogen

 

 

g Deaths (m+f)
E Exposed (E)
E
.".:.’
=
1..
u
g ,
Remission to Q Progress to
Exposed ’ Infectious

Deaths (m+f+d)

 

 

 

Figure 2.1. A conceptual model of the SEIE system

 

Day 1 ‘ Day 365
I I I I I I I I .
‘ Mortality i ‘ Mortalrty
. eq 2.4 . . . eq 2.4
Recruitment Change Transmrssron Disease Age class
and aging in eq 2, 5 Progression observed
eq 2.1 & 2.2 reservoir eq 2.6, 2.7, &
2.8

Figure 2.2. The sequence of events throughout a simulated year. Mortality is applied

over the solid line. Equations corresponding to processes in the diagram are noted.

43

 
    
 
 
 

 

 

 

 

 

 

 

 

 

 

60 l 56 54
I 5 million stock, E 52 5]
50 I 8 million stock, E
El 5 million stock, I 44
III 8 million stock. I "—
g 40 — 33 37 : I
0 1n '1
.1: .
g )4
E ‘ :Q
.E 1 25 E :
E 23 1 1
3 ‘ . ‘
a 1
o. i
4 l3 ,
10 : . ‘
0 l 2 3 4

Age
Figure 2.3. The percent of the wild Chinook population in exposed and infected health

classes, by age, for two different stocking levels at equilibrium.

44

 

 

 

 

30 -
Exposed
25 —
E
U 20 ~
5'.
E
m 15 - Infected
.5
§ 10 ~
3
o.
5 4
0 I I T T
0 50 100 150 200

Year

Figure 2.4. A 200-year simulation of wild Chinook exposed and infected prevalence at

age-4. Stocking is increased from 5 to 8 million for two years starting in year 85.

45

100 - 95

  
   

 

 

 

 

 
 
 
 

 

131 Wild E
90 T 82
hatchery E
80 - . '7“
g [I Wild 1 1’;
N 0.
3 70 ‘ lhatcheryl ES
2 60 ‘ x
a ..
2 50 45 ’z
.E i
a 40 ‘ i
E e
0 C
2 30 - 3
0 ‘ V
“‘ 20 .
10 8 8
n 2 2
good SD good SID natural satisfactory poor
replication
Hatchery practice
Panel A.
100 ~ 95
90 - 12 Wild E
80 _ a hatchery E
3’, [:1 Wild 1
a 2
Ta 70 lhatcheryl
§ 60 -
3
.1: 50 7
.E
a 40 “
5
2’. 30
U
°‘ 20 ,
10 a
0 a?
good SD good SID natural satisfactory poor
replication
Hatchery practice
Panel B.

Figure 2.5. The percent of the population in the exposed and infected health classes by
hatchery practice at equilibrium. Panel A shows age—0 fish and panel B shows age-4

(SID = stock independent, SD = stock dependent).

46

70.
60*
50*
404

3 0 4 Exposed

 

 

20 I Infected

Percent in Health Class

 

'04

 

 

0 50 100 150 200

Year

Figure 2.6. A ZOO-year simulation of wild Chinook exposed and infected prevalence at
age 4. Hatchery practices are changed from good stock-dependent to poor for two

years, starting in year 85.

47

DageO
Iagel

4523 53

   

mwowmummummmommuwmuowmwommommgommuw.

 

601

50
0
o-
O
O
0

320 5:8: E Eoocom

     

reservoir

reservoir no reservoir reservorr
reservoir

no
reservoir

and
Whitefish

only
Infected

and
Whitefish

only

Exposed

Figure 2.7. The percent of the wild Chinook population in exposed and infected health

and the effects of adding an environmental reservoir and Whitefish

9

among ages

classes,

to the model at equilibrium.

48

CHAPTER 3
FISH PATHOGEN SCREENING AND ITS INFLUENCE ON THE
LIKELIHOOD OF ACCIDENTAL PATHOGEN INTRODUCTION DURING
FISH TRANSLOCATIONS

3.1 Introduction

Fisheries managers often employ fish translocations to help achieve a variety of
objectives, including the improvement or rehabilitation of existing stocks, the
establishment of populations in new areas, the reintroduction of extirpated stocks, and
for use as biological pest control agents.1 Fish translocations, however, have been
implicated in undesirable and costly pathogen translocations (Stewart 1991; Reno 1998;
Murray et al. 2002). Consequently, there is increasing interest in conducting risk
analyses to evaluate the tradeoffs between stock improvement and disease risk (Stephen
2001; Bartley et a1. 2006). Pathogen transfer risk is influenced by human decisions and
management actions in addition to biological and environmental factors (Shogren and
Crocker 1999; Caley et al. 2006), and therefore key components to managing and
evaluating risk are programs that influence managers’ decisions, such as pre-
translocation screening of fish.

Pathogen screening does not eliminate risk and imposes additional costs on fish
translocation programs. First, there is the marginal cost of screening each additional
individual fish (e.g., diagnostic tests and opportunity cost of the agency personnel).
Second, many aquatic animal screening programs require the sacrifice of the screened

individuals, thereby reducing the number of individuals left for transfer. Third, the

 

' This paper focuses on fish translocation; however, the techniques and conclusions also apply to
hatchery management programs.

49

extra time that fish are held awaiting screening results prior to transfer can reduce their
survival. Given the benefits and costs of a screening program, an important question is,
“How should a screening program be designed?” Mathews et al. (2006) advocate full
veterinary screening of all mammals that are to be translocated, but acknowledge that
such screening is unlikely to be feasible, especially in the case where screened
individuals must be sacrificed, and is likely to be extremely costly. A common
approach in fisheries is to screen subsamples of the groups of organisms slated for
transfer (USFWS and AF S-FHS 2005). This often involves sacrificing the screened
individuals.

Managers interested in designing screening programs to help evaluate and
manage risk need to consider two questions: i) how does the number of fish screened
affect the likelihood that a pathogen is translocated; and ii) how long should a screening
program continue if no pathogens are found? The literature mostly has focused on
statistical approaches for estimating prevalence from increasingly complex sampling
designs (Williams and Moffitt 2001; Munoz-lanai et al. 2006), and less on how such
estimates can be used to manage risk.

In this paper, we review screening design and explain the implications of
pathogen screening for use in fish health risk analysis (not to be confused with
surveillance). We emphasize the relationship between screening assumptions and
objectives, while addressing the two questions raised above. We illustrate key concepts
using a case study of sea. lamprey (Petromyzon marinas) translocations in the Great
Lakes; these are undertaken as part of the sterile male release method for biological

control in the St. Marys River integrated sea lamprey management program (Schleen et

50

al. 2003; Twohey et al. 2003). We provide a critique of this screening program’s
implementation.

Ultimately, aquatic pathogen screening programs need to be designed to achieve
‘acceptably low’ probabilities of pathogen translocatiOn. 'An interdisciplinary approach
(e.g.,' a structured decision analysis sensu Peterrnan and Anderson 1999) is needed to
determine the acceptable probability of pathogen translocation and overall risk. Such
an analysis is beyond the scope of this paper. We present a narrower analysis that
clarifies the linkage among screening program design, assumptions, likelihood of

pathogen transfer, and risk analysis.

3.2 How Many Fish to Screen for Unwanted Pathogens

Verifying the absence of a pathogen is all but impossible; given a probability >
0 that the organism is present, the harder one looks, the more likely one is to find an
organism, (Regan et al. 2006). Consider the situation in which a batch of fish is
collected for the purpose of transfer. A subsample of fish drawn from the batch is first
screened for pathogens. Assume that if an infected fish is detected in the subsample,
then the remainder of the batch will not be transferred.2 The transfer proceeds if no
infected fish are detected in the subsample. Assume that screened fish are scarified and
not transferred. Further, assume that the subsample of fish represents (i.e., is a random

sample of) the transfer batch. That is, whether any given fish hosts a pathogen is

 

2 This is a fundamental assumption in our analysis. It is, however, necessary to assume that some
probability of pathogen transfer is acceptable. Therefore, it is conceivable that prevalence in the source
population would still be acceptably low such that the transfer would proceed when a small number of
fish in a large sample test positive. Williams and Moffitt (2001) and Munoz-Zanzi et al. (2006) provide
techniques for estimating prevalence and confidence bounds when a proportion of fish in a sample test
positive. We believe, however, that our assumption is typical of the way most screening programs
operate.

51

independent of whether any other fish hosts a pathogen, and the probability that any
given fish hosts a pathogen is the same, (i.e., identical), regardless of whether the fish
comes from the subsample or the batch that is transferred.

A Simple decision tree (Figure 3.1) illustrates that the relevant statistic is the
probability that a pathogen is present among the fish released, given that the pathogen
was not detected within the subsample. Therefore, the probability of transferring a
pathogen given screening is
(3.1) Pr(t) = [1 — Pr(d)] * Pr(b),
where Pr(d) is the probability of detecting the pathogen during screening, and Pr(b) is
the probability that the transfer batch contains at least one infected fish.

To calculate the probability of failing to detect a pathogen (i.e., [l-Pr(d)]) in a
given sample, the Poisson or binomial distribution can be used (Green and Young
1993; Dell’modame and Prati 2005). The Poisson distribution approximates the
binomial when the population (batch) is large, but for small batches, representative
sampling assumptions for both approaches may be violated (Dell’modame and Prati
2005). We base our analysis on the Poisson distribution, but see endnotes for
analogous equations for the binomial distribution. The necessary sample size, n, for
detecting at least one infected fish from a population with prevalence m at a specified
confidence level (probability), x is: 3
(3.2) n=—ln(1—x) m.

The true prevalence of pathogens in a wild population is often unknown and must be

assumed. This assumption can greatly affect the likelihood of translocating at least one

 

3 The formula using the binomial distribution is n = ln(l — x)/In(l — m).

52

infected fish. As the assumed prevalence level decreases, the number of fish that
needed to be tested to achieve a given confidence of detection increases at a rapidly
increasing rate, going to infinity as the assumed prevalence declines to zero (Figure
3.2). Similarly, as the desired confidence level increases, the number of fish that need
to be tested also increases rapidly.

Equation 3.2 can be rearranged to provide the probability of detecting at least
one infected individual in a sample of size n from a population with a known (or
assumed) prevalence m: 4
(3.3) x = 1 -exp(— mn) .

Thus, the probability of failing to detect a pathogen is Pr(d) =1 -x.

Equations (3.2) and (3.3) implicitly assume that the screening instrument’s
diagnostic specificity and sensitivity are perfect and equal to 1 (i.e., that infected
animals always test positive and uninfected animals always test negative). Equations
(3.2) and (3.3) can be generalized to account for the probability of obtaining a false
negative, p, and a false positive, q. Equation (3.2) provides the necessary sample size if
one requires confidence x given observable prevalence m. If there is a known bias in
the observed prevalence, resulting from an imperfect test specificity or sensitivity, then
m in equations (3.2) and (3.3) is replaced with observable prevalence
(3.4) m' = m(1—- p) + (1 — m)q.

If only false negatives are considered (q = 0), then observable prevalence decreases,
resulting in the need for a larger n. When only false positives are considered (p = 0),

the observable prevalence increases reducing n. Issues regarding sensitivity and

 

4 The formula using the binomial distribution is x = l — (1 — my .

53

specificity of tests are addressed in more depth by Dell’Omodarrne and Prati (2005) and
Munoz-Zanzi et a1. (2006).

Equation (3.3) also allows us to calculate the probability that a pathogen is
present in the translocation batch by replacing the subsample size with the number of
(untested) fish in the batch to be transferred (note that m, and not mi should be used for
this calculation). Table 3.1 provides calculated probabilities for a range of prevalence
levels and sample Sizes. For all prevalence levels, as the number of individuals
screened or in the batch increases, so does the probability that at least one infected fish
will be detected in the screening subsample or will be present in the transfer batch. For
example, if prevalence is 5%, then the probability of detecting at least one infected fish
in the subsample increases from 0.78 to nearly 1.0 as the sample size increases from 30
to 200 fish.

