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This is to certify that the

dissertation entitled

HABITAT USE AND COMPETITIVE BOTTLENECKS
IN SIZE—STRUCTURED FISH POPULATIONS

presented by

JAMES F. GILLIAM

has been accepted towards fulfillment
of the requirements for

Ph.D. Zoology

degree in

 

 

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Habitat Use and Competitive Bottlenecks

in Size-structured Fish Populations

By

James F. Gilliam

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Zoology

1982

ABSTRACT

HABITAT USE AND COMPETITIVE BOTTLENECKS
IN SIZE-STRUCTURED FISH POPULATIONS

BY

James F. Gilliam

This thesis deals with two current problems in the ecology of
size-structured populations. By size-structured populations, I mean
populations which consist of an array of sizes, and in which different
sizes have different ecological characteristics. The first problem is
the question of optimal habitat choice when habitats differ not only
in the foraging rates obtainable in them, but also in the mortality
risk to which an animal is exposed while foraging. The second problem
involves an interaction between two species, say A and B, in which A
preys upon B when A is large, but A competes with B when A is small.

The first chapter deals with the problem of foraging under
mortality risk. This chapter is completely theoretical. Using
optimal control theory, I derive decision rules for animals given a
"choice" of a number of growth rates and mortality rates, where the
choice of a higher growth rate during a day brings with it a higher
probability of being preyed upon. For juveniles in populations near
equilibrium, the rule is very simple: an animal should choose the
habitat or behavior which minimizes the ratio of mortality rate to
growth rate. For adults or animals in increasing or decreasing
populations, the results can be expressed as degrees of departure from

the rule just stated.

James F. Gilliam

The second chapter concerns the problem of the predator-prey
"bottleneck." The chapter contains some work on the implications of a
bottleneck's effect on the demography and population dynamics of the
system, but most of the chapter is an investigation of the potential

for this type of effect between the largemouth bass (Micropterus

 

salmoides) and the bluegill sunfish (Lepomis macrochirus). The

 

largemouth bass preys upon the bluegill over much of its life, but
when small the bass and bluegill consume similar prey. I conclude
that the bluegills probably reduce the growth rates of small bass in
one lake but not another, and quantify the effect of four bluegill

densities on bass growth rates in an experimental pond.

ACKNOWLEDGEMENTS

I wish to express my bountiful gratitude to my adviser, Earl
Werner, for providing an atmosphere for intellectual rumination
punctuated with poignant yet gentle insights. He has helped me to
begin to discriminate between details which elucidate and those which
obscure. Don Hall formed the other half of the dynamic duo of whose
company I never tired, and in whose presence my mind never had a
chance to lapse for long. Don and Earl know that to do good science
is, in part, to get wet and muddy, and we did a lot of good science.

I thank many people at Kellogg Biological Station and Michigan
State University. Laura Riley provided instant expertise on even
unfamiliar tasks; she has a job waiting in my lab if she ever leaves
KBS. Gary Mittelbach, Kay Gross, Leni Wilsmann, Tim Ehlinger and
other present and past students helped make my stay at KBS
provocative, often delightful, and always more than tolerable. John
Gorentz and Steve Weiss provided superb, professional computing
facilities and service. Charlotte Seeley provided the most skillful
and dedicated secretarial services of which I have ever seen or heard.
Art Wiest kept the lab lively and provided skilled and dedicated help
whenever asked. Carolyn Hammarskjold and Marilyn Jacobs were far more
efficient at providing library materials than I was at returning them.
M. Salmoides and L. Macrochirus provided both surprises and steady

service throughout.

ii

Debra and Matthew Gilliam helped reconvince me each time I saw
them that the benefits of an academic life for us make the costs
worthwhile. Jack and Bett Gilliam, and Bob and Betty Scott, have
supported us in many ways during the career bottleneck called graduate
school.

I thank Eric Goodman and Bill Cooper, two committee members who
remembered who I was after I moved from the MSU campus to KBS, for
their enthusiasm for my work when I made regrettably infrequent
contacts.

Financial support was provided by grants DEB-7824271 and
DEB-8119258 from the National Science Foundation.

KBS has probably spoiled me. I hope that the proverbial

lightning strikes twice.

iii

TABLE OF mNTENTS

Page
LIST OF TABLESOOOOOOOI.O0..00....00.0.0...OOOOOOOOOOOOOOOOOOOOOO vi
LIST OF FIGURES....0OO0.000000000000000000000000000.00.00.00.00. vii

CHAPTER 1 - FORAGING UNDER MORTALITY RISK IN
SIZE-STRUCTURED POPULATIONSoooooooooo00000000000000. I

INTRODUCTIONOOOOO0.0.0.0000...000......OOOOOOOOOIOOOOOOOOOO 1

THE NATURE OF THE TRADEOFF: A REPRESENTATIVE
PROBLEM ANDAGENERALIZATIONOOOOOOOOOOOOOOOOOOOCO0.0.000... 3

THE SELECTION CRITERIONoooooooooooooooooo00.000000000000000 13
A SOLUTION-cococoosooassesses-sococoooocoooooooooooooooooco 17
Optimal DeC1810n8 for Juvenilesoooo00000000000000.0000 21
Optimal DECISIODS for Adoltsocoooooo0.0000000000000000 32
INCREASING AND DECLINING POPULATIONSOOOOOOOOOOOOOOOOOOOOOOO 36
CONCLUSIONooooooooooooooooosuccesses-000000000000000000000. 38
LIST OF REFERENCESooooosocoocoocoo00000000000000.0000...coo 44
CHAPTER 2 - RECRUITMENT OF A PISCIVORE THROUGH A COMPETITIVE
BOTTLENECK: COMPETITIVE EFFECTS OF BLUECILL
SUNFISH 0N JUVENILE LARGEMOUTH BASS................. 47
INTRODUCTIONOOOOOOOOOOOOOOOOOOOOOOO0.00.000...0.......00... 47

GROWTH RATEASAEY PARAMETER...OOOOOOOOOOOOOIOOOOO0...... 51

MEmODSOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO0000...... 57

RESULTSOOOOOO...00.0.0.0...OOOOOOOOOOOOOOOOO0.0000000000000 60
Handling Times...OOOOOOOOOOOOOOOOOOOOOOIOOOOOOOOOOOOOO 60
Habitat US$00so...0.0000000000000000...ooooooooooooooo 66
Growth Rates.......................................... 69
Diets in Lawrence Lake and Three Lakes................ 75

iv

Page
A QUANTIFICATION OF THE EFFECT OF BLUEGILL
DENSITY ON BASS GROWTH RATESOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 88
CONCLUSIONOIOO0.0.0.000...00......000......0.00.00.00.00... 93

LIST OF REFERENCESIOOOOOOOOOOOOOOOIOOOOOOOOOOOOOOO...0.....102

APPENDIXOO0.00.00...OOOOOOOOOOOOOOOOOOOO0.00......0.0.0.00000000105

Table

LIST OF TABLES

Page

Diets of young-of—year largemouth bass in Lawrence
Lake, 1980, expressed as percent dry weight of
total diet. To be included in the table, an item
must have comprised at least five percent of the
diet on at least one date. Values rounded to the
nearest percent, and values less than 2 percent
omitted. Prey categories are, in order, copepods:
Cyclopoida, Calanoida; cladocerans: Bosmina,
Sida, Latona, Simocephalus; Chironomid larvae;

 

 

 

Baetidae;centrarCh1d fry...0.0000000000000000000000.0 76

Diets of young-of-year largemouth bass in Three
Lakes, 1980. Description as in Table 1. Acro =
Acroperus (a littoral cladoceran), Cypr -

cyprinid fiSbooooooooococoone...00000000000000.0000... 79

Diets of 25-60mm bluegill sunfish in Lawrence

Lake, 1980, expressed as percent dry weight of

the total diet on a date. Criteria for inclusion

as for Table 1, except Baetids also included.

Values rounded to the nearest percent, and values

less than two percent omitted. Ophr - Ophryoxus

(a littoral cladoceran), Caen = Caenidae

(mayfly nymph), Coen - Coenagrionidae, Tric -

Trichoptera, Amph = Amphipoda, Acar - Acarina,

Gast - Gastropods..................................... 87

Diets of 25-60mm bluegill sunfish in Three

Lakes, 1980. Description as for Table 3, plus
the following additional prey categories:

Bury - Eurycercus (a littoral cladoceran), Daph =
Daphnia, Chao - Chaoborus (a plaktonic

dipteran13rva>0000000OOOOOOOOOOOOOOOOOOO00.00.0000... 89

vi

Figure

LIST OF FIGURES
Page
CHAPTER 1

Size-specific growth (g1) and mortality (p1)
rates in habitats 1 and 2. b(s) is the birth rate..... 5

Hypothetical size-specific growth and mortality
rates for bluegill sunfish in Lawrence Lake............ 10

(a) Growth and mortality rates resulting from

mixed habitat use. (b) The relationships in

(a), replotted to express mortality rate as a

function of growth rate................................ 12

(a) Size-specific birth and maximal growth
rates. (b) Mortality rate, a function of
Size and "Chosen" grOWth rateOOOOIOOOOOCOOOOOO000...... 15

Graphical solution of the optimal growth rate
for juvenileBOOOOOOOOCOO...0.00.00.00.000.0.0.0000...O. 23

(a) Optimal size (sept) for switching from

habitat 1 to habitat 2. s* is the size at which

an animal maximizing its growth rate would switch.

(b) Optimal switching sizes between plankton and
vegetation. Fitness is maximized by foraging on

plankton between sizes so and 81, foraging among
vegetation between sizes 31 and 82, and

returning to the plankton at size 32................... 26

(a) Illustration of habitat specialization

when p is linear in g. The "corners" each represent
exclusive use of a habitat. (b) choice of an

intermediate growth rate (use of both habitats)

when is convex....................................... 29

vii

Figure

Page
CHAPTER 1

An example of added complexity in the u-g

plane: the effect of travel costs and short-term
learning on mortality and growth. Both factors
favor habitat specialization. (a) Energetic

travel costs are depicted as displacing the

growth rate downward. Short-term learning effects
are represented by a "sag” in the growth rate,
depicting a case in which an animal's foraging

rate while in a habitat is a function of the time
spent there. (b) The dashed line represents no
travel or learning cost. The travel cost "detaches"
the line and moves it to the left. The learning
cost makes it bend. Both effects make a mixed
strategy unlikely...................................... 31

Optimal growth rate for adults. The optimal
marginal risk (an/3g) for an adult is less than

that for a juvenile with the same u(g) for a

given size. Starting with the juvenile solution,
Bu/Bg - u/g, the adult relationship can be viewed

as (a) reducing Bu/ag until Bu/Bg - (u-b/V)/g,

or (b) drawing a new function, (p-b/V)/8, and
finding the tangent through the origin, at which
8(u-b/V)/3g - (u-b/V)/g. (a) and (b) are equivalent
because 3(u-b/V)/3g - Bp/Bg, since the derivative
is evaluated with the state and auxiliary variables,
and hence b/V, held constant........................... 35

CHAPTER 2

Percent reduction in survivorship through a

size (not an age) interval due to a reduction in

growth rate. L(Sl, 82) is the survivorship prior

to the reduction in growth rate. c is the value

by which the growth is multiplied...................... 55

Handling times on Simocephalus and damselflies
for largemouth bass and bluegills. Data points
are the individual values for the bass................. 62

 

Optimal prey weights for bluegills and largemouth
bass 0 Left wrve: bass. Right curve: bluegill o o o o o o 65

Habitat use by largemouth bass and bluegills.
"Microhabitat" represents location in or above
the vegetation. LV - lower vegetation, UV -
upper vegetation, 0-.25 - zero to 0.25m above
vegetation. "Position" represents distance

from ShoreOOOOOO0.0..OOOOOOOOOOOOOOOOOOOOOOOO00......O. 68

viii

Figure

10

ll

12

Page
CHAPTER 2

Mean largemouth bass lengths in a maximal

growth rate experiment (Ponds) and two lakes.

3Ll980 - Three Lakes in 1980. LL1980 -

Lawrence Lake in 1980.................................. 71

Instantaneous relative growth rates (day’l)

of largemouth bass in the maximal growth rate
experiment (Ponds), Three Lakes (3Ll980), a
school in Lawrence Lake (LLBOS), all bass
collected in Lawrence Lake (LL80) and Lawrence
Lake in 1979 (LL79). The arrow indicates the
point of transferral of the fish from one pond

to anotherOOOOOOOOOOOO0.00.0.0...00.000.000.00...O...O. 73

Diets of largemouth bass in Lawrence Lake.............. 78
Diets of largemouth bass in Three Lakes................ 31

Mean dry weight of prey in the stomachs of

individual largemouth bass in Lawrence Lake.

The X's represent bass with at least one fish

prey. The E's represent fish with empty stomachs...... 83

Mean dry weight of prey in the stomachs of
individual largemouth bass in Three Lakes.
symbOIS as in Figure 9000000ooooooooooooooooooooooso... 85

Final mean dry weights of bluegills and largemouth
bass, as a function of the density of bluegills
in an experimental pond-00000ooooooooooooooooooooooooso 92

Phase-plane representation of a predator and a

prey species, in which the offspring of the

predators first compete with the prey. See

Appendix for assumptions. H - prey, P - predator.

The prey and predator isoclines are represented

by H' - 0 and P' - 0, respectively. K a maximal

prey density. (a) The predator-prey phase plane

(b) The relationship of R1 to H and the relationship

of L to H for three densities of P: 0, P1, and P2......1oo

ix

CHAPTER 1
Foraging Under Mortality Risk In Size-Structured Populations

INTRODUCTION

For most animals, fitness is clearly related both to an ability
to obtain food and to avoid predators. However, many circumstances
arise in which these two components of fitness conflict. For example,
the most profitable habitats for foraging may also be the most
dangerous, or certain foraging tactics may make a forager more
vulnerable to predators. It is often concluded that predation risk
strongly affects the evolution of a forager's behavior (e.g.
Rosenzweig 1974; Sih 1980; Nelson and Vance 1979; see Stein 1979 and
Morse 1980 for reviews). Further, some work has shown that the
behaviors can be plastic since foragers show a reduction in foraging
rate when predators or predator models are present (Stein and Magnuson
1976; Milinski and Heller 1978; Caraco, Martindale, and Pulliam 1980;
Shaffer and Whitford 1981; Fraser and Gerri 1982). Werner, Gilliam,
Hall, and Mittelbach (1983) have shown that small bluegill sunfish
shift to energetically less profitable but apparently safer habitats
when a predator is present, and, further, that the shift results in
reduced growth rates for the survivors. Thus, the presence of a
predator introduces a cost to a forager, the risk of predatory death,

which the forager may partially offset by incurring a second cost, a

reduced foraging (growth) rate. Collectively, the above studies show
that at least some ants, fish, birds, and mammals do in some way
facultatively balance a tradeoff between mortality risk and foraging
rate.

Since habitat use and prey selection within a habitat are central
to competitive processes, a method of predicting foraging behavior
under predation risk is an essential element in a predictive theory of
community organization. In this paper I present a basis for
predicting behavior when the maximization of foraging rate and
short-term survivorship conflict. The major question asked is: Given
a "choice" of a number of foraging rates, each with an associated
mortality rate, which choice maximizes fitness? In particular, given
an array of habitats, each with an associated foraging rate and
mortality rate, in which one(s) should an animal forage?

Here, I investigate the above questions for iteroparous
size-structured populations. By "size-structured" populations, I mean
populations in which an individual's salient ecological properties,
notably its foraging abilities, vulnerability to predators, and
fecundity, can be regarded as functions of the individual's size
rather than its age. I also allow the animal's foraging rate and
mortality rate to be under behavioral control; that is, the animal's
daily foraging and mortality rates are functions of both its size and
its behavior (e.g. habitat use, times of feeding, swimming speed).
The fecundity rate is taken to be a function of the animal's size
(reproductive effort questions are not considered here). This
dependence of ecological characteristics on size rather than age

appears to be a reasonable representation for many animals which

exhibit indeterminant growth, as many fish, reptiles, amphibians, and
invertebrates.

I first consider optimal behavior in populations near equilibrial
population size, and then investigate the cases of increasing and
decreasing populations. Initially, I assume that the foraging,
mortality, and fecundity rates are not explicit functions of time.
That is, these rates are functions of time only to the extent that
body size is a function of time. The important biological consequence
of this assumption is that this implies a constant environment and
continuous reproduction. This allows an analytical solution to the
problem which provides a starting point from which consequences of

alternative assumptions can be explored.

The Nature of the Tradeoff:
A Representative Problem and a Generalization
As an example of the growth-mortality tradeoff, consider the
problem of ontogenetic habitat shifts by fish. As fish grow, they
typically exhibit marked shifts in prey utilization, as documented by
a huge literature on fish diets (Carlander 1977). Habitat shifts
often accompany the changes in diets. For example, juvenile bluegill

sunfish (Lepomis macrochirus) travel from shallow littoral nests to

 

the limnetic zone upon hatching and feed upon zooplankton (Werner,
1966). They then usually return to the littoral zone after reaching a
size of about 20 mm standard length and prey primarily upon insects
and crustaceans. In lakes in southwest Michigan, the bluegills
typically forage among littoral vegetation until reaching 60-100 mm
length (3 to 4 years old), at which time they switch habitats again,

and most of their time is subsequently spent foraging on open-water

Figure 1. Size-specific growth (g1) and mortality (“1) rates in habitats

l and 2. b(s) is the birth rate.

$3

 

w.MNfiw’#flu

 

 

EIVH

zooplankton (pers. obs.; Mittelbach, 1981).

Mittelbach (1981) has presented evidence that the bluegills in
the littoral zone would usually exhibit higher growth rates if they
instead foraged in the open limnetic zone. He suggested that small
fish may remain among vegetation because the structure affords
protection from predators, and that larger fish can utilize the open
water with less danger because their vulnerability is lower than small
fish. Further, Werner, Gilliam, Mittelbach, and Hall (1983) have
experimentally confirmed that small bluegills shift to greater use of
a vegetated habitat in the presence of predators versus their use in
the predators' absence, but larger non-vulnerable bluegills do not
show the habitat shift.

A simple representation of vital rates associated with
ontogenetic habitat shifts is shown in Figure 1 (see werner 1982 for a
similar treatment of habitat shifts when habitats do not vary in
mortality risk). At a particular resource level, a fish of a given
size foraging in Habitat 1 is depicted to grow (gm/day, cal/day, or
mm/day) at a rate g1 and incur a mortality rate (probability of
death/day) of H]. In Habitat 2, the fish would experience growth and
mortality rates of g2 and Hz, respectively. Thus, Habitat 2 is always
the more dangerous habitat, but large fish can grow faster there. “1
and “2 are shown as declining functions of fish size, which will
usually be true if larger fish are less vulnerable to predators,
though ml or U2 may increase over some range if the predators are
positively size-selective. As drawn, the growth functions, g1 and g2,
indicate that, on a particular resource, the growth rate first

increases and then declines to zero. This happens when the animal's

gross foraging rate increases with size, but at large sizes metabolic
costs increase faster than increases in foraging capabilities. The
details of the growth and mortality functions are not important to the
conceptual development of the problem; Figure 1 simply represents a
seemingly common case.

Given the relations shown in Figure 1, a small fish would both
maximize its growth rate and minimize its mortality rate by foraging
in habitat 1. This would maximize its survivorship and size at each
age and is clearly optimal in this framework. As the fish grows, at
what size would a shift from habitat l to habitat 2 be favored by
natural selection? A fish shifting a size 3* would maximize its
growth rate, but at the cost of raising its mortality rate from
U1 (3*) to U2 (3*). Thus, at sizes greater than 3*, the animal is
faced with a growth rate-mortality rate tradeoff; maximization of
growth during a day conflicts with the minimization of mortality risk
(maximization of survivorship) during that day. Intuitively, one
might surmise that a delay in the habitat shift would be favored by
natural selection, since at sizes slightly larger than 9*, the fish's
mortality rate can be substantially lowered at the cost of a slightly
lowered growth rate. However, a delayed switch to Habitat 2 would
usually increase mortality at future ages, since the fish would be
smaller at each future age than it would have if it had switched at
3*. Also, the animal's fecundity at each future age will be lowered
by the delay in switching since the animal will be smaller at each
subsequent age. Clearly, the short-term tradeoff between mortality
rate and growth rate can be restated to be a tradeoff between

short—term survivorship (during a day) and long-term survivorship and

fecundity (over the rest of the animal's life).

Figure 2 represents a case similar to Figure l, but depicts one
habitat (in this case, a vegetated habitat) as being safer (lower u)
but less energetically profitable (lower 3) than another habitat (an
open water habitat with planktonic prey). This may crudely
approximate the options often available to bluegill sunfish. Here,
the question is not just when to switch habitats, but whether to
switch, and how many times.

Now suppose that an animal may choose to split its time between
the two habitats (the case for three or more habitats can also easily
be constructed). Figure 3a depicts, for a fish of a given size, the
overall daily growth and mortality rates as a function of the time
spent in one of the habitats. Here, Habitat l is "dangerous“ but
"rich", and Habitat 2 is "safe" but "poor". The growth rate, g, is
drawn as an increasing, concave function of the time spent in the rich
habitat. The concavity is due to a concave relation between daily
foraging rate and growth rate (Webb 1978). Other shapes of u and g
are certaily plausible. For example, 3 may decline over some range in
Figure 3a if there are substantial travel costs between habitats;
different shapes of curves can be easily accommodated in the model
presented in this paper.

Since u and g are both functions of behavior (the proportion of
time in Habitat 1) for a fish of a given size, we can plot U directly
as a function of g, as in Figure 3b. This makes the tradeoff
explicit; the animal must, in effect, choose a growth rate (via its
habitat choice), and each increase in growth rate (a benefit) carries

with it an increase in mortality (a cost). At other points in an

Figure 2. Hypothetical size-specific growth and mortality rates for

bluegill sunfish in Lawrence Lake.

RATE

10

"plankton

 

 

gplankton

 

Qveg
FISH SIZE ,3

Figure 3. (a) Growth and mortality rates resulting from mixed habitat
use. (b) The relationships in (a), replotted to express

mortality rate as a function of growth rate.

12

P ._.<._._m<I 2.
m2: “.0 ZO_._.mOn_Omn_

o

 

 

3v

 

A3

 

BIVH

13

animal's life history, u may be a nonincreasing function of g. For
example, fish smaller than 3* in Fig. 1 simultaneously maximize
growth and minimize mortality by foraging in Habitat 1.

The problem can now be generalized by defining three functions,
as shown in Figure 4. The animal's birth rate, b(s), is a function of
the animal's size. Also, there is a function, gmax(s), which gives
the animal's maximal growth rate obtainable under existing resource
levels. The animal's mortality rate on a particular day is a
function, u(g,S), of the animal's size and the growth rate the animal
"chooses" (Figure 4b). It will be convenient to refer to a "choice of
growth rate" on each day, but, of course, this phrase really means
that the animal behaves in a certain way (e.g. through its habitat
choice, diel activity pattern, foraging speed, distance from a
refuge), and that behavior, together with the animal's size, results

in a growth rate and a mortality rate for that day.

The Selection Criterion
On each day an animal must adopt some behavior, and the behavior
adopted will have associated growth and mortality rates. Taken
together, the series of decisions across an animal's (or cohort's)
lifetime results in a growth path. There exist an infinite number of
possible growth paths, each with an associated survivorship and

fecundity schedule, and the present task is to find the particular

 

growth path which maximizes fitness.

The criterion of fitness used here is this: In a
density-dependent population at equilibrium, find the series of
decisions which result in R0 = 1 (r = 0) when all other series of

decisions results in R0 < 1 (r < 0). Here, r is the instantaneous

14

Figure 4. (a) Size-specific birth and maximal growth rates.
(b) Mortality rate, a function of size and "chosen"

growth rate.

RATE

 

 

15

(a)

b(s)

9 max(S)

u [9.8] (b)

\ Q max(5)

16

rate of increase of the population, and Ro is the net reproductive

rate,
R, = affix)b(x>dx. (1)

where £(x) = survivorship to age x, and b(x) = fecundity at age x. A
morph with R0 = 1 just replaces itself, and ”invading" morphs with
R0 < 1 are eliminated.

This criterion is used implicitly in density-dependent models
without age-structure, where the criterion is equivalent to the
maximization of population density. The maximization of total
population density cannot usually be used as a congruent criterion for
size-structured populations, although Hastings (1978) has shown that
it can be for the special case of the density-dependence at each age
(size) depending only upon the total population size, and Charlesworth
(1980) has shown that if a "critical age group" can be identified, the
maximization of the density of that age group is equivalent to the
criterion, R0 = 1 when all alternatives yield Ro < 1. The analysis in
this paper assumes nothing about the details of density-dependence
except that, for a population exhibiting an arbitrary series of
behaviors, the population will establish an equilibrial size (and age)
structure at which Ro - 1, and the task is to depict the choice of a
growth path (series of behaviors) which is noninvasable by alternative
choices when Ro - 1 for the "established" path.

I describe below the series of decisions which maximize R0 for a
given level of resources and mortality pressure, and note that the

solution holds for the particular case of the resource levels and

 

mortality pressures existing at equilibrium. That is, a population

17

exhibiting optimal behavior will have the property that R0 = 1 at
equilibrium, and all invading behaviors yield RO < l. The case of

increasing or declining populations will be considered subsequently.

A Solution

Mathematically, we want to find a function, g(x), which maximizes
an integral, R0. This kind of problem can be solved with optimal
control theory, a technique with its roots in the calculus of
variations. Optimal control theory is certainly largely unfamiliar to
most ecologists, though optimal control theory has been used in
ecological contexts by some authors, especially recently (Katz 1974,
Leon 1976, Sibly and McFarland 1976, Clark 1976, Mirmirani and Oster
1978, Oster and Wilson 1978, McCleery 1978, Vincent and Pulliam 1980,
Cab 1980, Goodman 1982). Suggested introductions are contained in
Intriligator (1971), Takayama (1974), and the original work of
Pontryagin 35 El' (1964), as well as the papers cited above.