When the prevalence of a pathogen is high, the situation is straightforward;
infected fish will be detected during screening and the translocation will be halted
(Figure 3.1, top branch and Figure 3.3). Similarly, when the actual prevalence is near
zero, the situation is again straightforward; the pathogen is unlikely to be detected in
the screened sample and is unlikely to be present the translocation batch. In such a
case, the fish transfer proceeds with a low likelihood of pathogen transfer (Figure 3.1,
bottom branch and Figure 3.3). Moderately low prevalence rates (approximately, <
0.05) present a more challenging situation, since the pathogen may go undetected in the
subsample, but is likely to be present in the larger transfer batch (e.g., when prevalence
= 0.01 and sample size = 60, there is a 45% chance that an infected fish is in the

transfer batch — see the middle branch of Figure 3.1, and Figure 3.3).

54

All else equal, screening smaller subsamples results in higher probabilities of
translocating at least one infected individual over a wider range of prevalence. The
ability to decrease the likelihood of a translocation event by increasing subsample size,
however, decreases as prevalence increases. If prevalence is sufficiently high, then
smaller subsarrrples are nearly as likely to detect a pathogen as are larger subsamples
(Figure 3.3). Increasing subsample Size generates the greatest effect at moderately low
prevalence levels (Figure 3.3).

Figure 3.4 illustrates the interaction between numbers of fish screened and
numbers of fish transferred, assuming a given true source prevalence (solid lines, true
prevalence = 0.05; dashed lines, true prevalence = 0.01). The figure shows that the
probability of transferring an infected fish decreases as the number of fish screened
increases, for a given number of fish in the transfer batch. The probability of
transferring an infected fish increases, however, as the number of fish transferred
increases, for a given number of fish screened. Decreases in true prevalence increase
the probability of transferring an infected fish for a given number of fish screened and a
given number of fish transferred. This is illustrated by considering the situation where
60 fish are screened and 240 fish are transferred (point A on Figure 3.4). If the true
prevalence were 5%, there is a 5% chance of transferring an infected fish, whereas if
the true prevalence were 1% then the likelihood of transferring an infected fish climbs
to 50% (Figure 3.4).

The utility of a screening program for preventing the transfer of an infected fish
depends on pathogen prevalence. In the absence of a screening program, high

prevalence is more likely to result in a transfer because at low prevalence there is a

55

lower probability that infected individuals are in the batch. However, as the number of
fish screened increases, the isoclines in Figure 3.4 cross, because at high prevalence

screening is likely to detect infected fish and thus stop the translocation.

3.3 How Long to Continue Screening for Pathogens

At some point the aquatic pathogen screening program should be terminated if
pathogens are not found in multiple successive screening events and there is no
evidence that a new (existing) pathogen has been introduced (re-introduced). Regan et
al. (2006) develop a framework for evaluating invasive plant search programs that is
applicable to this pathogen screening question. The length of time a screening program
should persist, given that no infected individuals are found, Should be based on three
factors: i) the cost of the screening program, CS; ii) the expected cost of a pathogen
translocation, CT; and iii) the probability of facilitating a pathogen translocation
(Pr(t) ¢ 1). Regan et al. (2006) offer a ‘rule of thumb’ calculation that balances these

factors:

—CS
(3.5) y _ ln( C'T*ln(Pr(t)))/ln(Pr(t))

where y is the number of screening events (given a constant interval) that screening

 

Should continue when no pathogen is being detected. This criterion can be applied to
screening programs and ‘disease free’ certification programs. This approach assumes
that there are no other data that conflict with the results of the Sereening program, i.e,
new accounts of pathogen introductions. This rule implies that when the expected
damages from translocation are small, CT< -CS/ln(Pr(t)), then no screening should

take ‘ " ' place.

56

3.4 A Case Study

We now present the case of sea lamprey transfers to illustrate screening program design
considerations for risk management and demonstrate the application 'of the methods
discussed above. Sea lampreys are parasitic fish that invaded the North American
Great Lakes during the early 20th century and have been the Object of a pest ‘control
program since 1955 (Smith and Tibbles 1980). The sterile male release technique
(SMRT) is a component of the sea lamprey control strategy for the St. Marys River,
which is a major sea lamprey spawning area and therefore an important source of
parasitic sea lamprey for Lake Huron (Schleen et al. 2003). The SMRT provides social
and economic benefits through increased production of lake trout (Salvelinus
namaycush) (Lupi et al. 2003). Sterilized male sea lampreys are released to compete
with wild fertile males to attract female mates, thereby reducing female reproductive
output. A high ratio of sterile to fertile males is desirable for cost-effective control of
sea lampreys (Haeseker et al. in press). Managers go to considerable effort to obtain
adult male sea lampreys for sterilization. Greater than 85% of the collection comes
from upper Great Lakes sources close to the St. Marys River (Twohey et al. 2003), but
some sea lampreys used in the SMRT program are collected in Lake Ontario, which is
isolated from the upper Great Lakes by Niagara Falls.

Interest in screening sea lampreys for pathogens before translocation, to prevent
unwanted introductions, has intensified due to the discovery of “pathogens of c'oncem”
in Lake Ontario. Specifically, a Screening program has been initiated to screen for
Heterosporis sp., a microsporidian parasite associated with percids (Bergstedt and

Twohey 2005) that has not been detected in Lake Huron. Heterosporis sp. is known

57

from Lake Ontario, and sea lampreys there potentially feed on infected fish. Therefore,
there is concern that the movement of sea lampreys from Lake Ontario to the St. Marys
River couldfacilitate the spread of Heterosporis sp. to valuable fish Species in the
upper Great Lakes (Bergstedt and Twohey 2005). The sea lamprey’s actual disease
status andgrole in pathogen transmission are unknown. Sea lampreysfrom the Great
Lakes have been found to host two other pathogens of concern that are already present
in Lake Huron fishzyAeromonas salmonicida (causing furunculosis) and Renibacterium
salmoninarum, (causing bacterial kidney disease) (Eissa et al. 2006). These pathogens
are already present in Lake Huron fish, and are not seen as a reason to prevent sea

lamprey transfers.

Screening effort for Lake Ontario sea lamprey and the likelihood of pathogen
translocation

The Great Lakes Fishery Commission (GLFC) has adopted a policy of
screening 60 fish prior to transfer (G. Christie, GLFC, personal communication).5 In
2004, sea lampreys were collected from Lake Ontario tributaries for sterilization and
transfer. The Aquatic Animal Health Laboratory at Michigan State University
examined a subsample for the presence of emerging and restricted pathogens identified
by the Office International des Epizooties (OIE) and the American Fisheries Society
(AFS) using the AFS aquatic animal health survey procedures (G. Christie, GLFC,
personal communication). All fish tested negative for Heterosporis sp., resulting in the

sterilization and transfer of 600 fish to the St. Marys River (Klar and Young 2004).

 

5 In 2004, 119 fish were actually screened (approximately 60 from two different rivers). Since 2004,
however, only 60 fish total have been screened annually. We‘therefore use the 60 fish number in our
analyses because it is current policy, and it is a common level of screening effort.

58

Neither the maximum acceptable probability of Heterosporis sp. translocation
nor the expected true prevalence of Heterosporis sp. in Lake Ontario for Heterosporis
sp. is documented for the sea lamprey transfer program. Therefore, we assume an
initial value of 5% prevalence of Heterosporis sp. in Lake Ontario sea lampreys to
illustrate the calculations described above. Assuming perfect Specificity and
sensitivity, screening 60 fish from the batch affords a probability 'of 0.95 of detecting
21 infected fish (Equation 3.3). If the prevalence of Heterosporis sp. in'the batch of sea
lampreys were exactly 5%, then testing 60 sea lampreys would result in a 5% chance of
failing to detect the pathogen in the screened subsample despite being present in the
batch (Equation 3.3). For a prevalence of 5% and a batch size of 600 fish, the
probability of the presence of at least one infected fish in the batch is essentially 1.0
(see Table 3.1). We then use Equation (3.1) to calculate the probability of a pathogen
translocation as (1-0.95)*1.00 = 0.05.

These calculations are sensitive to the assumption made about the assumed
prevalence, m. Had the actual prevalence of Heterosporis sp. in Lake Ontario sea
lampreys been 1%, then we could be 45.1% confident that at least one infected fish
would be detected in a 60-sea lamprey subsample (Equation 3.3, Table 3.1). There
would have been a 99.8% chance that at least one infected fish was present in the batch,

implying that there would have been a (l—O.451)*O.998 = 54.8% chance that at least

one infected individual was moved from Lake Ontario to the St. Mary’s River. These
calculations illustrate the importance of explicitly determining the maximum acceptable

likelihood that a pathogen is transferred and assumptions about the prevalence in the

59

source population. Managers need to be aware of all the factors affecting the risk of a
translocation decision and carefully consider carefully their implications.

The likelihood of pathogen translocation given a screening program is also
sensitive to assumptions about diagnostic sensitivity and specificity (Figure 3.5), and
this issue has received less attention (but see Dell’Omodarme and Prati 2005, and
Munoz-Zanzi et al. 2006 for exceptions); The sensitivity and specificity of sea lamprey
screening results were not reported to Great Lakes managers (G. Christie, GLFC,
personal communication). Therefore, to illustrate the implications of sensitivity
assumptions for the sea lamprey example, we assume, hypothetically, that the
sensitivity of the test were 0.7 and specificity were 1, implying a probability of a false
negative = 0.3. Thus, if the true prevalence were 5%, the observable prevalence would
be 3.5% (Equation 3.4). This would increase the chance of failing to detect an infected
individual in the subsample from 5% to 12.2%. The corresponding probability of
pathogen translocation for a 600- fish batch would be 12.2%. Alternatively, it would be
necessary to screen 86 fish, as opposed to 60, to provide a 5% chance of transferring at
least one infected fish (see Figure 3.5’s dotted lines). Lower levels of assumed
prevalence accentuate the effect of false negatives (in Figure 3.5, notice the increase in
distance between the solid lines associated with 5% prevalence and the dashed lines
associated with 1% prevalence).

Assumptions about specificity also affect decisions. Specificity < 1 can reduce
the probability that pathogen will be transferred because such tests inflate the likelihood

of a transfer-blocking positive result. This, too, needs to be accounted for when

60

designing screening programs, as a cost results when otherwise beneficial transfers are

forgone.

Determining the duration of a Lake Ontario sea lamprey pathogen screening program

We now demonstrate how to determine how long a screening program should
continue given successive failures to detect pathogen in screened samples and assuming
that there is no independent evidence for new pathogen introductions. The GLFC pays
a direct cost of about $30 per sea lamprey screened (G. Christie, GLFC, personal
communication), which does not include indirect costs such as transportation. The cost
of transferring a pathogen has not been estimated, so for this example we compute the
duration of the screening program over a range of costs. The probability of pathogen
transfer has already been addressed.

Figure 3.6 illustrates how the likelihood of translocation, the cost of screening,
and the cost of translocation of at least one infected fish interact to determine the
number of years to continue screening, assuming no new pathogen introduction (see
Equation 3.5), and leads to four observations. First, the economic impact of a disease
event has little effect on the length of time the screening program should continue when
zero infected individuals are detected, except in situations when such costs are minor.
Second, if the expected damages from a pathogen translocation are small (< $601 if the
true prevalence were 5% and the screening sample size were 60), then no screening
should take place. Third, using Table 3.1 and Figure 3.6, we see that the cases where n
= 60 and m'= 5% (solid line) and n = 300 and m = 1% (dotted line) have the same

probability of pathogen translocation. In the latter case more fish must be screened to

61

achieve this low probability, the screening program is more costly, and thus Should
terminate slightly sooner due to higher costs. Finally, in comparing the cases where n =
60, and m = 5% (solid line) versus m = 1% (dashed line), the lower source prevalence
in the latter case increases the probability of transferring a pathogen (with a 600 sea
lamprey transfer) to 54.8%. This implies that screening programs that result in lower
confidence of pathogen detection and higher likelihood of pathogen translocation
should persist longer.