In the terminology of optimal control theory, the growth rate,
g(x), is a control variable, whose value at each age is to be chosen
so as to maximize the objective functional, R0. In addition, two
state variables can be defined which describe the state of a cohort at
age x. These state variables are the size at age x, 3(x), and the
integral of previous mortality rates, D(x) = 6x u(y)dy, where y is a
dummy variable. These two state variables completely specify £(x) and
b(x) since £(x) = e'D(x) and b(x) a b [s(x)]. Across age (time), the
state variables change at a rate determined by the control variable
and the state variables themselves:

x

31—3 = d ’ggffldx = 11(X) = u[S(X). g<x>1 <2)

 

18

‘35 = g(x) (3)

X

Thus, at each age, the life historical effects of the choice of growth
rate are expressed through the effects on the state variables.

There is also a constraint on the growth rate, g(x) S
gmax [s(x)]. This can be incorporated either by explicitly entering
the constraint in the problem, or by imposing an artificial penalty
that effectively prevents g(x) from exceeding gmax[s(x)]. The latter
method keeps the mathematics more transparent in this case and has no
effect on the biological results, as will be seen in graphical
solutions. The penalty is imposed by letting u(g, 3) turn arbitrarily
sharply upward at gmax(3)' In Figure 4b, this could be depicted by
drawing u(g, s) as turning very sharply along the edge defined by
gmax(s), rather than terminating the surface as now drawn. Thus,
strictly speaking, g(x) has no formal upper bound, but the penalty
(greatly increased mortality rate) is set so great that in practice it
will not exceed gmax(5)° The mathematics also require that b(s) be
continually differentiable; accordingly, b(s) is taken to be very
sharply sigmoid near the size of first reproduction, rather than a
discontinuous function as depicted in Figure 4a.

The Hamiltonian function can now be formed:
H = e-D<X> b[s(x)] + AD<xI utg<x). s<x>1 + A8<x)g(x) <4)

The first term is the integrand of the objective functional. The
second and third terms are each the product of an auxiliary variable
(also called a costate variable, adjoint variable, or multiplier), and

a state variable's rate of change. 3AD(x) is an auxiliary variable

19

associated with the state variable D(x) and multiplies u[g(x), s(x)],
the value of dD/dx (see Eq. 2). Similarly, As(x) is associated with
the state variable s(x) and multiplies g(x), the value of ds/dx

(Eq. 3).

Pontryagin's Maximum Principle states that H is maximized at each
x by the Optimal g(x). Further, it is known that dH/dx = BH/Bx. For
this problem, 3H/3x = 0 since x does not appear explicitly in the
equation (e.g. b and u depend only upon the state and control
variables, not explictly on age itself). Thus, the Hamiltonian is
said to be autonomous and is in fact constant across all ages.

The fact that H is constant for all x can be used to good
advantage in determining a solution. Since H is constant, its value
can be determined by finding its value at any particular x, in this
case when x is very large. It is known that when the objective
functional's (Ro's) limits of integration are from say, 0 to T, where
T is finite, the values of the auxiliary variables at T are equal to
zero if the final values of the state variables are unspecified (in
this problem, we do not specify required final values for s and D).
This is true for any arbitarily large but finite T. Surprisingly,
perhaps, this result does not always hold for T =‘”, and
counterexamples have been found (e.g. Takayama 1974, p. 625). It does
appear that *D(”) a 0 and A8(m) - 0 for this system, but the general
conditions for which this would hold are not known. However, we can
circumvent this problem by taking the integral (R0) from 0 to T, where
T is very large but finite, rather than from 0 to “a If T is
sufficiently large, we can safely take e‘D(T)b(T) = 0, and since AD(T)

= AS(T) = 0, it follows that H(T) = 0. Thus, for all x, H = 0 for the

20

optimal growth path, and H < 0 for suboptimal paths.

reminiscent of the condition, r = 0 (R0 = 1) for the

flusis

optimal path and

r < 0 (Ro < 1) for all other paths.
The problem can now be stated succinctly within the structure of
The goal is to find

Pontryagin's Maximum Principle. the path of the

control variable, g(x), which maximizes R0 = éTe-D(x) b[s(x)]dx,

subject to the constraints on the state variables (Eqs. 2 and 3). For
this problem, dropping the arguments of the functions for brevity,
H = e’Db + AD D+ Asg = 0 (5)
Further, since H is maximized by the choice of g,
-§E = 0 = ADEB-3+ A for each x (6)
as g S
The changes in the auxiliary variables are described by
d)‘D 23H
__._x = — 8—D: e Db (7)
dxs _ 3H _ "D (11) A 3M (8)
.__. - - .__ _ .__ _ -D"‘
dx 33 d3 33

The usual way of solving this type of problem would be to solve
the differential equations, Eqs. 2, 3, 7, and 8 with two known initial
values, D(O) - 0 amd 3(0) 8 so, and two known final values,3\D(T) -
AB(T) - 0. However, the observation that the Hamiltonian is constant
and equal to zero allows us to circumvent this process and obtain

useful results without having to specify the forms of the functions.

One way to describe the relationship between u and g at the

21

optimal solution is to find Bu/Bg, which will be called the "marginal
risk," at the optimum. Solving Eq. 6 for Bu/Bg yields 3u/3g - -
As/AD. Solving Eq. 5 for As (this is where H = 0 is a crucial

observation) and substituting into a“lag = - As/AD yields

3 + e'Db/A
—B-= u D for each x (9)
as g

 

This equation is the central result of this paper. The meaning of
this equation will be assessed below separately for juveniles and

adults.

Optimal Decisions for Juveniles

 

For juveniles, b = 0 by definition. Hence Eq. 9 reduces to

an u
33 g ( )

In Figure 5, it can be seen by inspection that this relationship holds
at only one value of g, denoted 80pt' That is, straight lines drawn
through the origin depict points at which U/g is constant, and the
particular line which is tangent to u defines the point at which

Bu/ 3g = u/g.

Figure 5 suggests an alternative statement of the condition for
optimality. SOpt may be recognized graphically as the value of g
which minimizes u/g. This can be confirmed by solving a(u/g)/3g = 0
for au/ag, which yields Eq. 10. Thus a simpler statement of the rule
is “minimize p/g at each age (size)."

The meaning of this rule becomes clearer when related to an
expression for survivorship to a particular size. It has been shown

(VanSickle 1977, Warner, Gilliam, Hall, and Mittelbach, 1982) that the

22

Figure 5. Graphical solution of the optimal growth rate for juveniles.

23

 

 

 

 

24

survivorship to a particular size, 31, can be described by
81.119;

Manse" 8: 3‘8) d8 . (11)
where so - size at birth. An animal which minimizes u(s)/g(s) at each
size thus maximizes its probability of reaching each size, in
particular the probability of reaching reproductive size.

The simplicity of the rule (minimize u/g) suggest that animals
may evolve an ability to make decisions approximating optimal
behavior. For example, if an animal must choose between two habitats,
the first of which confers twice the growth rate as the other, the
animal should just choose the first habitat unless it is twice as
dangerous (u twice as large). That is, "choose habitat 1 if ullgl <
“2/82" can be rewritten as, "choose habitat 1 if gI/gz > ul/uz." The
animal does not have to judge the absolute levels of u and g in each
habitat, only the relative values of growth and "danger". Given the
repeatedly demonstrated ability of animals to make some kind of
decision when foraging rate and mortality rate conflict, this
particular rule does not appear difficult to follow, whether through
the evolution of inflexible behavior or the evolution of an ability to
facultatively balance costs and benefits through some process. The
simplicity of the rule also suggests that tests of this model are
achievable if growth and mortality rates can be experimentally
determined for alternate habitats.

The problem of ontogenetic habitat shifts can now be solved for
juveniles. Figures 6a and 6b are solutions to the problems in Figures
1 and 2. Here, the switches are shown to be discrete, though a period

of use of both habitats might be predicted in some cases (see below).

Figure 6.

25

(a) Optimal size (SOpt) for switching from habitat 1 to
habitat 2. 3* is the size at which an animal maximizing
its growth rate would switch. (b) Optimal switching sizes
between plankton and vegetation. Fitness is maximized by
foraging on plankton between sizes so and 31, foraging
among vegetation between sizes 31 and 32, and returning to

the plankton at size 32.

RATE

RATE

26

(a)‘

uI/s.
112/92

 

 

 

 

FISH SIZE ,3

(b)

    
 
  
   

(ll/g) veg

(N/Q)plankton

 

  

 

Q plankton

S0 S1 82
FISH SIZE .s

27

Other problems can be solved if the options can be depicted in the U-g
plane (either as a continuous function, as first modelled, or as any
other set of points, since "minimize U/g" is the general rule for
juveniles). For example, if u and g can be depicted as a function of
movement rate, distance from a refuge, or duration of a crepuscular
foraging bout, the Options can be plotted in the u-g plane. The
optimal solution can then be found graphically by increasing the slope
of a line through the origin until it meets an option.

When would discrete habitat shifts be predicted? Graphically,
mixed habitat use might occur when D(S) is a convex function, but use
of only one habitat would be expected if u(g) is a linear or concave
function. This is illustrated in Figure 7. In the original statement
of the problem, the concavity of u(g) was taken to be a result of
growth inefficiencies, i.e. that growth rate is a concave function of
foraging rate, at high foraging rates. When growth is strongly
food-limited, p(g) may be approximately linear. Thus, we might expect
discrete habitat shifts during periods in the life history in which
growth is strongly food-limited.

Other factors which would favor habitat specialization include a
cost (in energy or mortality) of transit between habitats and reduced
foraging efficiency, due to short-term learning effects, when two
habitats are used. Both of these effects are illustrated in Figure 8.
The displacement of g downward for a mixed strategy represents a
travel cost, and the "sag" in g represents a foraging rate depression
from short-term learning effects, i.e. over some range, the animal's
foraging rate in a habitat is an increasing function of the time spent

there during that day. Mathematically, U is no longer a function of

28

Figure 7. (a) Illustration of habitat specialization when u is
linear in g. The "corners" each represent exclusive use of
a habitat. (b) choice of an intermediate growth rate (use

of both habitats) when u is convex.

 

 

 

 

29

 

 

 

 

 

Figure 8.

30

An example of added complexity in theIJ-g plane: the
effect of travel costs and short-term learning on mortality
and growth. Both factors favor habitat specialization.

(a) Energetic travel costs are depicted as displacing the
growth rate downward. Short-term learning effects are
represented by a "sag" in the growth rate, depicting a case
in which an animal's foraging rate while in a habitat is a
function of the time Spent there. (b) The dashed line
represents no travel or learning cost. The travel cost
"detaches" the line and moves it to the left. The learning
cost makes it bend. Both effects make a mixed strategy

unlikely.

 

 

 

 

31

 

3v

_. ._.<._._m<I 2.
m2; “.0 ZO_._.mOn_OmQ

 

 

 

 

 

BLVH

32

g, but the graph in the u-g plane is sufficient.

Having seen that the optimal behavior is the behavior which
maximizes survivorship to each size, and that an animal does this by
minimizing u/g (or maximizing g/u) at each size and age, it is worth
noting what the strategy does not do. First, it usually does not
maximize the probability of survivorship to each age, £(x). For
example, if au/as a O and 3u/3g > O for all sizes, an animal would
maximize its survivorship (to each age) by not growing at all.

Second, it is not the strategy which "projects" the maximal biomass of
the cohort into the next day. This strategy can be approximated for
small u and g as maximizing y - (l-u)(s+g). Setting ay/ag - 0 yields
Bu/Bg a (1-u)/(s+g). This strategy would result in acceptance of a
higher marginal risk (higher Bu/ag) by small animals and lower
marginal risk by large animals than the optimal strategy. Third, it
is not the same as maximizing the expected gain in cohort biomass
during a day. This can be approximated by maximizing y = (l-p)(g),
and setting ay/ag = 0 yields Bp/ag = (l-p)/g. Since 1-11> g for small
u, this strategy would result in acceptance of a higher marginal risk

by all sizes.

Optimal Decisions for Adults

 

For adults, the condition is Eq. 9. For small values of e‘Db/AD,
the condition is very close to the condition for juveniles; as
e’Db/AD increases, the condition diverges from that for juveniles.
What is the meaning of the term e‘Db/XD? The expression e'Db is
the rate of offspring produced by a cohort of age x. It turns out
(see below) that AD can be interpreted as the negative of the expected

number of future offspring produced by the cohort. Thus, the term

33

e'Db/AD is a measure of the ratio of present and future reproduction.
To see that AD is the expected number of future offspring
produced by a cohort, first recall that AD(T) - 0. From Eq. 7,

A - A
D(T) D(O) +'£T e‘D(X)b(x)dx. Since this integral is just R0 = l .

AD(O) - -1 (- -Ro). Thus AD(x) begins at -1, is constant for
juveniles, and upon onset of reproduction approaches zero as age
progresses. At age x, AD(x) - AD(O) + 5x e'D(Y) b(y)dy -

- gCZ-D(y) b(y) dx + ofx e'D(y) b(y)dy, where y is a dummy variable.

Thus,
mm = - xf"°e=-D<>'> b(y)dy (12)

This shows that AD(x) is the negative of the cohort's expected
reproduction.

Defining R(x) as the cohort's expected future reproduction,
R(x) = - AD(x), so Eq. 9 can be rewritten as

_33 a U-e’Db/R

as g (13)

Alternatively, define V(x) as the reproductive value of an animal of
age x. Then V(x) - R(x)/2(x) - R(x)/e‘D(x) - - AD(x)/e'D(x), and

Eq. 9 can be rewritten as:

filL..E_Z_llfl (14)
38 8

Equations 13 and 14 emphasize that the solution for adults is
similar to that for juveniles, but the acceptable marginal risk is
"discounted" by the ratio of present to future reproduction. A
graphical relationship is represented in Figure 9.

For juveniles, the optimal growth rate at each size depends only

34

Figure 9. Optimal growth rate for adults. The optimal marginal risk
(Bu/33) for an adult is less than that for a juvenile with
the same u(g) for a given size. Starting with the juvenile
solution, au/Bg - u/g, the adult relationship can be viewed
as (a) reducing Bu/ag until Bulag - (u-b/V)/g, or (b)
drawing a new function, (u-b/V)/g, and finding the tangent
through the origin, at which 3(u-b/V)/3g - (u-b/V)/g. (a)
and (b) are equivalent because 3(u-b/V)/3g a Bu/Bg, since
the derivative is evaluated with the state and auxiliary

variables, and hence b/V, held constant.

 

35

p-b/v

 

 

 

 

g
V.
I
.mu n3 __
u. g V \I
_ / u m
._ b __ b g
“.9 _ u. g _ .0
Lo
nu. AU AU .mw
I I
/ /
- -u/- - - - - - - -ur- - - -. - - -
I /
I //
/
11/7 /
III /
/.I /
[I %
¢ .1
I.
I II II II II II II I II II II I
I.
/ l/
/ 1,1,1. /
// Ill
// III
/

 

36

upon the growth and mortality rates extant at that size, but for
adults, AD(x) (or R(x) or V(x)) must also be determined. The value of
AD(x) depends upon the growth, mortality, and birth rates at all other
adult sizes. Since much more information is required to depict
adults' optimal behavior, quantification of optimal growth rates in
experiments will often be more difficult for adults than for

juveniles.

Increasing And Declining Populations
In an increasing or declining population, the task is to find the

behavior which maximizes r in the equation,

1 '3 {)Te-I‘X £(
x)b(x)dx (15)

This problem is mathematically closely related to the
density-dependent case, though conceptually the problems differ. In
the density-dependent case, operationally, r was set equal to zero,
the strategy which maximized the integral in Eq. 15 was determined,
and it was assumed that density-dependent processes must adjust R(x)
and b(x) so as to make Eq. 15 true (i.e. R0 - 1). In the problem of
increasing or declining populations, the strategy which maximizes the
integral is found, where the optimal strategy is specified in part by
r, and then r is adjusted so as to make Eq. 15 true (see Leon 1976,
Goodman 1982).

Mathematically the problem is to find the growth path, g(x),

which maximizes

T
J I of e-rx e"D(x)b(x)dx (l6)

37

subject to the constraints on the state variables, Eqs. 2 and 3. The
problem can be substantially simplified by introducting a new state
variable, z(x), defined as z(x) - rx + D(x). Then Eq. 16 can be
written as

T
J: {, e-z(x)b(x)dx (17)

Noting that dz/dx = r + u, the Hamiltonian and associated equations

are
H = e-zb + Az(r+u) + Asg (18)
%3= 2%4‘As (19)
-E;; s —«%% = e-zb (20)

The introduction of z(x) has allowed the problem to be written in an
autonomous form rather than the nonautonomous form in Eq. 16. In
fact, as before, the Hamiltonian is constant and equal to zero.
Solving Eq. 19 for Bu/Bg yields 3u/3g - - As/Az, and solving Eq. 18

for As and substituting into the above expression for Bu/Bg yields

a“ (u + r) + e'2 b/Az
15E... 3 (22)

 

The behavior of A2 is analogous to the behavior of 8D in the
equilibrial density-dependent case. For the value of r which results
in J - 1, 12(0) - -l, dz/dx - O for juveniles, Xz approaches zero as x

becomes large, and *2 (T) = 0. Also, the reproductive value, V,

38

equals -(Xz/e-z); the term e-zb/Az in Eq. 22 can be written as b/V.
Thus, this "density-independent" case differs from the equilibrial
density-dependent case only in that r # O in Eq. 22.

For juveniles, the rule thus becomes "minimize the quantity

(11+ r)/g at each age (size),' where r is the rate of increase of the
optimal strategy. When r > 0, the minimization of (u + r)/g becomes
more nearly equivalent to the maximization of g; i.e. an animal should
accept greater mortality risk and decrease its time to maturity (and
fecundity at each adult age) relative to the density-dependent,
equilibrial case, (r - 0). When r < 0 , the opposite is true; the
animal should reduce its daily mortality rate with the effect of
slowing its growth rate and lengthening its age at maturity. Thus,
compared to juveniles in a density-dependent population near
equilibrium which maximize fitness by maximizing survivorship to each
size, animals in an increasing population should tend towards being

"bold" "growth maximizers," and animals in declining populations

should tend towards being "timid" "mortality minimizers."

Conclusion

The ways in which animals respond to predators have been of
interest to animal behaviorists and ecologists for some time, but the
development of a testable predictive basis has lagged behind the
recognition of the question's evolutionary and ecological importance.
Certainly, the major impediment to the development of such a construct
has been that the tradeoff between growth and mortality involve
benefits and costs expressed in different units. The simplicity of

the central result for juveniles, "minimize u/g, is encouraging given

the initial complexity of the problem. It is also encouraging that

39

this simple rule is "central" with respect to temporary disturbances
from equilibrium; in increasing populations, natural selection will
favor greater risk-taking, and in declining populations, it will favor
less. Of course, the particular form of perturbations may influence
long-term strategies (e.g. Turelli and Petry, 1980), but the rule
appears to be a good starting point for density-dependent
size-structured populations near equilibrium.

In addition to quantitative predictions of habitat use, the model
suggests some qualitative results which are perhaps not obvious a
priori. First, as developed earlier, rather discrete habitat shifts
are expected at sizes at which growth is strongly food-limited, but
not at which growth is not strongly food—limited in at least one
habitat. Second, if "background" mortality rates in two habitats are
increased by a constant, an optimizing animal may shift from the safer
to the more dangerous habitat. Graphically, this can be seen in Fig.
7(a) by uniformly raising or lowering u(g). However, if u is
multiplied by a constant, the solution is unchanged. Third, given the
same u(g) function for a juvenile and an adult, the adult should adopt
a lower marginal risk (an/3g) than juveniles. However, this does not
necessarily mean that the adult would, for example, forage for a
shorter time, since it must produce offspring ("maintain" the.
prescribed b (8)). A model incorporating reproductive effort could
provide more complete predictions of adult foraging behavior relative
to juveniles (see below).

Some previous work provides a basis for describing optimal
behavior in populations which are not size-structured. Pulliam, Pyke,

and Caraco (1982) have derived and tested a predicted tradeoff between

40

feeding and scanning for predators by flocking birds, under the goal
of minimizing mortality from either predators or starvation. Katz
(1974) described annual patterns of foraging rates which would
minimize the total time spent foraging during the year, and listed
some alternative criteria. Craig, DeAngelis, and Dixon (1979)
presented a similar model with the objective of minimizing total time
spent handling prey. Pearson (1976) developed a model of optimal prey
selection under the objective of maximizing the energy gain per
mortality cost during a foraging bout; this criterion maximizes the
total energy gained over the lifetime of an animal with no age— or
size—dependent properties, and is reminiscent of the minimization of
u/g in juveniles near equilibrium. Hopefully, the present paper,
together with the above work, will provide starting points which
approximate the tradeoff in many species.

When discussing the problem of habitat selection under mortality
risk, a question often raised is whether animals are capable of making
the sort of decision suggested by the theory developed here. The
question can be divided into two parts. First, can an animal make an
appropriate decision, given all necessary information (mortality and
growth rates of options)? Second, can an animal obtain the necessary
information in the first place?

To answer the first question, experimentation is necessary. Some
tests with centrachid fish are in progress. .A priori, given that the
fish have accurate information on alternatives, I see no strong
impediment to the evolution of appropriate behaviors. The behavior of

many animals is obviously influenced by "fear" and "hunger,' and these

motivations must be integrated in some way to effect a behavior. I

41

would expect natural selection to select for particular integrative
processes which maximize fitness. The theory presented here is a
beginning in the description of the particular integrative process
which does maximize fitness.

The second question, that of obtaining information on which to
make decisions, certainly can present major difficulties to animals.
To respond facultatively, an animal must assess at least relative
mortality risk and foraging rates associated with different options.
It does appear that many animals can assess and compare different
foraging rates, and behave in such a way as to approach maximization
of their foraging rate (e.g. Krebs 1978; Werner, Mittelbach, Gilliam
and Hall 1983), but the assessment of mortality risk probably presents
a more difficult "sampling" problem. Especially when predators are
seldomly encountered, information on mortality risk may be both rare
and highly variable. When information is impossible or expensive to
gather and process, and/or the benefit of facultative responses is
small due to constant conditions, rather inflexible behaviors may
evolve. When information can be gained at small cost, and/or
conditions change often, evolution of more plastic behavior may occur,
as has now been documented for several species.

The bluegill may exhibit both types of behavior at different
points in its ontogeny. As outlined earlier, upon hatching from eggs
laid in nests in the shallow littoral zone, bluegill fry swim to the
limnetic zone of lakes. At about 10-20 mm standard length, they
return to the littoral zone. Since visual sensory abilities are
limited at these very small sizes, the fry may not assess the presence

of predators at all, and the timing may be inflexibly set by natural

42

selection. However, at larger sizes (ca. 35 mm, one year old fish in
the experiment by Werner 35 El, 1982b), bluegill sunfish are known to
assess predation risk and respond facultatively. Direct underwater
observations suggest to me that in lakes and ponds with somewhat clear
water, fish routinely encounter predators on a daily basis, and likely
form some sort of estimate of danger in various habitats. In lakes
with less opportunity for visual assessment, habitat selection may be
less flexible, and might depend in part upon an evolved affinity for
physical structure. The affinity, which might decline with fish size,
could be integrated with information on foraging rates to effect a
habitat choice.

In the model presented in this paper, it has been assumed that
u(g,s) is time-invariant and that fecundity, b(s), is continuous and a
function of size. This has allowed an interpretable analytical
solution from which the impact of alternative assumptions can be
explored. For example, seasonality of growth rates and/or mortality
rates for a given behavior can be summarized by making n a function of
age as well as growth and size. Seasonality in birth rate can
similarly be introduced. If particular forms of the seasonality are
assumed, specific cases are solvable in principle. One might expect,
for example, that animals would accept a higher marginal risk at times
immediately preceeding a predictable decline in resource levels if an
increased size during the decline confers some benefit; e.g. if the
mortality rate in winter is negatively size-dependent, an animal might
accept a higher marginal risk in the preceeding summer. Similarly, if
resources are expected to increase in the near future, allowing a

lower mortality rate for a given growth rate, an animal may "wait out

43

the storm" by temporarily accepting a low marginal risk (or even
shrinking). Reproductive effort questions could also be explored by
both making g and b control variables, and compactly describing the
relationships between growth, fecundity, size, mortality, and perhaps
age in a single function, g(g, b, s, x).

Hopefully, the beginning presented here will prove useful in
exploring community-level interactions between size-structured
populations. The development of Optimal foraging theory in the 1970's
has provided what appears to be a powerful tool in the prediction of
resource utilization by animals unconstrained by mortality risk. If
the theory developed here and modifications of it prove useful in
predicting habitat use, the essential tools will be available to
predict both habitat use and prey utilization within a habitat by
species in a community. This would provide a more powerful
alternative to exhaustive experimentation and "intuition" in exploring
the impacts of species introductions (or removals) and resource
changes in a community, and could provide predictions of evolutionary

directions in extant or perturbed communities.

LIST OF REFERENCES

LIST OF REFERENCES

Caraco, T., S. Martindale, and H. R. Pulliam. 1980. Avian flocking
in the presence of a predator. Nature 285:400-401.

Carlander, K. D. 1977. Handbook of freshwater fishery biology. Vol.
2. The Iowa State University Press, Ames.

'Charlesworth, B. 1980. Evolution in age-structured populations.
Cambridge University Press, Cambridge.

Clark, C. W. 1976. Mathematical bioeconomics: the optimal
management of renewable resources. John Wiley & Sons, New York.