These calculations are based on the assumption that the pathogen prevalence in
the source population is not increasing, but is stable or declining. If there is evidence,
independent of the screening program, suggesting that pathogen prevalence is
increasing in the source population (e. g. due to the invasion of a new host or detection
of the pathogen through other means), then screening efforts should persist. In such
cases — where the prevalence is increasing or a pathogen has been introduced — it would

be expected that infected individuals would begin to appear in the screening subsample.

3.5 DiscussiOn and Conclusion

Fish translocations create a likelihood of pathogen translocation, which may
have adverse consequences for society. This risk can be managed, in part, through pre-
translocation screening programs, and the effective and efficient design of these
programs is important for two reaSons. First, the design will affect the likelihood of
pathogen detection and thus translocation decisions. Second, screening programs are
not Cost-free, and allocating resources to screening diverts them from other beneficial

uses including other methods of managing fish pathogen transfer risk (e. g. disinfection

62

or treatment). This is not to say resources Should not be used for pathogen screening,
but that tradeoffs and the value of information gained must be considered.

Our analysis demonstrates that the translocation of an infected fish is most
likely to occur at moderately low prevalence levels (see Figure 3.3). This is because
the probability of failing to detect infected fish through screening declines as pathogen
prevalence increases, while the probability that an infected individual would be in a
batch for translocation declines as prevalence falls. The probability that the screening
program prevents a translocation event is the intersection of these two probabilities.
Fish screening is most likely to provide benefits when pathogen prevalence is relatively
high, (i.e., > 5%), in which case, relatively small subsamples may be adequate (see
Figures 3.3). Increasing subsample size, however, may be beneficial because the
probability of a pathogen translocation event decreases as subsample size becomes a
substantial fraction of the translocation batch (Figure 3.4). This may, however, be
infeasible when screening is lethal. As expected, decreases in diagnostic sensitivity
increase the likelihood of transferring at least one infected fish, but sample size and
assumed prevalence interact with sensitivity to determine the Size of the effect (Figure
3.5). Information about the sensitivity and specificity is vital to decision makers for
interpreting results and designing screening programs.

Our analysis also examined the length of time a screening program Should
continue given no new pathogen detection and no evidence of new pathogen invasions.
The probability of transferring a pathogen, which “is determined by assumptions about
the observable pathogen prevalence (m) and management decisions, followed by the

cost of screening, have the greatest impacts on the length of the time a program

63

continues.6

Pathogen translocation costs are also important to consider but have a
lower impact'on how long to Continue the program, particularly when they are large
relative to screening costs, which will typically be the case. We provided a formula for
determining the minimum expected costs of a pathogen transfer that justifies screening.
These costs are usually modest (<$601 in the case’study).

Aquatic disease and fish health activities must be evaluated within the context
of fisheries or ecosystem management objectives (Stephen 2001). Historically, fish
health policy has presumed that the consequences of any pathogen translocation are
unacceptable (Stephen 2001). Simplistic goals such as estimating and “minimizing
disease prevalence” are not appropriate. Such goals do not account for the opportunity
cost of using resources to manage disease that could have been used elsewhere in the
fisheries management program, and do they adequately address real management
tradeoffs.

We have presented a case study where sea lamprey transfers create the potential
for pathogen transfer with negative consequences, yet the overall consequences of these
translocations may be positive for the fishery, as a consequence of the benefits that the
sterile sea lampreys provide to the overall health of Lake Huron fisheries. Heterosporis
sp. has not been detected in the St. Marys River or Lake Huron, and SMRT related sea
lamprey suppression has improved Lake Huron fisheries (Lupi et al. 2003; Klar and

0

Young 2004; G. Christie, GLFC, personal communication). This result prompts one of

 

6 Formally, the marginal impacts are the derivatives of y with respect to each variable in Equation (5).
These are 6y/6CS = I/(CS ln(Pr(t))), 6y/6CT= -l/(CT ln(Pr(t))), and

(iv/6 Pr(t) = —l +1n(- CS / (CS ln[Pr(t)D)ln[Pr(t)]'z Pr(t)‘I . The marginal effect of a change in either
cost is the same in absolute value, but dependent on the cost, with larger costs resulting in effects smaller
in absolute value. Given screening should begin and under realistic conditions, CT> CS (this will

always hold if Pr(t) 2 e'1 and will hold for a range of Pr(t) < e"). Also, |6y/6Pr(t)| > lay/6C7] for given
values of Pr(t), CT, and CS.

64

two possible interpretations. Either the screening program is adequate, or managers
have been lucky. Good outcomes do not always imply good past decisions; thus a full
review of the screening program for sea lampreys in the context of overall fishery
objectives is merited. We found no evidence that managers had documented likely
prevalence for Heterosporis sp. in Lake Ontario or specified a maximum acceptable
likelihood of pathogen transfer - we encourage managers to address this issue explicitly
as a part of screening program design. In general, fish health diagnostic professionals
should provide, and decision makers should request, information on the performance of
the diagnostic tests, specifically diagnostic sensitivity and Specificity so that decision
makers have a clear context for interpreting screening program results and modifying
screening programs.

The concerns raised in the case study are not unique. Managers and researchers
seek to understand the issues associated with control of aquatic pathogen spread. For
example, Great Lakes fishery managers are grappling currently with disease issues
associated with the potential reintroduction of extirpated sculpins (Cottus ricei and
Myxocephalus thompsoni) and ciscoes (Coregonus spp.) (Eshenroder and Krueger
2002). Recent, positive screening results for R. salmoninarum, however, may prevent
these reintroductions (G. Wright, Chippewa-Ottawa Resource Authority, personal
communication). In this case, the likelihood of a pathogen translocation and potential
damages must be considered in terms 0f broader fishery objectives — i.e., what are the
translocations likely to achieve for the overall community or ecosystem? Thrush and
Peeler’s (2006) ‘contingency planning’ model for the spread of the monogenean

parasite Gyrodactylus salaries in Britain may best illustrate the potential for screening

65

programs to prevent pathogen translocation damages. Although they do not explicitly
include screening, they identify the ability to detect and prevent the movement of
infected fish as the main component in preventing a “major outbreak.” Including
simulated screening programs, based on the principles presented here, in such models
could help identify efficient screening policies.

In this paper, we outline the screening design characteristics that affect the
likelihood of a pathogen detection event and explain how these can affect pathogen
transfer risk. Both over-allocation and under-allocation of resources to screening can
have adverse consequences — striking a balance requires a sophisticated consideration

of risk.

66

Table 3.1. The probability that a given pathogen would be present in a sample of a

given size (or detected in a subsample with specificity = sensitivity = l), for different

prevalence levels among source fish.

Number of fish is in the sample

 

 

30 60 150 200 300 500 1000
0.001 0.030 0.058 0.139 0.181 0.259 0.393 0.632

o
g 0.01 0.259 0.451 0.777 0.865 0.950 0.993 1.000
g 0.02 0.451 0.699 0.950 0.982 0.998 1.000 1.000
g” 0.05 0.777 0.950 0.999 1.000 1.000 1.000 1.000
E 0.1 0.950 0.998 1.000 1.000 1.000 1.000 1.000

 

67

Probability of a
pathogen being
present in the
transfer batch, b.

Probability of
detecting a pathogen
by screening, d.

Outcome

 

No transfer and therefore
pathogen transferred with 0
probability

No transfer batch

 

detect, d

 

 
   

Pathogen transferred with

Present, b probability (l-d)*b

   

Fail to
detect, l-d
Pathogen not transferred with

N01 Present, 1'1? probability (1-d)*(1-b)

Figure 3.1. Decision tree giving the probability of transferring a pathogen given
screening. If a pathogen is discovered upon screening, there is no translocation. Even
if no pathogen is detected upon screening, it still may be present in the batch to be
transferred. The probability of transferring a pathogen given screening is equal to the

joint probabilities of Pr[(l-d)]*Pr[b].

68

100000 ~

 

 

 

10000 -
<1)
N
I}; l
O
'5. 1000 a
E
«I
V)
b 150
a 100 ~
3 60 1
8
Z
10 ~ fl
I T ""‘.' " '"zf— "_ F _. "FF—“I AT " ,-,-__ ' HT— "T“ ’# | T”"‘ 47“" '1"
O 0.02 0.05 0.1 0.15 0.2 0.25
Prevalence

Figure 3.2. Number of fish that need to be tested to achieve 95% confidence that a
pathogen is absent from the transfer batch at a given prevalence level (assuming
sensitivity = specificity = 1). The numbers of fish required at 2% and 5% prevalence
are identified with dotted lines. Note that the y-axis is on a log scale. Equation (3.2)

was used to generate this graph.

69

1.00 a

0.90 - - Number of fish screened
60
0.80 - _ _ _ .150

— —300

 

0.70 r
0.60 7
0.50 d
0.40 d
0.30 7
0.20 r
0.10 7

 

Probability of transferring an infected fish

 

 

0.00 I I I I i ‘ I I
1.0x10'6 1.0x10'5 1.0x10'4 1.0x10'3 1.0x10'2 1.0x10" 1

Prevalence of pathogen in the lake or maximum acceptable prevalence

Figure 3.3. Probability that an infected fish would be transferred given different levels
of source prevalence and screening effort, assuming translocation of 600 fish. Note that

the x-axis is on a log scale. Equations (3.1) and (3.3) were used to generate this graph.

70

600 ’ i I
0.75 '

500 ’
0.5

 

400 r l

300 -

 

 

Number of fish transferred

200

 

 

100:

 

 

 

100 150 A 200
Number of fish screened
Figure 3.4. Probability (labeled on the contour lines) of translocating at least one
infected fish, given the screening and transfer batch numbers. Solid lines represent the
cases where the true prevalence in the transfer batch is 5%, and dashed lines represent
the cases where the true prevalence in the batch is 1%. Point A indicates how the
probability of transferring at least one infected fish increases by 10 times when the true

prevalence is 1% rather than 5%. This graph was generated using Equations (3.1) and

(3.3).

71

 

 

\ o .
..C‘. \ \ \0.5
IE \\ \(23 \
E .\ . \
é \ \ \ \

\ ‘ x

.E \\ \ \
a) \ \
8 \ \ \ \\
,_. \ \ \ \
g \ \ \ \
3 \ \ \ \\
H \ \ \ \
3) ‘ ‘01 X
c: \ V \
o ‘ \ \
“a \ \ s
5 0\ \ \
I-r \ \ ‘\
a: \ \ .
O s \\
A? \\ \\
z " s
a ‘4“
..D ‘
O
L:
Q.

 

 

 

 

 

50 100 150 200 250 300
Screening sample size

Figure 3.5. Probability of translocating at least one infected fish, given the screening
sample size (x-axis) and the probability of a false negative screening result (labeled
contour lines). Solid lines represent the cases where the true prevalence in the transfer
batch is 5%, and dashed lines represent the cases where the true prevalence in the batch
is 1%. The dotted lines illustrate the adjustment needed in sample size and the increase
in the likelihood of a pathogen transfer when the probability of a false negative
increases from 0 to 0.3 (representing a sensitivity of 0.7). Figure generated using

equations (3.3) and (3.4).

72

 

prevalence 0.05 and 60 fish screened

 

 

 

14 " — - prevalence 0.01 and 60 fish screened
12 - - - 'prevalence 0.01 and 300 fish screened __ __ _. .—
10 7 / r ’
/ /
>‘ 6 ~ /
l
4 el
I ..........
2 4 """"""""""""""""""""""""""
O r 1 I I I
0 1 2 3 4 5

Cost of a disease outbreak ($ in millions)

Figure 3.6. Effects of screening costs, the cost of a disease outbreak, and the
probability of pathogen introduction on the number of years that a screening program
Should continue, given that no infected individuals are detected. The cost of the
screening program is calculated as $30 multiplied by the number of fish screened. The
probability of transferring at least one infected fish, assuming 600 or more fish are
translocated, was determined using Table 3.1. We assume that the translocation of a
Single infected fish is sufficient to cause damages. Equations (3.3) and (3.5) were used

to generate this figure.

73

CHAPTER 4
A BIOECONOMIC MODEL OF CHINOOK SALMON STOCKING AND
ALEWIFE CONTROL TO MAXIMIZE ECOSYSTEM BENEFITS THAT

ACCOUNTS FOR INSTITUTIONAL CONSTRAINTS

4.1 Introduction
Great Lakes fishery managers work within a system that generally seeks to maximize
the welfare to society through the management of fishery resources. Formally, the
Great Lakes Fishery Commission (2001) states as one of its two major responsibilities,
“...to recommend measures which will permit the maximum sustained productivity of
stocks of fish of common concern.” We interpret this as to maximize net benefits
associated with fish stocks of common concern, because some stocks of common
concern may provide non-consumptive values, while others may actually create
damages (negative benefits). Two non-native fish stocks of concern are Chinook
salmon (Oncorhynchus tschawytscha) and alewife (Alosa pseudoharengus). Chinook
salmon are stocked in the Great Lakes to create a sport fishery and control the invasive
alewife population, thereby increasing social welfare. Alewife can be a nuisance
species damaging infrastructure, natural amenities, and can negatively affect native fish
populations.

Salmon and alewife management is a dominant issue on Lakes Ontario, Huron,
and Michigan. Salmon provide fishing benefits and act as a biological control agent.
Increasing salmon populations lead to decreases in alewife populations, and this

appears to be a “win-win” situation. Alewives, however, comprise the majority of the

74

Chinook salmon diet (Madenjian et al. 2002), and 'Holey et al. (1998) state that the
recreational Chinook fishery may depend on sustaining a large alewife forage base.
Thus, there is an intertemporal tradeoff whereby efficient management must balance
the benefits from the fishery and near-term alewife reductions against the long-term
costs associated with the invasive alewives and maintaining a recreational salmon
fishery.

Alewives invaded Lake Ontario as early as 1873, but were not detected in Lake
Michigan until 1949 (O’Gorman and Stewart 1999). Alewives imposed costs on
society by fouling beaches and drainpipes following their invasion (O’Gorman and
Stewart 1999). Initially, alewife abundance and expansion was likely curtailed by
native predator populations that subsequently succumbed to high fishing levels and
mortality caused by the invasive sea lamprey (Petromyzon marinas) (O’Gorman and
Stewart 1999). Alewives diminished the ability of the Great Lakes to provide
ecosystem services. It is believed that alewife predation on lake trout (Salvelinus
namaycush) fry impedes native lake trout restoration (Krueger et al. 1995; Madenjian et
al. 2002). Moreover, alewife predation has been implicated in the decline of yellow
perch (Percaflavescens) populations (Shroyer and McComish 2000), perhaps the most
widely targeted Sport fish in Lake Michigan (Wilberg et al. 2005).

Managers began stocking salmon into Lake Michigan in earnest in 1965 in part
to control alewife populations (Madenjian et al. 2002). Chinook salmon are the main
Pacific salmon stocked into Lake Michigan, and today create a valuable sport fishery

(Hoehn et al. 1996). Alewives provide a benefit to the recreational fishery, because

75

alewives appear to be a required input, as prey, for Chinook salmon production
(Stewart and Ibarra 1991).

Alewives provide both benefits and costs to society that managers concerned
with social welfare must balance. Alewives provide benefits in the form of forage for
the Chinook salmon. Alewives, however, generate costs (ecosystem disservices) to
society through damage to infrastructure, natural amenities, and predation on the eggs
and fry of valued native fish populations. The idea that species can generate both
benefits and costs depending on the population’s size has been explored generally
(Rondeau 2001) and Specifically in the cases feral pig management (Zivin et al. 2000)
of urban deer (Rondeau and Conrad 2003) and elephant management (Horan and Bulte
2004). These authors show that this “multiple-use species” problem (Zivin et al. 2000)
has the potential to result in non-convexities that lead to multiple bioeconomic
equilibria. Multiple bioeconomic equilibria imply that the optimal outcome or basin of
attraction for the system may depend on the initial conditions. However, in the
previously Studied cases, managers were assumed to directly control the population
size. In the case of alewife management, managers indirectly influence the alewife
population through the stocking of salmon.

It is often assumed that, all else equal, more salmon improves social welfare by
providing more angling opportunities, but stocking more fish is not always better. This
is because there are direct costs of stocking and indirect costs of depleting the prey
base. Managers must achieve an intertemporal balance. Lake Michigan Chinook
salmon stocking was recently decreased in part due to concern that the prey base had

been reduced. However, the idea that the ‘correct’ stocking level for predator fish must

76

be connected to the prey fish stock level is not a new idea (Jones et al. 1993). Complex
stochastic simulation models have been developed to inform stocking decisions in the
Great Lakes (Jones et al. 1993; Rand 1994; Jackson 1997; Szalai 2003; Jones and
Bence in review). These models include multiple species, bioenergetics, age and size-
structure, and uncertainty. These efforts have focused on describing the distribution of
outcomes that result from stocking strategies. However, such models have not
explicitly optimized stocking with respect to an economically or socially defined
objective, and they do not consider that angler behavior responds to changes in fish
stocks. Morey and Waldman (1998) report that angler behavior is tied to catch rates,
which likely correspond to fish stocks. Moreover, the complexity of these simulation
models can make understanding and generalizing the results difficult.

Human welfare and the state of ecosystems are jointly determined by ecological
and economic processes (Shogren et al. 1999; Sanchirico and Wilen 2001). We
develop a simple model that captures the relationships between anglers, Chinook
salmon, and alewives. We then solve for an optimal stocking program from the
agency’s perspective — one that maximizes social welfare, defined as the sum of
discounted net benefits from the open access salmon sport fishery and averted damages
from the alewife stock. In this case the agency is not a true social planner because the
agency takes angler behavior as given. This can be thought of as an institutional
constraint (Dasgupta and Maler 2003), but makes the nature of the solution “second
best.”1 This paper examines the tradeoffs associated with the stocking program in an

analytical fashion, and develops general rules that'can help guide stocking decision

 

' A “first best” solution would require that managers control angler behavior, and therefore could
optimally manage salmon and alewife harvests in addition to stocking.

77

making. The inclusion of human behavior in such rules is important if managers are
concerned about human welfare or the effects that humans have on the ecological
system (Settle et al. 2002). Finally, this paper contributes to the bioeconomic literature
by linking the literature on non-convexities (Tahvonen and Salo 1996; Rondeau 2001;
Dasguta and Maler 2003) with the literature on imperfectly targeted controls (Clark
1990; Fenichel and Horan 2007a, b; Horan et al. in press).

We begin by defining a bioeconomic objective. Then, we model decentralized
economic angler behavior and biological processes. Next, we present optimality
conditions for a general model, and show that the optimal stocking strategy is a
nonlinear feedback rule that is a function of the state of both salmon and alewife
populations. We then Specify a full model and use phase-plane analysis to present
simulation results and sensitivity analyses. We conclude with a discussion of the
implications for salmon stocking policy in Lake Michigan and in general, and we

discuss the linkage between imperfectly targeted controls and non-convexities.

4.2 A bioeconomic model of salmon stocking

A fishery management agency’s charge can be interpreted as maximizing the
discounted social net benefits associated with a fishery resource. Discounted social net
benefits (SNB) are defined as the present value of the difference between how much
salmon anglers gain from the fishery (total angler net benefits, TAB) minus how much
society invests in the fishery (stocking in kilograms of salmon, w) and any damages

caused by the fishery.

78

(4,1) SNB = CRT/1130) — D(a(t))— vw(t))e"p’dt
0

where p is the discount rate, v is the constant marginal cost of stocking a kilogram of
salmon, D is a damage function, and a is the stock of alewives.

Alewife damages include ecosystem disservices (predation on the eggs and fry
of native fishes) as well as the fouling of beaches, water intake pipes, and other
amenities. Assume that alewife damages, D(a), increase with increases in the alewife
biomass, D'(a) > O, and do so at an increasing rate, D"(a) > 0.2 This kind of
nonlinearity with biomass or abundance is likely a common trait for species that are

considered invasive pests.

A model of angler behavior

Total angler benefits, TAB, are the sum of the net benefits received by all
individual anglers at time t. Assume that all anglers have unique costs to fishing c, but
all anglers have the same angling preference or “inverse demand” function, Q(m, z(s)),
where z(s) is the quality of the fishery, experienced by all anglers, represented by catch
per day, and catch per day increases with increases in salmon stock, 8, and m is the
number of days an individual angler fishes per season.3 Following Anderson (1983) the
inverse demand function is assumed to have the following characteristics. The inverse

demand fimction is downward sloping with increases in days, Qm(m, z(s)) < 0 (Qm

 

2 This seems reasonable at the relevant biomass levels for alewife. There will be a level of alewife at
which alewife cease to cause marginal damages, but this level is likely higher than the stock sizes
considered.

3 The current mode captures the general dynamics of angler behavior and we believe that the qualitative
results of the model are robust to the assumption that anglers only differ in costs. In reality all anglers
also have unique benefits fi'om fishing and skill levels resulting in different catch rates. However, the
incorporation of these differences is left for future analysis.

79

represents the partial derivative of Q with respect to m; this notation will be maintained
throughout). Increases in quality shift the inverse demand function upwards,
Q;(,)(m, z(s)) >0. An individual angler’s net benefits, a money metric of utility, from

fishing m days is written as

(4.2) U (m,Z(S).c)= jQ(§,z(s))dg—cm
0

where g is the dummy of integration.

In a recreational fishery, the individual angler has two choices i) whether or not
to fish in a given season, and ii) how many days to fish given that he chooses to
participate (McConnell and Sutinen 1979; Anderson 1983). An angler enters the
fishery if his net benefits, U 2 0 and will continue to increase the number of days that
he fishes as long as his marginal utility from an additional day of fishing is positive, Um
2 0. Given that an angler participates, his individual net benefits is maximized when
(43) Um = Q(m,z(s)) — c = 0
implying that he will fish m* = m[z(s), c] days. Thus, we assume that anglers behave to
maximize their utilities so that U = U (m*, z(s), c).

To determine the level of effort in the fishery, we recognize that each angler has
a unique cost to fishing and think ofc as a cost type. Each cost type is treated as a
“micro-unit” (Hochman and Zilberman 1978). Cost types are ordered in increasing
order, such that the last cost type to enter the fishery is E . When the distribution of

cost types is continuous this is equivalent to the amount the last angler to enter pays for

angling. This creates a cost threshold for entry so that only anglers with costs less than

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E will enter. To solve for E , we identify the cost at which the last angler is indifferent
about entry and receives U = 0. The condition
mil!
(4.4) U(m*,z(s),E) = IQ(m, z(s))dm — Em* = 0
0
can be solved forE .
Cost types, 6, are distributed over the interval [0,00] with the probability density
function 11;. If N is the total number of potential anglers, then the actual number of
anglers in the fishery, n(s), depends on salmon biomass.
E(m*,s)
(4.5) n(S) = N We
0

Total angler benefits, TAB(s) and total catch, h(s), in biomass are also functions of
salmon biomass and are derived similarly

E(m* s)

(4.6) TAB(5)=N ill/("1549501111610
0

E(m*,s)

(4.7) h(s>=N l(m*z<s>w)dc
0

Fish population models

The fishery manager must take into account the dynamics of the fish stocks. Indeed, an
ecosystem management approach must be concerned with the dynamics of multiple
species and the interactions among those species. Let the dynamics of the harvestable

Chinook salmon stock be defined in terms of biomass as

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(4.8) s = 6(an + b)— s6(s, a)- h(s)
where b is the total biological recruitment and is independent of the stock size, i.e., b
represents fish produced in nature rather than by stocking. This is motivated by an
assumption that limited spawning habitat will be saturated (implying strong density-
dependent mechanisms). In baseline numerical simulations b = 0, presuming a salmon
population entirely dependent on stocking; 6(a) is a scaling function that scales biomass
at stocking or biological recruitment to harvestable biomass or recruitment to the
fishery. The scaling function is included because small non-catchable fish are stocked,
and these fish must survive and grow before anglers have an opportunity to catch them.
At higher alewife levels more young salmon survive and the average fish is larger. The
mortality rate of fish in the fishery, (3(s, a), is a function of salmon and alewife biomass.
General results only require the assumptions that salmon mortality is a

decreasing function of alewife biomass, 65(s, a)/6a < 0, and that as alewife biomass

. . . . . 2 2
Increases, ultimately salmon reach a mrnrmum mortality rate, 6 5(s, a)/(6a) 2 0. We

also assume that 66(a)/aa > 0 and 626(a)/(6a)2 < 0. Specific functional forms are

specified in the simulation section.
The alewife population is defined in terms of biomass and follows logistic

growth.