Craig, R. B., D. L. DeAngelis, and K. R. Dixon. 1979. Long- and
short-term dynamic optimization models with application to the
feeding strategy of the loggerhead shrike. Am. Nat. 113:31-51.

Fraser, D. F. and R. D. Cerri. 1982. Experimental evaluation of
predator-prey relationships in a patchy environment:
consequences for habitat use patterns in minnows. Ecology
_§§:307-313.

Goh, B.-S. 1980. Developments in agricultural and managed-forest
ecology. Vol. 8. Elsevier Scientific, Amsterdam.

Goodman, D. 1982. Optimal life histories, optimal notation, and the
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Katz, P. L. 1974. A long-term approach to foraging optimization.
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Krebs, J. R. 1978. Optimal foraging: decision rules for predators.
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Leon, J. A. 1976. Life histories as adaptive strategies. J. Theor.

McCleery, R. H. 1978. Optimal behaviour sequences and decision
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44

45

Milinski, M. and R. Heller. 1978. Influence of a predator on the
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Mirmirani, M. and G. Oster. 1978. Comptition, kin selection, and
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Oster, G. F. and E. O. Wilson. 1978. Caste and ecology in the social
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46

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University.

CHAPTER 2

Recruitment of a piscivore through a competitive bottleneck:

Competitive effects of bluegill sunfish on juvenile largemouth bass

INTRODUCTION

Fish, as many other animals, usually exhibit dramatic niche
shifts during their ontogeny. These shifts usually involve changes in
diet and often habitat use, and results in an individual's
encountering a series of different competitors and predators as it
grows. This complexity renders suspect any blanket classification of
a species as being another's predator, prey, or competitor since the
interaction may be very age- or size-specific.

In this paper, we investigate a case in which a species, the
largemouth bass, competes with at least some size-classes of another
species, the bluegill sunfish, early in its life, and then preys upon
the same species at larger sizes. Werner (1977) and Keast (1977) have
noted that many fish species which eventually become piscivores pass
through stages during which they consume mainly zooplankton and/or
insects, as commonly do all sizes of species on which the piscivorous
individuals prey. This suggests the possibility of a competitive
"bottleneck" in which an increase in the density of the "prey" species
may decrease the growth and survivorship of small individuals but

increase the growth and survivorship of large individuals in the

47

48

"piscivorous" population, therefore altering the population's age and
size structure. This effect would be enhanced if the "piscivorous"
species' morphology rendered it a poor competitor on small
invertebrate prey. The possibility of the complete exclusion of bass
or other "piscivores" from a community due to a competitive bottleneck
is raised, similarly to the size-specific developmental bottlenecks
between crustacean competitors described by Neill (1975). While the
exclusion of a species is a very dramatic example of a competitive
bottleneck, more subtle effects on a species' age and size structure
may commonly occur, affecting not only the "predator's“ population
structure but its impact on other members of the community. The
survival of small fish is often thought to strongly influence the
strength of a year-class of fish (e.g. Cushing 1974, Kramer and Smith
1962), and reductions in the growth rate of small fish might strongly
affect early survival (next section).

The interaction we investigate is between the largemouth bass

(Micropterous salmoides) and bluegill sunfish (Lepomis macrochirus),

 

 

two dominant fish in lakes through much of Eastern North America.
There is a huge literature on the life histories and especially diets
of both of these species (reviewed in Carlander 1977), but nearly all
of the studies deal with a single species or do not provide
information on the nature of size-specific interactions (but see
notable exceptions below). The bass-bluegill system has usually been
viewed as a predator-prey interaction, but at small sizes largemouth
bass feed upon invertebrates, as, typically, do all sizes of
bluegills. We know of no detailed work on the simultaneous diets of

specific sizes of small bass and bluegills other than the work by

49

Mullan and Applegate (below) and the work reported in this paper.
However, single-species studies indicated the potential for extensive
overlap in resource use.

Life histories of course vary somewhat betwen lakes, but the
basic pattern is as follows. Largemouth bass spawn at a water
temperature of about 16°C and the male tends the nest and fry until
the fry reach a length of about 10-25 mm Standard Length (SL, the
length from the anterior tip of the mandible to the end of the
vertebral column; essentially, the length excluding the caudal fin).
The bass usually switch to progressively larger prey as they grow,
first consuming small zooplankters (e.g. rotifers, Bosmina, copepod

nauplii), then larger zooplankters (e.g. Daphnia, Sida, copepods)

 

initially at 8-12 mm, then sometimes insects or amphipods at initially
10-30 mm, and finally switch to fish prey at 25-90 mm. In Michigan,
bass are commonly about 100 mm long at one year of age and are largely
piscivorous. Bluegills first spawn at a water temperature of about
20°C, which in Michigan commonly occurs about three or four weeks
after the spawning of the bass. Unlike the bass, bluegills commonly
continue to spawn for several weeks after the initial activity.
Bluegill fry are not tended by the male after leaving the nest and
apparently travel to the limnetic regions of lakes and feed upon
zooplankton until they reach 8-20 mm SL, at which time they usually
return to the littoral zone. They then consume zooplankton, insects,
and other invertebrates thereafter, switching habitats on a seasonal
and size-specific basis (Mittelbach 1981). In Southwest Michigan,
bluegills are commonly about 30 mm SL at age I, 60 mm at age II, and

80 mm at age III.

50

In addition to the diet similarities noted above, there is some
more direct evidence that competition occurs between juvenile
largemouth bass and other fish. Applegate and Mullan (1966) and
Mullan and Applegate (1968) documented the diets of bass and other
fish in two reservoirs and noted that in the newer reservoir bass
between 20 mm and 40 mm SL utilized chironomid larvae as a "bridge”
between diets of zooplankton and fish. In the older reservoir, the
chironomids were essentially absent from the bass diets although
bluegills and other centrarchids consumed midges throughout the year.
The bass grew faster in the new reservoir. Von Geldern (1971) noted
an inverse relation between the density of adult threadfin shad in a
California reservoir and the size of largemouth bass year classes; he
suggested that competition for food or interference with spawning
success may be responsible. Von Geldern and Mitchell (1975) and Fast
.25 El. (1982) showed that the first-year growth rates of largemouth
bass decreased dramatically following the introduction of threadfin
shad into California reservoirs, and that the growth rates of older
bass increased concurrently.

This type of interaction may be common in other fish, although it
remains largely uninvestigated. Crossman (1959) and Larkin 35 El’
(1957) reported some effects of the introduction of the redside shiner
into Paul Lake, British Columbia, which included a decrease in the
growth rate of rainbow trout less than about 120 mm fork length and a
probable increase in the growth rate of rainbow trout over about 200
mm. Svardson (1976) noted that species introduced into Scandinavian
lakes were more likely to substantially impact resident species if the

invader was more planktonic than the resident. One explanation

51

offered by Svardson is that the invader competed with the young of the
resident species. Maly (1976) has suggested that a competitive
bottleneck may occur among c0pepods, and Neill and Peacock (1980) have
shown experimentally that an increase in algal productivity increases

the recruitment of herbivorous Cyclops nauplii to the carnivorous

 

adult population.

This paper is organized as follows. First, we develop a simple
representation of the effect of reduced individual growth rates on the
recruitment of small bass to the piscivorous population. Then the
potential for competition is assessed by four measures: the relative
handling times on invertebrate prey, a survey of habitat use,
measurement of growth rates in two lakes with a comparison to maximal
growth rates when food is abundant, and documentation of diets in two
lakes. Finally, the hypothesis that the growth rate of small bass is
a function of the density of bluegills is tested in a pond in which

small bass are exposed to a gradient of bluegill densities.

GROWTH RATE AS A KEY PARAMETER

What follows is a simple abstraction of the effect of a reduced
growth rate on the survivorship through some size interval, such as
the size at commencement of feeding and the size at commencement of
piscivory. If the growth and mortality rates are taken to be
functions of the animal's size rather than its age, as appears to be
the case for fish at least at early ages (e.g. Ricker 1979), a simple
expression may be derived relating the survivorship through the size
interval following changes in the size-specific growth and mortality

rates.

52

The survivorship of a fish from age x1 to age x2 can be described

by
- -&T2 u(x) dx
L(x1, x2) = e , (l)

where x = age and (x) = instantaneous mortality rate at age x (see
e.g. Hassell, 1978, Appendix I). If the mortality rate is explicitly
a function of size rather than age, we can rewrite u(x) as u[s(x)].
We can then change the variable of integration from age to size to
obtain an expression for survivorship from size s1 to size 32.
Equation 1 can thus be rewritten as

- fx(82) 11[s(x)]dx
le<s1), x<s2)1 = e “(81’ (2)

Changing the variable of integration yields

_ £52 u(s)/(dS/dX) d3
L(Slp 82) = e l (3)

Since ds/dx is the growth rate, g(s), we have

- £82 u(s)/g(s)ds
e 1 <4)

L(Sl, 82)

VanSickle (1977) has derived this equation by a different method.
Intuitively, Eq. 4 represents survivorship across a size interval
because u(s)/g(s) is the instantaneous probability of death at a
particular size, since that probability is the product of D(s) (i.e.
deaths/time) and the inverse of the growth rate (a measure of the time
spent at that size).

Assume some arbitrary "baseline" state at which some values of

u(s) and g(s) occur (e.g. bass in the absence or a given density of

53

bluegills). Now assume some change multiplies the growth rate by a
factor, c (c < l for a growth reduction). Then the new survivorship,
L', can be expressed as

-.fs2 U(s)/cg(s) ds
L' - e 81

- <1/c) {:2 u(8)/8(8)d8
= e

" {:2 u(8)/g(8) NC
= e 1 (5)

Since the expression in brackets is just L(sl, 82), this yields

(omitting the size range for compactness),

L' = L(1/C) (6)

This equation relates simply the survival through the size
interval before and after the reduction in growth rate. For example,
if the growth rate is reduced to one third its former level, the new
survivorship is just the cube of the former survivorship.

The qualitative message from Eq. 6 is this: at parts of the
animal's life history at which survivorship is already low, reductions
in the grOWth rate have a much larger impact on survivorship than at
points at which survivorship is initially high. For example, if c -
1/3 and L - .90, then L' - .73, a reduction of 19%. However, if L =
.01, L' - .000001, a decrease of 99.9902. The effect of the decrease
in growth rate is depicted in Figure 1 for various values of c and L.
Little is known of the size-specific mortality rates of fish during
the first year, but it is clear that the survivorship between first

feeding and piscivory lies near the origin in Figure 1 for bass

54

Figure 1. Percent reduction in survivorship through a size (not an
age) interval due to a reduction in growth rate. L(Sl, 52)
is the survivorship prior to the reduction in growth rate.

c is the value by which the growth is multiplied.

55

a.

 

IOO

 

 

 

0
Au

-

O

4

_cwco>_>5w E :oZozoom Eoocon

L (31- $2)

56

(Zweiacker st 31. 1975) and other fish (Braum 1978).

The impact of an increase in the size-specific mortality rate,
u(s), can also be assessed by the above method. By a derivation
analogous to that shown above, if u(s) is multiplied by a factor k,
Eq. 6 still applies if k/c is substituted for l/c, and Fig. 1 also
applies if the same substitution is made. Large bluegills and other
centrarchids are known to prey upon bass fry (e.g. Mullan and
Applegate, 1967; pers. obs.), so an increase in their density may have
a substantial "direct” effect on bass survivorship through a size
interval in addition to the "indirect" effect through the depression
of growth rate.

This representation is of course a vastly simplified
representation of the complex biological processes occurring during
the early life history of fish. Eipper (1975) reviewed many biotic
and physical factors which are thought to affect early survival of
bass and other fish, and there clearly are abiotic factors such as
wave action which can cause large variation in survival. The
deterministic treatment presented above is intended only to provide a
starting point from which to assess the impact of decreased growth
rates within the context of many other factors thought to affect
mortality, and to motivate the investigation of growth rates as a key
factor in an idealized deterministic system. The approach is most
directly applicable to systems in which density-dependent, biotic
processes are strong, but in more stochastic systems it can also
provide a starting point for assessing the effect of reduced growth

rate on the mean value of survivorship.

57

METHODS

Handling times were measured for fish feeding on Simocephalus and

 

Coenagrionid damselfly nymphs. These two prey were chosen as
representative of littoral zone cladocerans and insect nymphs,

respectively. The Simocephalus were cultured in the laboratory and

 

sorted with sieves to select individuals of 1.8-2.2mm length, mean =
2.0mm. The damselflies were collected from ponds at the Kellogg
Biological Station. Two sizes were used: 5.0-7.0mm long, excluding
the anal gills, and 9.0-11.0mm long. These sizes will be called 6mm
and 10mm damselflies, respectively.

Handling times on these prey were measured for bass of lengths of
25, 35, 45, 55, 70, and 138mm SL, and for bluegills of lengths 25, 35,
45, 55, and 80mm SL. The handling times were measured in ZOO-liter
aquaria while a fish foraged for a given prey size among a dense bed
of Chara. The handling times were measured with an event recorder and
were taken to be the time between the initial attack of a prey to the
resumption of searching behavior.