(49) a = m[l ——I:—) — P(s,a)

Where r is the net recruitment rate of biomass in the limit as stock size approaches zero
(recruitment minus non-predation mortality) and K is the alewife carrying capacity.

The function P(s, a) is the Chinook salmon predation rate on alewife. General results

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may be reached with a few assumptions. As salmon biomass increases, salmon
consume more alewife, 6P(s,a)/ as > 0. A unit increase in the salmon biomass may
lead to a constant rate of increase in the amount of biomass consumed or intra-specific

competition may lead to a decline in the amount of alewife consumed per salmon as

. . 2 2 . . . . .
salmon biomass increases, a P(s,a)/(as) S 0. For Simplicity assume no intra-specrfic

competition, 62P(s, a)/ (ds)2 = 0, so that P(s, a) = sP(a), where P(a) is the biomass of

alewife consumed by a biomass unit of salmon. Assume salmon consume more alewife
as alewife biomass increases such that P'(a) > 0. However, the rate at which salmon
consume more alewife biomass as alewife biomass increases may decline with
increasing alewife biomass, P”(a) S 0. That is, the response of salmon to increases in
alewife biomass may be saturating, but sP(a) should be related to 5(s, a). This is made

explicit in the simulation below.

4.3 Optimizing social welfare through stocking

The agency is a social decision maker in a “second best” world due to the ecological
and institutional constraints. The agency can not manage all aspects of the system, as
would be the case in a “first best” world, but only controls the stocking program. The
stocking program is therefore not perfectly targeted and leads to joint production of
ecosystem services and disservices. When managers alter the salmon stock through
stocking, the change in the salmon stock affects the alewife stock and angler behavior.
Angler behavior and the alewife stock, however, have feedback effects on the salmon
stock. The agency’s problem can be defined as choosing w to maximize (4.1) subject to

equations (4.8) and (4.9). This requires that the agency explicitly consider equations

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(4.6) and (4.7). The agency’s problem can be solved using the maximum principle
(Clark 1990). Write the Hamiltonian as

(4.10) H = TAB(s)—D(a)—vw+/ls + pa

In the Hamiltonian, A and u are the co-state variables associated with the salmon and
alewife stocks respectively. Coastate variables represent the marginal user cost
(benefit) associated with the salmon (alewife) stock. This is the cost (benefit) incurred
when the salmon (alewife) capital stock is reduced by one unit. An alternative way of
thinking of the marginal user cost is as the Shadow value, which is the benefit (cost) of
increasing the salmon (alewife) stock by one unit.

The marginal impact of stocking salmon is given by
OH
(4.11) —— = —v + 16(a)
6w

The right-hand-side (RHS) of expression (11) is the coefficient of stocking from the
Hamiltonian. The Hamiltonian is linear in the control variable, w. If 6H/aw > 0, then
stocking always increases the value of the Hamiltonian and stocking Should be set at
the maximum limit (an exogenously imposed limit that may represent hatchery
capacity). On the other hand, if aH/aw < 0, then stocking always decreases the value of
the Hamiltonian and stocking should be constrained to zero. These are constrained
solutions. Another possibility is the case when aH/bw = 0, and condition (4.11)
vanishes. This is known as the singular solution denoted w*. In this case It, the shadow
value for the salmon stock (the value of having an additional salmon in situ), is exactly

equal to the marginal cost of stocking scaled for growth, A: v/6(a). Regardless of

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whether the solution is constrained or singular, the solution to problem (1) is a feedback

rule and a function of the stock of alewife.

w=0, ifll9(a)——v<0

(4.12) w: w=w* if/16(a)-v=0

w = wmax if /l€(d)i— v > 0
Conrad and Clark (1987, p.76) state that linear control problems guarantee the
optimality of constrained solutions. In the linear control problem they describe, a
constrained solution is pursued to the equilibrium, and then the singular solution is
pursued to maintain the system at equilibrium (in their harvesting model this is a
constant escapement policy). This illustrates a case of rapid adjustment and then
maintenance at equilibrium. This result relies on the fact the control variable was
perfectly targeted at the state variable and had no other side effects. When the control
variable does not perfectly target the state variable, as in the case of the agency’s
salmon stocking program, more complex feedback rules may emerge (Mesterton-
Gibbions 1987; Horan and Wolf .2005). In this case, it is optimal to pursue the singular
solution when the system is out of equilibrium. This happens due to the nature of joint
production and imperfect control. When control perfectly targets the system,
adjustment can be rapid as the control completely determines the state of the system,
but when control is imperfect, has Side-effects, and only partial determines the state of
the system, then adjustment must be sloWer to “smooth out” the non-target effects.

This makes nonlinear feedback rule optimal.

Regardless of the type of solution, an optimal program requires that two adjoint

equations associated with the co-state variables be satisfied at each point in time

85

(Conrad and Clark 1987; Clark 1990), where terms with subscripts denote partial

derivatives with respect to the subscript

- H
(4_13) x1 = p —% = p). — TABS + k[5(s,a)+ s§S(s,a)+ h'(s)]+ ,uP(a)

,[t 2 p2 - ail—I = pa + D’(a)— 71((b + w)6'(a)— 56a (s,a))
(4.14) _ ”(I "213% ”sP(a)

These conditions prevent intertemporal arbitrage opportunities: if these conditions are
not satisfied, then welfare gains can be made by reallocating stocking across time, in

which case the solution would not be intertemporally optimal.

Interpreting the optimality conditions

Before proceeding to simulation results, we examine the details of the optimality
conditions to highlight tradeoffs associated with the stocking program. The first order
and arbitrage conditions may be manipulated into two “golden rule” equations that
must hold at each point in time (Conrad and Clark 1987; Clark 1990):

(415) p =%+—21:TABS —%P(a)— [5(s,a)+ s53(s,a)]—h'(s)

(4.16) p = fl — —D—'(£)- + i ((b + w)6'(a)— sé‘a (s,a))+ {I — a) — sP'(a)
rt .11 .71 - K

These golden rule equations highlight tradeoffs associated with increases or decreases
in salmon and alewife biomass. In equation (4.15) p, the discount rate, is the manager’s

rate of time preference and can be thought of as the opportunity cost of providing an

86

additional unit of biomass to the salmon stock. The RHS of equation (4.15)
collectively represents the marginal benefit of stocking an additional unit of salmon
biomass. The first term on the RHS is a capital gains term that will be zero at
equilibrium. The second RHS term is the total angler benefits gained from a larger
salmon stock and is always positive. This term indicates that all else equal anglers are
better off with a larger salmon stock because a larger salmon stock-increases fishing
quality, and thus enables more fishing days and more angler participation. The third
RHS term is the marginal impact of an increase in the Chinook population on the
alewife resource on which the Chinook population depends. This term is negative.
Increasing the Chinook population decreases the alewife population and is costly to
Chinook production. The fourth and fifth RHS terms together are the effect of the
Chinook stock on its own marginal growth. The fourth term, the term in the square
brackets, is the direct effect of the salmon stock on its own growth rate. This term is
always negative because recruitment is independent of the stock so increases in stock
only increase mortality. This term results because an increase in the salmon population
increases the salmon mortality rate, by reducing the prey base. This can be thought of
as the cost of natural mortality. The final RHS term represents the change in salmon
mortality with an increase in the salmon stock due to changes in angler behavior. This
term is negative; an increase in the salmon stock increases the total effort and thus
increases the catch in the fishery reducing future fishing opportunities ceteris paribus.
This can be thought of as the cost of fishing mortality.

Equation (4.16) is also a golden rule equation, but has a different interpretation

than equation (4.15)} The lefi-hand-side of the equation is also the discount rate, i.e.,

87

the manager’s rate of time preference, but is interpreted as the marginal cost of
diverting resources from elsewhere in the economy to manage the invasive alewife and
their associated damages. Alternatively, p is as the rate of return earned by forgoing
stocking in order to preserve the prey base, this is appropriate in a case where alewife-
induced damages are small. The RHS’of equation (4.16) can be thought of collectively
as the net gains from alewife management (or the net loses from depleting the prey
base). From the perspective of invasive Species management, the first RHS term is a
capital loss term, but from the perspective of managing the prey base it is a capital
gains term, either way it will be zero at equilibrium. The second RHS term represents
the marginal social benefits associated with a smaller invasive alewife population (for a
native non-pest species this term is zero). The third RHS term is the cost of reducing
the prey population to the recreational chinook fishery in terms of lost Chinook salmon
production. This is a cost because it reduces the rents to the recreational fishery. The
fourth RHS term is the effect of the alewife stock on its own marginal grth as a
result of it being a reproducible liability or asset; this may be thought of the base rate of
return to controlling alewife. This term can be positive or negative. The final RHS
term in equation (4.16) is the marginal benefit (cost) of decreases in the alewife (prey)
population on the ability of Chinook salmon to control (deplete) alewife numbers
ceteris paribus. That is, because for P'(a) > 0, decreases in the alewife population
make chinook salmon more efficient at controlling alewife numbers and provide a
benefit (this is cost if the prey does not cause damages), ceteris paribus. From a
biological perspective, when there are fewer alewife, salmon are not saturated, and a

hungry salmon will consume more prey.

88

Solving for the nonlinear feedback rule

The approach for finding the nonlinear feedback rule for the singular solution is
similar to Fenichel and Horan’s (2007a) procedure for finding a partial singular
solution (their model has two control variables). First, set equation (4.11) equal to zero
and. solve fork = Ma). Next, take the derivate of A with respect to time yielding N0) =
Ms, a) and substitute this into condition (4.13). Solve the resulting equation for u =
n(s, a) and take the time derivative u'(t) = u,(s, a, w). Notice that the time derivative of
u is a function of w. Finally, set the time derivative of [.1 equal to condition (4.14) and
solve for w. The solution is a nonlinear feedback rule where w = w*(s, a). A specific
feedback rule requires further specification of the general functions used so far (see
function specification for simulation section).

This feedback rule contrasts with current stocking policies and with feedback
policies under consideration. Management agencies often attempt to identify constant
stocking policies that do not respond to the state of the system. Constant stocking
policies would be optimal in the case where condition (4.11) > 0, and thus the
maximum stocking level were pursued (or in the case where no stocking takes place
because condition (4.11) < O). A constant stocking policy could also be optimal if the
system where at the socially optimal equilibrium; however i) this is unlikely at present,
and ii) even if the system were to arrive at the socially optimal equilibrium it may be
perturbed away from this point. Moreover, the optimal constant stocking rate in this
case would still be a function of the salmon and alewife population. These results
support the current efforts underway to develop state dependent stocking rule that are

based on more complex and biologiéally realistic models (Szalia 2003, Jones and Bence

89

in review). However, to date those efforts have not accounted for the response of
anglers to changing stock sizes, and have only consider feedback policies based on

alewife or salmon stocks.