Dry weights of the prey and the fish were calculated from
regressions of the form W = aLb, where the length measurement was in

mm and the weights in mg or gm. The parameters were: Simocephalus

 

(mg), a - .01737, b = 2.17 (Ivanova and Klakowski 1972); coenagrionids
(mg), a = .00124, b - 2.74 (unpublished); bass (gm), a = 0.00000170, b
= 3.2157 (regression for bass from Three Lakes and Lawrence Lake);
bluegills (gm),a = .00000676, b = 3.043 (Lawrence Lake).

The habitat use of the fish was recorded in Lawrence Lake
(described below) on July 25 and 26, 1978. The method was as

described in Werner, gt il’ (1977) and was taken along transect II in

58

Werner st 31. Divers counted all bass and bluegills along a 100m
transect and classified the location of each fish on two axes: the
"position" and the "microhabitat." The "position" is a measure of the
distance from the shore and hence the depth of the water. Position 1
was closest to the shore and was on a shallow (0.5m water depth) bench
without vegetation cover. Positions 2 and 3 were in about 1m of water
depth on the bench and at its point of contact with the beginning of
the sloping bottom, respectively. Positions 4 and 5 were at depths of
2m and 4m, respectively. Except for Position 1, the area was covered

with a dense growth of primarily Scirpus subterminalis. The

 

"microhabitat" measurement refers to the position of the fish in or
above the vegetation. The fish were recorded as being in the lower
vegetation, upper vegetation, or at some height above the vegetation
at 0.25m intervals.

Fish were collected for growth and diet studies by seining in
Lawrence Lake, Barry County, Michigan, and Three Lakes, Kalamazoo
County, Michigan. Lawrence Lake is a mesotrophic lake, 4.9 hectares
surface area, and 12.6m maximum depth. Three Lakes refers to a chain
of three lakes; we collected fish in the middle lake of the chain.

The lake is more productive than Lawrence Lake, is about 10 ha in
surface area, and has a maximum depth of 10m. The vegetation along
the sampling area on the south side of the lake is predominantly
‘Ehara. Fish were collected between 0900 and 1100 h on a weekly basis.
The fish were killed with an overdose of tricaine methanesulfonate and
preserved in 10% formalin. Stomach contents were identified and prey
length measurements taken. Length-weight regressions were used to

convert the measured lengths to weights.

59

The maximal growth rates of bass were estimated by placing bass
fry from a holding pond into a fishless pond at the Kellogg Biological
Station. Approximately five hundred 15.5 to 19mm bass fry were
introduced into a circular pond (29m diameter, 1.8m deep) with
abundant zooplankton and insect nymphs. Bass were sampled weekly for
growth and resources qualitatively monitored for depletion. Some
resource depletion became apparent by July 9, at which time 198 bass
were transferred to another pond. The stomachs of the bass were
distended throughout the experiment. The length (mm)-weight (gm)
power function regression had the parameters, a = .0000007863, b =
3.491.

The instantaneous growth rates were calculated for fish from
Lawrence Lake, Three Lakes, and the maximal growth experiment. The
calculation of the instantaneous rate between day i and day j was
calculated from r = [1n (weight at i)-ln(weight at j)]/(days between
samples). In each case, the growth rate of a fish of the mean length
in the cohort was calculated. Thus, on each date, the mean length of
the cohort was calculated, the weight of a fish of the mean length was
calculated, and that value was entered into the growth rate equation
together with the weight of the mean fish length of the previous
sample. This growth rate was then estimated to have applied to a fish
of the standard length which was at the midpoint of the two standard
lengths used in the calculations.

The bass-bluegill competition experiment was performed in an
experimental pond of the same dimensions as the ponds used in the
maximum growth experiment. To simulate conditions in local lakes, a

ring of cattails was removed from the pond, and the dense growth of

6O

Potamogeton was thinned to approximate densities in local lakes. The

 

pond was then quartered with partitions made of 3mm fish seines, and
bluegills of a mean length of 34.3 mm (sd = 2.80) were introduced in a
gradient of densities: 300, 500, 1000, and 1500 per quadrant. 250
bass (27.5mm, sd = 2.59) were placed into each quadrant. Samples for
growth and diet calculations were taken weekly. Only a cursory
treatment of the experiment is given here; a more complete account is

in preparation.

RESULTS
HANDLING TIMES

On prey ranging from 2mm Simocephalus to 10mm damselflies, a bass

 

of a given standard length took a greater mean time to handle a prey
than did a bluegill of the same length. Figure 2 shows the
regressions for both species and the original data for bass. The data
points corresponding to a (prey length)/(predator length) < 0.1 are

for Simocephalus and represent the mean handling time per prey during

 

a foraging bout; the points corresponding to a length ratio greater
than 0.1 refer to damselflies and are handling times on individual
prey items.

The regressions of handling time (H) on the ratio of prey length
to fish length (PL/FL) were fitted by a nonlinear regression algorithm
(program P3R in the BMDP computer programs series). This allowed the
regression line to better approximate the mean handling time rather
than the mean of the logarithm of handling time, which differed as a
result of the large variances in the data. The fitted equations were
as follows: bass, H = 1.68exp[6.438(PL/FL)], serial correlation

coefficient - 0.20, n - 388; bluegills, H = 0.80exp[7.904(PL/FL)],

61

Figure 2. Handling times on Simocephalus and.damselflies for

 

largemouth bass and bluegills. Data points are the

individual values for the bass.

Handling Time (secs)

100.0 "1

50.0 -‘

10.0“I

5.0 -‘

0.5 "

 

 

62

Bass

Bluegill

I I I I I
0.1 0.2 0.3 0.4 0.5

Prey Length/Fish Length

63

correlation = 0.16, n = 129. Regressions of handling time on the
ratio of prey weight to fish weight (PW/FW) are: bass, H =
l.80exp[178.3(PW/FW)], correlation = 0.38, n = 388; bluegills, H =
O.80exp[516.7(PW/FW)], correlation = 0.26, n = 129, where both the
prey weight and the fish weight are expressed as mg dry weight.

The Optimal prey weight is taken to be the weight which minimizes
the ratio of mean handling time to prey weight for a given fish
weight; the variance in handling times is not considered in this
calculation, although in a more detailed treatment of foraging rates
the variance might alter the optimal value somewhat (Gilliam,
unpublished). The regressions of the form H = (a)exp[b(PW/FW)] have
the convenient property that the value of PW/FW which minimizes H/PW
for a given fish weight occurs at (l/b), or PW = FW/b. Therefore, the
above regressions estimate the optimal prey weight for bass to occur
at FW/l78.3 = 0.0056FW and at 0.0019FW for bluegills.

Figure 3 shows the optimal prey weights for bass and bluegills as
a function of their standard lengths, calculated from the above
relationships. For present purposes, the utility of the figure is to
suggest that a bass of a given length has the same Optimal prey weight
as a bluegill that is about ten percent longer. This is one way to
estimate the potential for the source of the most direct competition
from an array of bluegill sizes present.

It should be noted that in Figure 3, the Optimal prey weight
exceeds the weight of a 10mm damselfly for bass larger than 33mm and
bluegills larger than 36mm. Since invertebrate prey of that size are
usually rare in lakes, both fish species may in practice be limited

largely to invertebrates below the optimal prey size. If this is the

64

Figure 3. Optimal prey weights for bluegills and largemouth bass.

Left curve: bass. Right curve: bluegill.

65

2:5 59.3 :2“.

Dow om om on om om 0v

— _ _ _ _ — —

Bronson. EEoF llllllllllllllllllllll

 

I'lJ

 

(5w) iufiieM Reid newndo

66

case, the opportunities for segregation of bluegill diets by prey size
may be limited for bluegills of about 36mm and above. Thus, Figure 3
suggests that bass of about 33mm and perhaps smaller may compete
directly with most sizes of bluegills present in the littoral. It
should also be noted that the curves are extrapolated for optimal prey
weights above the weight of a 10mm damselfly, 0.68mg. Thus, while the
curves are appropriate for the original purpose of suggesting the
optimal sizes of invertebrate prey likely to be found in lakes, and
hence the fish sizes with similar optimal prey sizes, the curves may
not be appropriate for describing the optimal prey sizes of fish above

about 35mm.

HABITAT USE

Figure 4 describes the habitat use by young-of-year bass and four
size classes of bluegills in Lawrence Lake on the afternoons of July
26 and 27, 1978. The patterns were similar for the two dates and the
data are combined. In the microhabitat dimension, the bass were found
mainly in the upper half of the vegetation (predominantly Scirpus

subterminalis about 0.5mm tall). Young-of-year bluegills (O-25mm size

 

class) were found among and just above the vegetation. The 26-50mm
bluegills were distributed similarly to the bass, and larger bluegill
classes shifted progressively higher in the water column. Across
positions, the bass were distributed across all positions except the
unvegetated portion of the shallow bench. The 26-50mm and 51-75mm
bluegills showed a similar distribution to the bass. These results
are in general agreement with previous surveys (Werner 35.2}! 1977,
Hall and Werner 1977). Qualitative observations in 1979 and 1980

yielded the same general pattern. Mittelbach (1981) found that

Figure 4.

67

Habitat use by largemouth bass and bluegills.
"Microhabitat" represents location in or above the
vegetation. LV - lower vegetation, UV - upper vegetation,
0-.25 = zero to 0.25m above vegetation. "Position"

represents distance from shore.

PERCENT

60"1

40"

20"

 

60-

40"

20-1

 

60‘

40"

20'l

 

60"
40--l

20‘

 

60"
401

20‘

 

1:. E—

YOY BASS

 

 

36
26-50

51-75

86
76-125

LV uv 0‘ >
.25 .25

MICROHABITAT

 

 

 

 

 

1 .2 3 4 5
1

POSITION

69

bluegills larger than 75mm did not forage extensively in the littoral
zone in July; this is also consistent with the present data.

These results show that there is not extensive habitat
segregation betwen bass and 25-75mm bluegills in the littoral zone,
and that the 25-50mm (predominantly one-year-old) bluegills utilize

essentially the same areas as the bass.

GROWTH RATES

Figure 5 depicts the growth curves for cohorts of bass in Three
Lakes, Lawrence Lake, and the "unlimited growth" experiment in ponds.
On all dates, the fish in the ponds were substantially larger than in
the lakes. The sizes are plotted as lengths since fish length is a
more familiar quantity than weight; if the size were expressed as
weight, the difference between the lakes and the ponds would appear
even larger.

While Figure 5 might suggest a substantial depression of growth
rates below their maximum, the data are confounded by a difference in
spawning times in the lakes. By expressing the growth rates as a
function of the fish size rather than time, specific sizes at which
the growth rates are depressed can be identified.

The estimated instantaneous relative growth rates are shown in
Figure 6. The growth curves are somewhat irregular, as expected with
the estimation of growth rates from mean sizes of bass caught at
tabout one-week intervals. However, the data yield useful information
on the extent of growth depression at different sizes.

Three growth curves are shown for Lawrence Lake. The curve
designated LLBOS is a school of bass which we were able to locate

regularly early in 1980. After the school's diSpersal, we were able

70

Figure 5. Mean largemouth bass lengths in a maximal growth rate
experiment (Ponds) and two lakes. 3L198O - Three Lakes in

1980. LL1980 = Lawrence Lake in 1980.

71

now 93. 2:- c3-
NN up n mm mw w mm mp m mm 0F

>92
om

 

 

ommpum

mocoa

low

ION

low

low

low

Iom

ION

 

low

(will) 13 ueew

Figure 6.

72

Instantaneous relative growth rates (day'l) of largemouth
bass in the maximal growth rate experiment (Ponds), Three
Lakes (3L1980), a school in Lawrence Lake (LLSOS), all bass
collected in Lawrence Lake (LL80) and Lawrence Lake in 1979
(LL79). The arrow indicates the point of transferral of

the fish from one pond to another.

73

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_ _ _ _ _
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74

to identify individuals from the school over the next two weeks since
those fish formed a distinct, large size class relative to bass which
were spawned later. The calculated instantaneous relative growth
fluctuate somewhat, but it appears that the cohort grew at about
one-half the rates of the bass in the pond between the lengths of
about 22mm and 42mm. The second curve is designated by LL80. It is
calculated from the lengths of all fish captured on a date in Lawrence
Lake during 1980. At sizes < 28mm, the curve is identical to the
LLSOS curve. The "dip" at 36mm and peak at 41mm are not judged to
reflect real changes in the growth rate since the size-frequency
distribution was bimodal and the changes can be attributed to sampling
error in the proportion of fish near each mode. The overall trend
suggested by LL80 is that growth rates are depressed in Lawrence Lake,
and that the depression occurs across all sizes. The third curve,
LL79, is constructed from bass collected in 1979. Over the size range
collected, growth rates were substantially depressed.

The data for Three Lakes shows a substantially different pattern
from that of Lawrence Lake. At lengths up to about 33mm, the bass
grew at approximately maximal rates. At sizes > 35mm, growth rates
were somewhat depressed. In Three Lakes, therefore, neither inter-
nor intraspecific density-dependent growth appeared to occur at sizes
less than about 33mm.