4.4. Function specification for simulation

We Specify explicit functional forms for the remaining implicit functions in the model
in order to examine the optimal solution numerically. A numerical approach is required
because the angler behavior and the nonlinear feedback rule for stocking are too
complex to analyze analytically. Moreover, when to employ the nonlinear feedback
rule and when to pursue a constrained solution inherently requires numerical solutions
(Arrow 1968). There are two features of this problem that can lead to non-convexities
and multiple equilibra i) the multiple-use species problem (Rondeau 2001; Horan and
Bulte 2004) and ii) the potential for complex human-ecological relationships (Crepin
2003). For problems with non-convexities, “what is required is the sheer brute force of
computing welfare along candidate programs and comparing them” (Dasgupta and
Maler 2003). The following functional forms were chosen to be as simple as possible
while capturing desired aspects of the relationships that are consistent with theory and

economic and biological knowledge.
Economic functions

The exact nature of the alewife damage function is unknown. However, given the

assumptions that D’(a) > 0 and D"(a) > 0, we use a'second order approximation to a

90

. 2 . .
convex function D(a) = Da , where D rs a damage parameter. The angler “inverse

demand” function takes the form

(4.17) Q(m, z(s)) = x1 — xzm + x3qs

where q is the catchability coefficient. The distribution of cost types is assumed to be
log-normally distributed (Just and Antle (1990) recommend this distribution for micro-
parameter models as a default) with a mean of n and standard deviation 0. Given this
specification we derive specific relationships corresponding to' the general ones
established above.

*= x1+x3qs-c

 

(4.18) m
3‘2

(4.19) E = x1+x3qs

Ecological functions

We define the function 6(s, a) as

(4.20) 6(5, a) = a — y(sP(a)/s).

In equation (4.20) salmon mortality declines linearly with increases in the biomass of
alewife consumed per biomass unit of salmon. That is, (1 is the instantaneous annual
mortality rate with zero alewife, a-y is the instantaneous annual mortality rate at
satiation. Assume that salmon have a type-II predator response function with

parameters [3 and (0.

 

(oa
P =
(4.21) S (a) Sl+,6a

91

Equations (4.20) and (4.21) satisfy the condition above in the general model and are
coupled, so there is a direct connection between salmon survival and prey consumption.

The alewife population also affects the stocking survival and recruitment to the

fishery. We model this with a scaling function, 9(a) = ya¢ , where 0 < (b < 1 and y are

parameters.

Mathematica 6.0 (Wolfram Research) was used to implement the model. The
parameter values used are listed in the appendix. We have used the best available data
on the salmon-alewife-angler system. Some processes, however have been condensed
to simple functional forms to improve tractability. For example, many biological
relations are often assumed to depend on size of individuals, and given that fish change
many magnitudes order in size, this required that some parameters be rescaled to apply
to the average of aggregated individuals represented in our model by the biomass of
alewife or salmon. The quantitative rescaling required judgment. Consequently, our
specific numerical results should be used as a guide to help navigate the often
perplexing outcomes of more complex models. Nonetheless, the model allows us to
describe tradeoffs associated with salmon stocking programs and identify important

interactions between non-convexities and imperfectly targeted controls.

4.5 Optimal management

The case of no alewife damages

We begin with the special case when alewives do‘not cause damages. This case is
interesting in its own right because in many systems the prey fish is native and does not

have adverse effects. It also serves as a baseline to illustrate the importance of

92

considering the ecosystem damages (disservices) caused by invasive alewife when
developing a stocking strategy. Given an initial state of the world, so and an, the agency

planner must choose a stocking program. This program may include constrained or
Singular values of w (i.e., w = 0, w = w , or w = w*(s, a)) at different pornts 1n time.

The choice of when to apply which type of solution is a common problem when
multiple populations are managed with imperfectly targeted management measures
(Mesterton-Gibbons 1987; Fenichel and Horan 2007a; Horan et al. in press). Phase

plane analysis is a useful tool for answering this question.

. max
Dynamics when w = w

Consider the dynamics if stocking were to always occur, i.e., w = wmax (Figure 4.1). In

this case the biomass of salmon would build up, and the high biomass of salmon would

reduce the biomass of alewife. If the system were to start at (0, K), then a constant

max . . . . .
w policy leads to maxrmum number of salmon that can be 1n the system (thrs is the

salmon limit). Indeed from any initial condition, pursuit of a maximum stocking
strategy will lead to alewife eradication.4 Recall, that alewife are necessary for the
survival of salmon and that the salmon that are stocked are not yet available to the
fishery. With no alewife in the system any salmon that are stock are expected to die

prior to entering the fishery.

 

4 Eradication results from the assumption of a type-II predator response function.

93

Dynamics when w = 0

Now consider the dynamics if stocking were forgone all together (Figure 4.2). If the
system were to start at (0, K), then with no stocking, the dynamics are trivial and the
system stays at (0, K). Given the assumption the salmon do not reproduce natural w = 0
is only possible after some stocking has occurred. The system is divided into two
basins of attraction by a separatrix that leads to the origin (Figure 4.2). For starting
points above the separatrix the system will ultimately arrive at (0, K) if stocking is

always forgone. If, however, stocking iS ended at a point below the separatrix, then the

. . . . . max .
alewrfe population 1S eradlcated. AS With the w case, thlS means that the salmon

fishery can not persist.

Dynamics when w = w*(s, a)
Finally, consider the non-linear feedback rule w = w*(s, a) (Figure 4.3). First, we
identify the state space where the non-linear feedback rule from the singular solution is

feasible. To do this, we identify extreme solutions where the singular solution implies

a boundary, i.e., w*(s, a) = 0 and w*(s, a) = wmax (see Fenichel and Horan 2007a and

Horan et al. in press for a similar approach). Boundaries are plotted in all figures as
dashed lines. Hence, the phase space can be divided into four regions i) above the w =
w’"‘”‘ boundary where stocking at the maximum rate is implied, ii) an interior region
where the non-linear feedback rule is used to determine the amount of stocking, iii) the
area below the w = 0 boundary in Figure 4.3A (except for a small region close to the

origin) where it no stocking is implied, and iv) a second interior region between the

94

w = 0 and the y-axis (Figure 4.3B). The singular solution, w = w*(s, a), can only be

pursued 1n the interior regrons between w = w and w = 0 boundaries.

The dynamics when w = w*(s, a) are defined by the intersections of the s = 0
and a = 0 isoclines. These intersect in five places creating five equilibria. There are
two trivial equilibria at (0, 0) and (0, K), and three interior equilibria; one is at point A
(Figure 4.3A) and two are in the lower left comer near the origin where a second 8 = 0
isocline exists but loops back to the y-axis. These are labeled points B and C (Figure
4.3B). Multiple interior equilibria are typically associated with non—convexities and

come in odd numbers (Tahvonen and Salo 1996). The existence of an odd number of
equilibria is verified by taking the limits 6a/65=0and 65/63 =00 at the origin.

Moving from the origin along the s = 0 isocline, the 5‘ = O isocline must cross the
a = 0 isocline an odd number of times (Figure 4.3A and 4.38). To check the local
stability properties of the equilibria we determine the eigenvalues for the linearized
system. The eigenvalues associated with equilibria A and C are imaginary with real
parts indicating that equilibria A and C are unstable foci (Conrad and Clark 1987).
That implies that application of the singular solution leads to a path that spirals away
from these equilibria. Therefore, the system can not be maintained in the upper interior
region. These equilibria only represent optimal steady states in the case where the
initial conditions for the state variables are equal to the state variable values at the
equilibrium. One eigenvalue associated with equilibrium B is positive and the other
negative, indicating a conditionally stable equilibrium or saddle point (Conrad and

Clark 1987). A unique path known as a saddle path or separatrix may be followed to

95

arrive at this equilibrium point; however, deviations from the saddle path will lead
away from the equilibrium (Figure 4.3A and 4.3B).

The saddle path bifurcates the system, and divides alternate basins (Dasgupta
and Maler 2003). This saddle path is different from the separatrix in Figure 4.2, but
divides the system in a qualitatively similar fashion. Locating the saddle path is useful
even though, following the saddle path to the saddle point may notbe optimal in the
case with multiple equilibria. The saddle path'is piece-wise continuous as it approaches
equilibrium B from a constrained region. Moving backwards in time from the
equilibrium B along the saddle path, initially a Singular solution would be pursued unit
the point where the w = 0 boundary in intersected (Figure 4.3B). Upon crossing the
w = 0 boundary the w = 0 solution is pursued. That is, the piece—wise saddle, accounts

for the constraint of stocking zero salmon.

Optimal stocking strategies
There are two basins of attraction, above and below the saddle path. Whether the
system initially lies above or below the saddle path influences the optimal stocking
strategy. Therefore, optimal management from starting points below and above the
saddle path must be considered separately. This is the general nature of problems with
non-convexities.

First consider the solution for points below the saddle path. Recall that saddle
path is piece-wise continuous accounting for the w =4 0 constraint. Also, recall that
managers do not control harvests, and therefore setting stocking to zero does not lead to

a “jump” to the saddle path. This is because the only way for salmon to be removed

96

from the system is through harvests, which are not controlled and through natural
mortality. Therefore, the salmon population responds slowly to management. Below
the saddle path, managers optimally stock zero salmon, but this does not reduce the
salmon stock fast enough, and the remaining salmon consume all of the alewife.5

Now consider starting points above the saddle path. Within this region, there
are potentially two ultimate outcomes: (i) movement to the saddle path, which moves
the system'to equilibrium B, or (ii) cycling around equilibrium A, provided a stable
limit cycle exists. These two solutions can not be simultaneously optimal, implying
that if a Stable limit cycle is encountered the optimality of the saddle point can be ruled
out (see Liski et al. 2001 for details). Consider a general starting the point, (0, K). We
consider this to be a general starting point because all other starting points above the

separatrix are closer to either potential outcome. From (0, K), an interior path that

. . . . max .
eventually goes to the wmax boundary rs optrmal.6 At this pornt a w = w 1S pursued

until the dynamics, governed by the phase arrows, push the system back into the
interior, prior to hitting the saddle path. In the interior region, the singular solution is
pursued until the w = 0 boundary is intersected. Pursuit of the w = 0 rule leads back to

the interior region, where the singular solution is again applied. This leads back the w

= wmax boundary, creating a cycle. This cycle can be numerically compared to a path

in which wmax is pursed to separatrix and then followed to equilibrium B. The

discounted social net benefits for the stable cycle are greater than those for a program

 

5 Total consumption of alewife is a result of the type II predator response function.
6 It is possible there to apply the w = W” early and move rapidly to the right on the phase plan, but this
makes little difference given the high level of stocking implied by the w = w*(s, a) rule.

97

that moves first the separatrix and than to equilibrium B for all initial conditions above
the separatrix.

Thelogic behind the optimality of the stable limit cycle can be described two
ways. First, the “second best” nature of the problem requires Slower adjustment. In a
“first best” world, the planner would have control over salmon stocking and harvesting
and alewife harvesting. Hence, control would be perfectly targeted and controls would
only have direct effects leading to rapid “jumps” to a saddle path (e.g. Conrad and
Clark 1987). Stocking in the only control option and has many indirect effects on the
system. These indirect effects require that adjustments take place more slowly. This is
akin to convex adjustment costs in investment models (Liski et al. 2001).

The second explanation is that the optimal solution is a cycle due to the
production effect of alewife. Alewives are important for producing salmon angling
benefits. Salmon stocking begins to increase alewife biomass has begun to increase.
At this point stocking salmon and increasing alewife biomass are compliments in
production. Increasing alewife biomass allows for higher levels of stocking. Over
time, salmon biomass builds ups because managers do not control the rate which
salmon are removed from the system. Managers recognize that increased further
increases in stocking will lead to over exploitation of the prey. Managers substitute
alewife biomass for stocking at high alewife biomass. This leads to a reduction in
stocking. Maximum stocking is achieved prior to the maximum biomass of alewife for
the same reason. The long-term planner knows that salmon will reduce the alewife

biomass and that salmon will also attract more anglers.

98

Optimal management with alewife-induced damages
Describing the phase plane
We follow the same procedure for the case with alewife-induced damages (ecosystem

disservices). We begin by drawing the completed phase plane (Figure 4.4). The

individually constrained cases of w = 0 and w = wmax are the same as before (Figures

4.1 and 4.2). There is a noticeable change in the s = 0 isocline. Indeed, there is a

single continuous 5‘ = Oisocline originating at the origin. This isocline changes
because the incentives for stocking salmon have changed. Specifically, managers stock

salmon to control alewife related damages in addition to providing angler benefits. The

a = 0 isocline, however, has not changed because there is no direct control over the
alewife population. Furthermore, the additional consideration of alewife related
damages causes the stocking boundaries to Shift downward relative to the no damage
case, and there is only one continuous w = 0 boundary. This occurs because more
salmon are stocked, at a given level of alewife, to reduce alewife biomass thereby
reducing alewife-induced damages.

In this case there are also three interior equilibria, A, B, C (Figure 4.4A and
Figure 4.48). Linearizing the system at these equlibria and finding the eigenvalues also
indicate that the equilibria are an unstable focus, conditionally stable saddle point, and
unstable focus respectively. The additional consideration of alewife induced damages
causes point A to shift down and to the left relative to the no damage case. This is an
intuitive result, because alewife now create damages they are managed at lower

population level and can support a smaller‘population of salmon

99

Optimal stocking strategies

Damages do not eliminate the need to consider the effect of the initial state of the world
on the optimal stocking strategy. There are two possibilities for initial conditions above
the saddle path i) rapid movement to the piece-Wise continuous saddle path and then
pursuit of the piece-wise continuous saddle path towards equilibrium B or ii) cycling
around equilibrium A if a stable cycle exists. Notice that that saddle path leaves the
interior region (Figure 4.4B). Afterithis point, w = 0 along the saddle path. When
damages are large enough a stable cycle does not exist around equilibrium A. Prior to
completing even the tightest stable cycle around equilibrium A, applying w*(s, a) leads
to a path that intersects the piece-wise continuous saddle path. If a w*(s, a) were
pursued beyond the saddle path, then the system would enter the alternate basin of
attraction and could not return. Hence, for all initial conditions above the saddle path,

it is optimal to apply the maximum stocking level to move as rapidly as possible to the

saddle path (the generally dynamics for a w = wmax are same as in Figure 4.1). Then,

the piece-wise continuous saddle path is pursued, initially forgoing stocking until the w
= 0 boundary is crossed from below (Figure 4.43). At that point, stocking commences
according to the w = w*(s, a).

Managers face the same problem for all initial states of the world below the
saddle path that was described for the no damage case. Specifically, stocking has
already been optimally reduced to zero, but salmon mortality is not high enough to

rapidly reduce the salmon population. There are no alternative controls to move the

100

system back to the saddle path. Alewives are eradicated and the salmon fishery can not

be sustained.7

Explaining the change in stocking strategy by accounting for ecosystem disservices

A critical difference from the no damage case is that alewife are a “multiple-use”
species (Zivin et al. 2000) when damages are included. Alewives jointly produce
ecosystem services and disservices. Following Horan and Bulte (2004) we consider ex
situ and in situ marginal net benefits associated with the alewife population at
equilibrium. The ex situ marginal benefits are equivalent to the shadow value, p, while
the in situ marginal benefits, (D, represents a suite of tradeoffs and are equal to the RHS
of the golden rule equation (4.16) less the base rate of return (Horan and Bulte 2004).8
Following Horan and Bulte (2004) if u > 0, then alewife are considered to be a
commodity, but if p < 0 then alewife are considered a nuisance. Alewife may be
thought of as a commodity when their net impact on social welfare is beneficial, and
this occurs if alewife generate more benefits as input in salmon production then they
detract by fouling beaches and drain pipes or preying on eggs and fry. If, however, the
net impact of alewife on social welfare is negative, then they are considered a nuisance.
Furthermore if (I) > 0 then alewife are an asset and overall society is better off with
more alewife. Conversely, if (I) < 0, then alewife are a liability and society is better off
limiting the alewife stock. In the case with no damages, we expect alewife to be both

an asset and a commodity because they provide benefits in terms of increased salmon

 

7 Eradication is dependent on the type-II predator response.
8 Recall that at equilibrium the term lit/y = 0 and the base rate of return term is r(l — Za/ K ).

.101

production. We verified that this is indeed the case at equilibrium A in the no damage
case.

In the current case, with damages, u > 0 at equilibria A and B. For equilibrium
A, however, (I) < 0, creating a case where alewives are Simultaneously a commodity
and liability. Horan and Bulte (2004) point out that this case is unlikely to be stable
and optimal because it can often be less costly to convert the liability into an asset.
This is achieved by increasing stocking to reduce the alewife stock. At the saddle point
equilibrium B, (I) > 0. Equilibrium B shifts up and to right, expanding its domain of

attraction.

4.6 Sensitivity Analysis

Sensitivity analyses are commonly used to examine how changes in a parameter affect
model results. This model has a large number of parameters and sensitivity analysis
could be performed for each. Doing so requires that new a phase plane be drawn and
analyzed for each parameter evaluated. In the interest of space we focus on three
parameters of interest, the discount rate, p, the amount of natural recruitment, b, and
effect of alewife on salmon recruitment to the fishery, (I). The discount rate was chosen
in order to understand how time preferences affect optimal management. Chinook
salmon naturally reproduce in the Great Lakes (Scott et al. 2003), and it therefore is
necessary to understand how the natural recruitment rate affects results. Finally,
sensitivity analysis was conducted with respect to (1) because there is little biological
data on which to calibrate this parameter. We explore model sensitivity in cases where

alewives do and do not cause damages.

102

The discount rate determines how the planner weighs net benefits in the near
term versus the future. Lower discount rates imply that the planner places relatively
more weight on the future. To examine changes in time preference, we increase the
discount rate from 5% to 10%. Both with and without damages this has no qualitative
effect on the results. In both cases equilibrium A shifts down and to the left and
equilibrium B shift up and to the right. The Shift in B is proportionally larger than the
shift in A. This implies that fewer alewife-induced damages are needed to cause the
qualitative shift in management strategies at a higher discount rate. This comes about
because the future salmon angling is less highly valued and there are fewer incentives
to preserve the prey.

There were no wild spawned salmon in the Great Lakes in the 1960s, when the
Chinook salmon stocking program began. Moreover, it was suspected that Chinook
salmon would not establish wild spawning populations. To test model sensitivity to
this assumption, we increased the amount of wild recruitment from 0 kg to 10,000 kg to
examine how wild recruitment may affect model outcomes. Wild recruitment has no

qualitative effect on the solution. Indeed, the equilibria points are at the same location

in s-a space. The stocking boundaries do move; the w = wmax and w = 0 boundary shift

downward. Without damages, only, equilibrium A is achievable; equilibrium B would
require negative stocking, but still creates a “saddle path” that bifurcates the system.
Finally, we examine the effect of the importance of alewife biomass to
recruitment to the fishery. This is determined by the parameter (b. We increase the
value of this parameter from 0.1 to 0.2. A large value of (I) enables greater fishable

salmon biomass per unit of salmon biomass stocked. When alewife cause damages the

103

w = wmax boundary shifts upward, implying that it is optimal to stock fewer salmon at

any point (s, a) because stocking lowers alewife biomass, resulting in a greater cost to
salmon production and increasing the fourth term in equation (4.15) in absolute value.
Furthermore, equilibrium A shifts up and to the right, while equilibrium B shifts down
and to the left. This implies that when recruitment to the fishery is more dependent on
alewife, all else equal, alewife-induced damages have to be greater for the system to
shift such that pursuit of equilibrium B is optimal. Specifically, in equation (4.16) the
term (b+w)0'(a) increases and in order for (16) to continue to hold, damages must
increase. The solution itself, however, is qualitatively unchanged.

Increasing the importance of alewife to recruitment to the fishery had a
qualitative influence on results when alewife do not cause damages (Figure 4.5). Only
equilibrium A continues to exist in the interior region. It is never optimal to stock at
the maximum level. Continued pursuit of the singular solution stocking strategy, w =
w‘(s, a), is a myopic solution and leads to a point on the other Side of the saddle path.
Starting at s = 0 and a = K the w = w*(s, a) solution is initially pursued. Continued
pursuit past point X (Figure 4.5) would lead to eradication of the alewife stock. This is
not optimal. At point X the far-sighted planner recognizes that it is optimal to switch
“prematurely” relative to the point where the boundary would be imposed (Clark et al.
1979; Clark 1990) and impose the w = 0 rule. This rule is applied until the system re-
enters the interior region by crossing the w = 0 boundary in the low right quadrant of i
the phase plane. At that point the w = w*(s, a) is applied until the system arrives at
point Y. At point Y the manger again recognizes that it is optimal to switch

prematurely to the w = 0 stocking rule. The arc X-Z is just above the bifurcating saddle

104

path; points above this line optimally lead to cycles around point A. For points below
this line, manages optimal stock zero salmon and have'no other control over the system.

Therefore, alewives are eradicated.

4.7 Discussion
The interaction of imperfectly targeted controls and non-convexities

The issue 'of non-conveXities and multiple equilibria in natural resource
economics is increasingly important (Dasgupta and Maler 2003). Many ecosystem
processes include thresholds and multiple basins of attraction. Most previous work on
ecologically driven non-convexities (e.g., Tohvonen and Salo 1996; Brook and Starrett
2003) asSumes convex-concave piecewise differentiable functional forms. In our
model, three trophic levels (anglers, salmon, and alewife) interact to Create an
ecologically driven non-convexity, though none are explicitly modeled with convex-
concave functions. The combination of decentralized human behavior (i.e., an open
access fishery), ecological processes, and stocking leads to multiple equilibria. We
note, however, that the findings of these previous authors are applicable and
informative to this case where non-convexities where not imposed a priori, but rather
emerged due to ecological-economic interactions.

Non-convexities are of interest because they imply that there are multiple
candidate equilibria, and the equilibrium that is pursued may depend on initial
conditions. In other words, there are multiple basins of attraction. There are two types
of bifurcation in this model. First, there is the previously discussed bifurcation line,

i.e., the saddle path. Below the saddle path there is a region of the state space were the

105

planner optimal stocks zero salmon, and has no additional control of the system. This
can be thought of as a positive feedback process (Brook and Starrett 2003). Below the
saddle path the system moves further and further from the region were stocking is not
desirable.

The other bifurcation is a bifurcation point and has to do with changes in
parameters. As alewife damages increase there is a value for D at Which the qualitative
nature of the optimal solution changes. This is associated with joint production, the
inability to perfectly target controls, and the “multiple-use species” problem (Zivin et
al. 2000; Rondeau 2001; Horan and Bulte 2004). Specifically, for small levels of
damage the cycle illustrated in Figure 4.3a shift down and to the left. Increasing
damages further, however, leads to a qualitative change in the nature of the solution.
The new solution type iS illustrated in Figure 4.4a.

In the current case, there is a single control for managing the angler-salmon-
alewife system. Alewives do not provide existence ecosystem disservices (though we
model it this way for convenience and for lack of a known intervention opportunity).
In reality, alewives cause damages to infrastructure and native fish populations. If
managers had other controls available to avert or remedy these damages or there
sufficient alternated prey items available to salmon the non-convexity associated with
“multiple-use species” problem might vanish. Interdependency and the inability to
perfectly target controls, however, are fundamental to the “multiple-use species” non-
convexity.

The model presented here focuses on imperfectly targeted controls and non-

convexities. The imperfect targeting of controls can be seen as an additional

106

institutional constraint (Dasgupta and Maler 2003) that is associated with the trouble of
managing recreational fisheries (Cox et al. 2002) and perhaps all renewable resources.
The need to incorporate institutional constraints to develop policy-relevant models can
not be overstated. Such institutional constraints will ofien lead to imperfectly targeted
control options that will lead to an increasing importance for ecologists and economists
to work together to determine how to optimally allocate natural resources in a world

full of non-convexities.