The growth data therefore suggest a possibility for a competitive
bottleneck affecting bass in Lawrence Lake, but provides evidence
against such an effect in Three Lakes for small sizes, though the

effect might occur at larger sizes.

75

DIETS IN LAWRENCE LAKE AND THREE LAKES

A summary of the diets of young-of-year bass in Lawrence Lake are
shown in Table 1 for each sampling date. As they grew, the fish
switched from a diet of copepods and Bosmina to a diet of littoral

cladocerans (Sida, Latona, Simocephalus) and insects and finally to

 

fish (centrarchid fry). The transition is depicted in Figure 7, in
which the diets of the cohort are plotted against the mean length of
the fish in the cohort on a date. The cladocerans act as a "bridge"
between small copepods and fish prey for bass between about 20mm and
30mm.

The diets of bass in Three Lakes were somewhat similar to those
in Lawrence Lake, as shown in Table 2. However, in Three Lakes,
Baetid mayfly nymphs formed the transition between the copepod and
fish diets. This is seen in Figure 8. Unlike Lawrence Lake, the
mayfly nymphs continued to contribute substantially to the diet after
the cohort reached 30mm in mean length.

In both lakes, the shifts in taxonomic categories coincided with
a shift to progressively larger prey. In Figure 9, the mean prey
weight in the stomach of each fish collected in Lawrence Lake is
plotted as a function of the individual's length. The dots represent
bass without fish prey; the x's represent bass with at least one fish
prey. The mean prey size is clearly strongly related to the size of
the fish, and over the summer the mean prey size spanned nearly five
orders of magnitude. Figure 10 shows the same relationship for Three
Lakes.

The presence Of the mayfly nymphs in the bass in Three Lakes

resulted in a smoother progression to larger prey than did the

76

Table 1. Diets of young-of-year largemouth bass in Lawrence Lake,
1980, expressed as percent dry weight of total diet. To be
included in the table, an item must have comprised at least
five percent of the diet on at least one date. Values
rounded to the nearest percent, and values less than 2
percent omitted. Prey categories are, in order, copepods:

Cyclopoida, Calanoida; cladocerans: Bosmina, Sida, Latona,

 

Simocephalus; Chironomid larvae; Baetidae; centrarchid fry.

 

Date n mean SL Cycl Cala Bosm Sida Lato Simo Chir Baet Cent

 

 

6-16 8 13.9 45 40 7

6-23 9 16.2 15 63 16

6-30 12 23.8 6 43 15 6 10 13

7-8 10 31.1 19 11 3 2 2 61
7-14 12 27.1 5 4 4 81
7-21 10 31.6 3 7 2 86
7-28 10 35.7 3 4 3 85
8-4 10 36.0 4 95
8-12 8 45.9 99
8-26 9 51.7 99

9-26 10 65.1 99

77

Figure 7. Diets of largemouth bass in Lawrence Lake.

78

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lNSOHBd

79

Table 2. Diets of young-of-year largemouth bass in Three Lakes, 1980.
Description as in Table l. Acro - Acroperus (a littoral

cladoceran), Cypr a Cyprinid fish.

Date n mean SL Cycl Sida Simo Acro Baet Cypr Cent

 

 

6-24 5 13.9 99

7-1 11 13.4 38 20 17 8 16
7-10 8 24.3 4 54 39
7-15 22 29.3 19 6 72
7-23 13 37.2 4 95
7-30 10 36.9 27 13 58
8-6 10 41.7 33 17 48
8-13 10 50.7 99
8-27 10 59.7 99

9-26 8 68.9 99

80

Figure 8. Diets of largemouth bass in Three Lakes.

81

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82

Figure 9. Mean dry weight of prey in the stomachs of individual
largemouth bass in Lawrence Lake. The X's represent bass

with at least one fish prey. The E's represent fish with

empty stomachs.

Prey Mean Dry Weight (mg)

10

10

10

'2

'3

 

83

 

Lawrence Lake at
x x
x x
x
x x
x x
x
X xxx x x x x x
x
x x
x X
x
x C
ox
x
x x xx
ox): xxx
0. . J. .
.‘°' . ° _
. . OX
E E
E E E EE EE
I I I I I I l j
10 20 30 4O 50 60 7O 80

Standard Length (mm)

84

Figure 10. Mean dry weight of prey in the stomachs of individual

largemouth bass in Three Lakes. Symbols as in Figure 9.

Prey Mean Dry Weight (mg)

102

10

10‘

10

10'

'2

3

 

85

Three Lakes

X
X X
x x x
XX X
X
Xx X X
Xx X
X§§ x
X xx x xx x
X
X XX
X X X
X
X
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10 2O 30 4O 50 60 7O 80 90
Standard Length (mm)

100

86

cladocerans in Lawrence Lake, as can be seen in a comparison of Figure
9 and Figure 10. Inspecting the mean prey weights of 20 to 30mm bass
in Lawrence Lake, it is seen that the weights fall mainly between 0.01
and 0.05mg. These bass were feeding mainly on Sida and other
cladocerans. The two bass in that size range with mean prey weights
of 0.1 to 0.2mg had Baetid nymphs in their stomachs, along with
littoral cladocerans. In Three Lakes, the mean prey sizes in the
stomachs of 20 to 30mm bass was shifted up by an order of magnitude
relative to Lawrence Lake. These mean weights reflect a diet of
Baetid nymphs, and the mean prey weights of 20 to 30mm bass in Three
Lakes is close to the Optimal prey weight of 0.4mg calculated from the
laboratory experiments. It appears that the presence of Baetids in
the diet of Three Lakes bass may be a major contributor to the
difference in growth rates between the Lawrence and Three Lakes fish.
The bluegill diets were substantially different from the bass in
both lakes, though this could not be anticipated from the handling
time data and the habitat use patterns, both of which suggested
substantial potential for diet overlap. Table 3 shows the diets of
25mm to 60mm bluegills in Lawrence Lake. Throughout the sampling
period, the diets consisted largely of chironomid larvae. However, a
substantial part of the diet consisted 0f.§1§2 and some Latona. Thus,
in Lawrence Lake, it appears that direct competition between bass and
bluegills occurs for the littoral cladoceran and chironomid larvae
resources. The low level of Baetids in the Lawrence Lake bass cannot
be attributed to their depletion by bluegills based on the diets of
the bluegills, since Baetids were hardly consumed over the sampling

period.

87

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88

Habitat surveys (Werner st 31. 1977 and unpublished data)
indicate that the density of 25 to 60mm bluegills is at least ten
times the density of young-of-year largemouth bass in Lawrence Lake.
Absolute consumption rates are not known for the fish, but even if the
25 to 60mm bluegills had only the same consumption rate (mg/day) per
fish as the 25mm bass, the total demand on the §1d3_population by the
bluegills would exceed the demand by the bass, given that §1d3_formed
12-522 of the bluegills' diets in July. This suggests that if the
density of §1d3_was depressed by foraging, the primary source of the

depression was from the bluegill population, and the levels of Sida

 

largely set by the bluegills rather than the bass. Phoenix (1976)
found that fish can strongly depress population levels Of littoral
zone cladocerans.

Table 4 shows the diets of 25 to 60mm bluegills in Three Lakes.
These fish also mainly consumed chironomid larvae. They also consumed
some littoral cladocerans and Baetids. While there was some overlap
with the bass' diets, it was not so large as in Lawrence Lake. Unlike
Lawrence Lake, neither the diet information nor the growth rates

strongly suggest the operation of a competitive bottleneck.

A QUANTIFICATION OF THE EFFECT OF BLUEGILL
DENSITY ON BASS GROWTH RATES
The prey partitioning between the bass and bluegills was
greater than the laboratory experiments and habitat use survey
suggested might occur, and it brings into question whether any
substantial competition is likely to occur. Therefore, the effect of
bluegill density on bass growth was quantified in an experimental

pond, as described in the Methods. Here, the main result of the

89

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"moduowoumo mono Hmcoauavvm mafiaoaaom on»

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90

experiment is presented; the experiment will be treated more fully
elsewhere (Gilliam, Wérner, and Hall, in prep.), but is included here
because of its direct relevance to the material presented so far.

Figure 11 shows the final mean weights of the bass and bluegills
after forty-seven days. The bluegill growth rates were strongly
density-dependent. The bass growth rate was not differentially
affected by bluegill density across the first two densities, then
declined markedly. In the presence of 300 bluegills, surviving bass
each gained an average of 1.0gm dry weight; in the presence of 1500
bluegills, they gained an average of 0.25gm. The initial mean length
was 27.5mm, and the final mean lengths were 56.3mm at the highest
density.

The reduction in growth rate occurred despite a degree of
resource partitioning similar to the lakes. At the lowest density,

the bluegills' diet consisted of chironomid larvae, 78%; Simocephalus,

 

7%; and cyclopoid copepods, 5%. No other prey composed more than 3%
of the diet. The bass' diets consisted of Baetidae, 35%;

Simocephalus, 14%; Ceriodaphnia, 11%; cyclopoid copepods, 10%,

 

calanoid copepods, 10%; and chironomid larvae, 7%. The major source

of overlap was on a littoral zone cladoceran (Simocephalus) and

 

chironomids, as in Lawrence Lake. At the highest density, the
bluegills' diets consisted of chironomid larvae, 65%; cyclopoid

copepods, 8%; Simocephalus, 4%; and gastropods, 4%. The bass' diets

 

were calanoid copepods, 42%; Baetidae, 27%; chironomid larvae, 11%;

and Simocephalus, 7%. Again, the main overlap was on Simocephalus and

 

 

chironomids.

91

Figure 11. Final mean dry weights of bluegills and largemouth bass, as
a function Of the density of bluegills in an experimental

pond.

92

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93

CONCLUSION

The results of the pond experiment demonstrate the depression of
the bass' growth rates by increased bluegill densities, despite
substantial resource partitioning between the species. This increases
our confidence in the existence of a competitive effect of bluegills
on bass in Lawrence Lake, and contrasts with the apparent absence of a
strong depression in Three Lakes. The difference in the bass' growth
rates in Lawrence Lake and Three Lakes corresponds with the existence
of the Mayfly "bridge" in Three Lakes. The growth rates of fish have
previously been found to be closely related to the mean prey size
consumed (Paloheimo and Dickie 1966, Martin 1970, Kerr 1971, wankowski
and Thorpe 1979), so a causal relationship is a reasonable hypothesis.
It is not known to what extent the paucity of Baetids in the bass'
diets in Lawrence Lake is attributable to their depletion by bluegills
or other fish, but their absence does appear to contribute to the
bass' reduced growth rates and forces competition with the bluegill on

the Sida resource.

 

There is agreement between the expected impact of the bottleneck
and the population levels of bluegills and bass in the two lakes.
Censuses of the fish populations of Lawrence Lake and Three Lakes by
Werner £5 31. 1977 showed that the density of bluegills was an average
of 411 per 100m in the littoral zone of Lawrence Lake, but only 326 in
Three Lakes. This coincided with a lower bass density in Lawrence
Lake than in Three Lakes, 54 versus 66 per 100m.

The level of resource partitioning between the species was
greater than anticipated at the beginning of this investigation. The

bluegills' extensive consumption of chironomids is contrasted by the

94

bass' low utilization. Applegate and Mullen's (1967) report of the
bass' use of the chironomids when the resource seemed to be abundant
suggests that chironomids can afford high growth rates to the bass.
Thus the low levels of chironomids in the bass' diets in Lawrence Lake
may signify another source of the bottleneck. Indeed, the chironomids
in the diets of the bluegills usually average between 0.1 and 0.5mg
dry weight, which is similar to the weights of the Baetids utilized by
the bass in Three Lakes. Preliminary laboratory observations have
indicated that bass lack an ability to cleanly extract chironomids
from tubes in sediments, while bluegills can do so. When feeding on
chironomids, bluegills would attack burrowed chironomids by sucking a
precise area of the sediments into its mouth, and then expelling the
sediments while retaining the chironomid. The bass, which has a
larger mouth than the bluegill (see illustrations and descriptions in
e.g. Werner 1977), would attempt to grasp the tube or a partially
exposed chironomid between the jaws, employing little suction; this
behavior was less effective than the bluegill's. The bass also
appeared to position itself less precisely than the bluegill; this
seems to be related to the more gibbose body shape of the bluegill
relative to the fusiform shape of the bass usually associated with
efficient and rapid swimming speeeds rather than maneuverability
(Alexander 1967).

Although the bass' larger mouth and fusiform body shape may
contribute to its poor utilization of chironomids, its morphology
clearly confers an advantage in capturing large or elusive prey, such
as fish. It is not known to what extent this ability may apply to

elusive invertebrates, such as the calanoid copepods consumed in the

95

pond competition experiment, but elucidation of this question may
contribute substantially to our understanding of prey selection by the
bass. Timms (personal communication) has found that bass feed at a
substantially higher rate on Baetids in aquaria than do bluegills, and
that the bass dart at the mayflies and engulf them, while bluegills
attempt to maneuver precisely and employ suction, which is often
unsuccessful.