Implications for salmon stocking policy
The economic-ecological interactions associated with the salmon fishery on Lake
Michigan are more complex than any model can fully capture, but the results presented
here may provide insight into the tradeoffs associated with the stocking program.
Therefore, this analysis can be used to inform stocking policy in concert with more
complex simulation models (e. g., Jones and Bence in review). We note four key policy
relevant findings i) the optimal level of stocking is a function of the current salmon and
alewife stock, ii) the optimal of alewife eradication depends on current conditionsg, iii)
consideration of ecosystem disservices caused by alewife qualitatively alter the optimal
program, iv) the maximum catch along the optimal management path does not occur at
equilibrium.

The optimal stocking level depends on the current state of the system. Often
linear control problems result in cOnstrained solutions where a maximum or minimum

level of control is applied to rapidly arrive at equilibrium (Conrad and Clark 1987). In

 

9 Alewife eradication in the model depends on the assumption of a type-ll predator response. This result
however, may be interpreted as extreme alewife suppression. Such extreme suppression may lead to
other alternative ecological equilibria.

107

cases where controls are poorly targeted, as is the case of trying to manage a salmon-
alewife fishery (which is effectively a tri-trophic system), adjustment is likely to be
sluggish and involve nonlinear feedback rules (see Horan and Wolf 2005; Fenichel and
Horan 2007aifor other examples). This suggests that stocking policies that respond to
the state of the world will provide greater social benefits than state-independent
stocking strategies. This supports earlier simulation results by Szalai (2003) and Jones
and Bence (in review). The optimal “equilibrium”, however, may be a stable cycle that
requires constant adjustment. If there is non-malleable capital involved in generating
fishery benefits, and if there are adjustment costs associated with varying catches, then
such cycles may be dampened. Finally, we note that the optimal approach path to the
equilibrium (or stable cycle) leads to harvest levels that exceed those experienced at
equilibrium (or along the stable cycle). This is because it is optimal to “over-invest” in
salmon initially to reduce the alewife population and capture benefits associated with
the prey stock early. An alternative way to think about this is that it is optimal to
experience some higher catch levels early in the fishery that are not sustainable.
Alewife damages create the possibility for a bioeconomic regime shift. Under
high alewife-induced damages, management effort optimally shifts to averting these
damages and the salmon fishery shrinks drastically, in terms of catch and total net
benefit for anglers from the Chinook salmon fishery. It is important to note, however,
that averted alewife damages are synonymbuswith increases in the benefits from other
fisheries such as lake trout and yellow perch. This are not modeled explicitly, but are

assumed to be part of the alewife damage function.

108

We find that even though in our model alewife eradication is technically
feasible (this may not be the case in reality) the optimality of alewife eradication
(conservation) depends on initial conditions. For high initial levels of alewife biomass
eradication is never optimal. Conversely, for low initial biomasses of alewife with
significant salmon stocks eradication may not only be optimal, it- may be inevitable. If
alternate prey existed to support the salmon fishery (it does not in our model), then the
region where eradiCation of alewife may be optimal might be larger; we leave this for
future work. If salmon have a type-III predator response for alewife, then an alternate
equilibrium with low alewife and low salmon may exist. This too is left for future
investigation.

Finally, we revisit the notion that salmon stocking creates a win-win situation
with regard to angling and invasive alewife management. There is a stocking program
that maximizes social welfare, but there are still tradeoffs, as is almost always the case
in natural resource management. The use of salmon stocking clearly can provide
benefits on both fronts, but the ultimate result may not be perceived as a win-win
situation, especially if different stakeholders have different preferences for salmon

angling and averting alewife damages.

4.8 Appendix

This appendix explains the calibration of the model. Unless otherwise noted biological
parameters are based on Szalai (2003) or the Lake Michigan Decision Analysis
(LMDA) Model (Jones and Bence in review). Most biological parameters for fish are

based on size, we use age and weight data to convert these for parameter relevant to a

109

 

“representative fish.” The instantiations net recruitment rate for alewife, r, was
calculated based on per biomass recruitment of alewife minus the natural mortality rate
for alewife as 5.41. The carrying capacity for alewife, K, was based on simulations of
the LMDA model with no stocking, and was estimated at 2.89 x 109 kilograms of
alewife (a). The predation parameters [3 and (o where calculated based on
bioenergetics, weight, and age data reported in Szalai (2003) as B = 1.67 x 10'4 a'l and
(o = 4.27 x 10'8 NM. The salmon mortality parameters a = 5.82 t'1 and y = 0.02 a”1 were
chosen such that salmon had a high instantaneous mortality rate in the absences of
alewives and were comparable to the imputed representative instantaneous mortality
rates reported by Benjamin and Bence (2003b) associated with comparable alewife
biomasses. Salmon recruitment to fishery parameters (1) = 0.1 and y = 58 1/at where
based on bioenergetics parameters in Szalai (2003) and then manipulated to fit historic
salmon level. Salmon natural recruitment was assumed to be zero, b = 0 kg.

The recreational angling behavior parameters included x1 = 49.08, x2 = 6.47, x3
= 69.41, n = 6.66, and 0' = 1.74. An additional parameter was added to the cost

function so that the lognormal distribution was shift to the right so that that it was

defined from $35.93 to infinity. These parameters were estimated by assuming 13% of

angling license sold in Michigan resulted in Lake Michigan salmon fishing trips, and
fitting a time series of the number of anglers, effort, catch, and salmon biomass
(Benjamin and Bence 2003b and Jones personal communication) to the angler response
model by minimizing the sum of the squared error between that project by the model
and the observations. The alewife damage function was ($1.78 x 10'”)a2. The daily

catchablity, q = 6.43 x 10'8 is based on Benjamin and Bence (2003b) converted for

110

“”1

 

. in“

different time units. The marginal cost of stocking a kilogram of salmon, v = $19.55
was based on personal communication with Gary Whelan (Michigan Department of
Natural Resources) and weight-at-age data (Szalai 2003). The maximum number of
anglers was assumed to be the largest value in the number of angler time series. The
maximum number of salmon that could be stocked was chosen to be slightly greater
than historic high stocking levels at 40,500 kg (Jones personal communication). The
discount rate, p, was assumed to be 5%, which is within the range of standard discount

rates used in natural resource economics.

11]

4.9 Figures

2.5x109

2.0x10" ;

1.5x10" -

1.0x10"

Alewife biomass (kg)

5x108

 

 

 

0 L _. .7 .........
o 5x10° 1x107 1,5xlo7 2x107 2.5x107 3x107

Salmon biomass (kg)

Figure 4.1. Phase plane illustrating the dynamics of stocking salmon at the maximum

rate, w = wmax. Salmon biomass is shown on the x-axis and alewife biomass is shown
on the y-axis. Dashed lines represent the stocking zero and stocking maximum
boundaries (in lower right comer the w = 0 boundary and the s = 0 isocline are so close
they appear as a single solid line). The salmon axis is extended to 1.5 times the historic
high for salmon biomass in Lake Michigan.

112

 

2.5x10"

2.0x10" -
se aratrix
1.5x 109

1.0x109

5x10“ -
l . . /

o ‘ ‘ - - - - -
0 5x106 1x107 1.5x107 2x107 2.5x107 3x107

Alewife biomass (kg)

 

Salmon biomass (kg)

Figure 4.2. Phase plane illustrating the dynamics of stocking zero salmon, w = 0.
Salmon biomass is shown on the x-axis and alewife biomass is shown on the y-axis.
Dashed lines represent the stocking zero and stocking maximum boundaries (in lower
right corner the w = 0 boundary and thes = 0 isocline are so close they appear as a
single solid line). The salmon axis is extended to 1.5 times the historic high for salmon
biomass in Lake Michigan.

113

2.5xio9

 

    

 

 

 

’63
egzoxio9 ~
(I)
,3; 9
.9 1.5x10
.0
3.1.: 1.0x109
3
.9.
< 5x108 j S]
Saddle path a mon
‘ limit
0 . . . . . - . . . .
0 5x106 1x107 1.5xio7 2x107 2.5xio7 3x107
Panel A Salmon biomass (kg)
4x106 ~
A S = .
:2 3x106 r
ET » /
E B /
.9. 2x106 - /w=0
fl a=0 /
.3 /
3 1 i"
< X 0 ,4'/
0 A. C A l 4 A A L A A J l A A A _L _.
0 20,000 40,000 60,000 80,000
Panel B Salmon biomass (kg)

Figure 4.3. Phase plane illustrating the dynamics of the optimal solution in the case
with no alewife damages. Salmon biomass is shown on the x-axis and alewife biomass
is shown on the y-axis. Panel B expands the lower left hand comer of panel A. The
salmon limit is the maximum number of salmon that can be put into the system
determined by stocking at the maximum allowable level from an initial point of zero
salmon and alewife at carrying capacity. Dashed lines represent the stocking zero and
stocking maximum boundaries (in lower right corner of panel A the w = 0 boundary
and thes = 0 isocline are so close they appear as a single solid line). The arcs with
arrows represent optimal solutions from the initial point of zero salmon and K alewives
and the separatrices. The small regions in panel B do not permit phase arrows, but
local dynamics can be inferred from the arrows on the saddle path. In panel A the
salmon axis is extended to 1.5 times the historic high for salmon biomass in Lake
Michigan.

114

  
   

 

 

 

 

25x109
A 2.0><109
01)
ea +
(I)
is , . 1
E 1.5x10
.2
..D
so; .
9
'2 ”km ‘ $=O
<c
5x108
. Salmon
* / limit
0 _.. - . . . . .
0 5x106 1x107 1.5x107 2x107 2.5xio7 3x107
Panel A Salmon biomass (kg)
2x108
’53
.54
3? 15x108
8
E
35 1x108
$9.3.
.33.) 5x107 L
<1 I
0 ’ AAAAAAAAA l A l x A A J

 

 

0 1x106 2x106 3x106 4x106 5x106

Salmon biomass (kg)
Panel B

Figure 4.4. Phase plane illustrating the dynamics of the optimal solution in the case
with alewife damages. Salmon biomass is shown on the x-axis and alewife biomass is
shown on the y-axis. Panel B expands the lower left hand comer of panel A. The
salmon limit is the maximum number of salmon that can be put into the system
determined by stocking at the maximum allowable level from an initial point of zero
salmon and alewife at carrying capacity. Dashed lines represent the stocking zero and
stocking maximum boundaries. The arcs with arrows represent optimal solutions from
the initial point of zero salmon and K alewives and the separatrices. In panel A the
salmon axis is extended to 1.5 times the historic high for salmon biomass in Lake
Michigan. The small regions in panel B do not permit phase arrows, but local
dynamics can be inferred from the arrows on the saddle path.

115

Salmon limit

 

2.5x109 .
é” 2.0x10" 1
m .
g x
o 1.5x109 C330 A
'5 ;‘ ~.
93 : “"-~- -—-——_.l"i0
E) 1.0x109 -
<

5x108 _ +1

0 AllIAAJAIAALAIALIIILIIAIAlLLllAll
0 5><i06 1x1071,5x1072x1072.5x1073x107
Salmon biomass (kg)

 

 

Figure 4.5. Phase plane illustrating the dynamics of the optimal solution in the case
with no alewife damages and increased dependence on alewife for recruitment to the
fishery; (1) increased from 0.1 to 0.2. Salmon biomass is shown on the x-axis and
alewife biomass is shown on the y-axis. The salmon limit is the maximum number of
salmon that can be put into the system determined by stocking at the maximum
allowable level from an initial point of zero salmon and alewife at carrying capacity.
The dashed line represents the stocking zero boundary. The arcs with arrows represent
optimal solutions from the initial point of zero salmon and K alewife. The salmon axis
is extended to 1.5 times the historic high for salmon biomass in Lake Michigan.

116

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