The foraging experiments on Simocephalus and damselflies also led

 

to an entirely unanticipated behavior which may ameliorate the
competitive effect of the bluegills somewhat. As reported more fully
elsewhere (in preparation), when foraging on these prey in Chara, the
bass would approach an area of the vegetation, lower the mandible, and
then close it rapidly, "blowing" water from its mouth. This would
often dislodge prey from the vegetation and resulted in the bass'
experiencing about twice the effective encounter rate (prey attacked
per second searching but not handling prey), though it also incurred
the reported higher handling cost than the bluegill.

It is not known to what extent this behavior is utilized in the
lakes by the bass. Even with this behavior, the reduction in the Sida
by the greatly numerically dominant bluegills would be expected to
contribute to the observed reduction in the bass' growth rates.
However, the "included niche” nature of the interaction with bluegills
suggests that the bass' contribution to the reduction of S393 levels
would not affect the bluegills as much as the bluegills affect the
bass. This is because §1d3_forms most of the diet of the bass using
that resource, but the bluegills usually utilize Sida_less than or to

about the same extent as chironomids.

96

The reduction of the bass' growth rates in Lawrence Lake and the
pond competition experiment occurred despite the absence of large
bluegills. Mittelbach (1981) has shown that bluegills larger than
about 75mm forage largely on limnetic zooplankton during the summer
months. Mittelbach elucidated some of the implications Of the
presence of limnetic zooplankton in providing a means of resource
partitioning between large and small bluegills, and showed that the
absence of limnetic resources would be likely to result in greater
resource depletion in the littoral zone and stunting of the bluegill
population. This often happens in shallow ponds without distinct
littoral and limnetic areas. If this occurred, the small bass would
also be expected to be impacted by the same processes of resource
depletion and greatly reduced growth rates experienced by the
bluegills.

The size-specific reductions in growth rates have substantial
implications for the size structure of the population of bass and
other fish experiencing size-specific interactions. Since
depression of a cohort's growth rate at a particular size results in
a reduction in survivorship through that size interval, growth rates
at subsequent sizes would increase if the growth rates are
intraspecifically density-dependent; this would happen whether or not
the large sizes prey upon the species causing the growth reduction at
a smaller size. Thus a competitive bottleneck would usually act to
increase the subsequent survivorship and fecundity of animals
surviving the bottleneck. Thus, although the bottleneck acts to
reduce the number of fish in the piscivorous population, it may or may

not reduce the biomass and production of the piscivores. To this

97

extent, the effect of a bottleneck is somewhat analogous to problems
approached in work on Optimal harvesting. The usual goal of
harvesting models is to find the amount of imposed mortality (usually
on the adult population, unlike the bottleneck) which maximizes
production. At low harvest rates, stunted populations of many slowly
growing individuals can result; at high harvest rates, low population
numbers of fast-growing individuals can result. Thus, the effect of
harvesting or a competitive bottleneck changes the demography of the
population, but the effects on total biomass and production depend
upon the severity of the density-dependent processes occurring.

The presence of a bottleneck is also expected to affect the
evolution of an animal's morphology. If the introduction of a
competitor severely lowers a species' survivorship through some size
interval, the species would be expected to evolve to increase its
growth rate through that size range. This could occur either by
character convergence (e.g. become a better planktivore) or by
character divergence (e.g. become an even better piscivore, and switch
to fish at an even smaller size). However, the evolution of
morphology at one size carries with it morphological changes at all
previous and subsequent sizes, since fish do not extensively rework
their morphology ontogenetically, except for a few examples of
metamorphosis. The ways in which fish and other animals
morphologically balance selective pressures at various points in their
ontogeny remains a virtually uninvestigated problem at the
intersection between the disciplines of evolutionary ecology and
development.

Finally, a graphical model is presented here which is motivated

 

98

bv the bass-bluegill interaction. The intent of the model is to begin
to assess the potential population dynamics and equilibria in a system
in which one species preys upon another, but whose young incur
increased mortality as the ”prey" population increases. This is done
by severely abstracting the bass-bluegill and similar systems to
obtain a conceptually manageable, simple graphical representation as a
beginning from which the impact of the reintroduction of meaningful
biological features can be assessed.

The develOpment of the model is presented in the Appendix and the
resulting predator-prey phase plane is shown in Figure 12a, where H =
prey and P = predator. The model ignores size structure except for
the existence of "juvenile" predators whose survivorship is a
decreasing function of H and P. It also ignores time lags in the
system.

In Figure 12a, the prey zero isocline, denoted by H' = 0, is
drawn as in "standard" predator-prey theory (Rosenzweig and MacArthur
1963, Tanner 1975). However, unlike other predator-prey theory, the
predator isocline (P' - 0) forms a “dome." Under the dome, the
predator population increases; outside it declines.

Figure 12a tells us that there are two alternate stable
equilibria to which the populations can move in this abstraction. The
equilibrium on the left leg of the dome represents the “standard"
predator-prey equilibrium. The other (at H = K, P - 0) represents
exclusion of the predator by the prey. Thus, depending upon the
initial population densities, the system could behave like a
"standard" predator-prey relationship or result in the extinction of

the predator. If the prey exist at H = K, the system is

99

Figure 12. Phase-plane representation of a predator and a prey
species, in which the offspring of the predators first
compete with the prey. See Appendix for assumptions. H a
prey, P - predator. The prey and predator isoclines are
represented by H' = 0 and P' - 0, respectively. K a
maximal prey density. (a) The predator-prey phase plane
(b) The relationship of R1 to H and the relationship of L

to H for three densities of P: 0, P1, and P2.

100

(a)

R-
(b)

H

 

 

 

 

P-

 

101

non-invasable by small numbers of predators, but the predator can
invade if H is temporarily reduced to a level between the "window" at
the base of the dome or if it arrives in initially large numbers and
the population trajectories are attracted to the left leg of the dome.
In this context, it is interesting to note that much effort has been
given to trial-and-error studies of initial stocking densities of bass
and bluegill in farm ponds. Implicit in the justification for such
studies must be the assumption that alternate stable states exist, or
that initial densities strongly affect the population trajectories so
as to provide desirable though temporary population structures.

The above model of the bottleneck problem shows that the prey can
exclude the predator if it reduces L to a point at which 1/L > R1.
This necessary condition was clear from initial considerations of the
problem, but the questions of stability of the populations in that
region, multiple equilibria, and the nature of the stabilities at each
equilibria point were not clear. The addition of age and size
structure back into the model sets up time lags and hence "inertia"
into the trajectories of the populations, but the abstraction provides
a starting point from which those complexities can be assessed. As
investigations of size-structured population interactions continue, a
combination of simplified theory and illuminating, sobering field and
laboratory data appears to be essential for substantial progress on

this sometimes almost paralyzingly complex problem.

LIST OF REFERENCES

LIST OF REFERENCES

Alexander, R. McN. 1967. Functional design in fishes. Hutchinson,
London. 160 pp.

Applegate, R. L. and J. W. Mullan. 1966. Food of young largemouth
bass, Micropterus salmoides, in a new and old reservoir. Trans.
Am. F1811. SOC. 29:75-77.

 

Braum, E. 1978. Ecological aspects of the survival of fish eggs,
embryos, and larvae. pp. 102-131. In: S. D. Gerking, ed.
Ecology of Freshwater Fish Production. New.

Carlander, K. D. 1977. Handbook of freshwater fishery biology.
Vol. 2. Life history data on centrarchid fishes of the United
States and Canada. The Iowa State University Press, Ames, Iowa.
431 pp.

»Crossman, E. J. 1959. A predator-prey interaction in freshwater fish.
J. Fish. Res. Board. Can. 16:269-281.

Cushing, D. H. 1974. In possible density-dependence of larval
mortality and adult mortality in fishes. pp. 103-111. In: J.
H. S. Blaxter, ed. The early life history of fish. .__
Spring-Verlag, New YOrk.

Eipper, A. W. 1975. Environmental influences on the mortality of
bass embryos and larvae. pp. 295-305. .In: R. H. Stroud and H.
Clepper (eds.). Black bass biology and management. Sport
Fishing Institute, Washington, D.C.

Fast, A. W., L. W. Bottroff, and R. L. Miller. 1982. Largemouth bass,
Micropterus salmoides, and bluegill, Lepomis macrochirus, growth
rates associated with artificial destratification and threadfin
shad, Dorosoma petenense, introductions at El Capitan Reservoir,
California. Calif. Fish and Game 62:4-20.

 

 

 

Hall, D. J. and E. E. Werner. 1977. Seasonal distribution and
abundance of fishes in the littoral zone of a Michigan lake.
Trans. Am. Fish. SOC. 106:545-5550

Hassell, M. P. 1978. The dynamics of arthropod predator-prey
systems. Princeton University Press, Princeton, N. J. 237 pp.

102

103

Ivanova, M. B. and R. Z. Klakowski. 1972. Respiratory and filtration
rates in Simocephalus vetulus (O. F. Muller) (Cladocera) at
different pH. Pol. Arch. Hydrobiol. 19:303-318.

 

Keast, A. 1977. Mechanisms expanding niche width and minimizing
intraspecific competition in two centrarchid fishes. pp.
333-395. In: M. K. Hecht, W. C. Steere, and B. Wallace (eds.).
Evolutionary Biology, Vol. 10. Plenum Press, New YOrk.

Kerr, S. R. 1971. Prediction of fish growth efficiency in nature.
J. Fish. Res. Board Can. 28:809-814.

Kramer, R. H. and L. L. Smith, Jr. 1962. Formation of year classes
in largemouth bass. Trans. Am. Fish. Soc._91:29-41.

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Maly, E. J. 1976. Resource overlap between co-occurring copepods:
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Mittelbach, G. G. 1981. Foraging efficiency and body size: a study
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£3;1370-1386.

"~ Mullan, J. W. and R. L. Applegate. 1968. Centrarchid food habits

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104

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Southeast Assoc. Game and Fish Comm. 26:530-540.

APPENDIX

APPENDIX

The abstraction results in the isoclines in Figure 12a and was
formulated as follows. The "prey" population level is described only
by their density, H; no age or size structure is assumed. The
"predator" population level is represented by the density, P, of
individuals large enough to consume the prey. The rate of change of H
is a function of H and P. Following standard predator-prey theory, in
Figure 12a the prey zero isocline is drawn as in Rosenzweig and
MacArthur (1963) and is labelled as H' - 0. The rate of change of P
is a function of H and P. P does not change if Ro - 1, and R0 is
taken to be L(H,P) ° R1(H). L(H,P) is the survivorship of the
predator's offspring (hereafter called "juveniles") to predatory size,
and is a decreasing function of H due to a bottleneck effect, and also
a decreasing function of P due to cannibalism (the feature of
cannibalism may be easily removed; see below). R1(H) is the expected
number of juveniles produced by a predator over its lifetime given
that it has reached predatory size, and is an increasing function of
H. It is assumed that the predator receives negligible benefit from
cannibalism and that the prey incur negligible cost from competition
with juveniles. Then P is stationary when L ° R1 - 1, or R1 - l/L.

In Figure 12b, R is drawn as are the values of l/L for three levels of
P. P is stationary at the intersections of R1 and 1/L; when R > 1/L,

P increases, and when R < 1/L, P decreases.

105

106

This allows the construction of the predator isocline. The first
two points of the isocline can be drawn as depicted by the dashed
lines; at very low P, P increases if H is between the two points just
plotted. If H is higher, P decreases because the low survivorship of
the juveniles outweighs the high reproduction of the predators. If H
is lower, the survivorship of the juveniles is high, but the
reproduction of the predators is low since the density of prey is low.
The next points on the predator isocline can be plotted at the value P
and the value of H at which 1/L(P1) - R1. Similarly, at a predator
level of P2, there is a single point.

The predator isocline thus constructed will be called the "dome."
At any point inside the dome, the predator increases; at any point
outside, the predator decreases. If cannibalism is not assumed to
occur, then L depends only upon H, and the dome does not close at the
top; instead, there are two vertical lines positioned at the first two
points drawn on the predator isocline.

There are three equilibrium points in Figure 12a (in addition to
H = 0, P - 0). The first is the intersection of the left leg of the
dome with the prey isocline. The second equilibrium is the
intersection of the right leg with the prey isocline. This
equilibrium is unstable and is reminiscent of the unstable
two-competitor case in Lotka-Volterra competition theory. The third
equilibrium is at H a K, P = 0, which represents the exclusion of the
prey by the predator. Thus in this mixed predator-prey-competitor
system, one equilibrium occurs at which the predator—prey interaction
appears to dominate the dynamics of the system, and two equilibria

occur at which the competitive effects are salient.

107

If the prey have little effect on L, the right leg of the dome
will intersect the H-axis at H > K.and the system will behave
essentially as a predator-prey interaction. The dome may also be
shifted to the left if the predator has alternative prey. Numerous

similar variations are left to the interested reader.

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