SPATIAL HETEROGENEITY AND THE STABILITY
OF A PREDATOR-PREY LINK

A Dissertofloh .
for the Dew” of DH. D.
MICHIGAN STATE UNWERSITY
Philip Haney" Crowley
' ‘ 1975

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SPATIAL HETEROGENEITY AND THE STABILITY

OF A PREDATOR-PREY LINK
presented by
Philip Haney Crowley

has been accepted towards fulfillment
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ABSTRACT

SPATIAL HETEROGENEITY AND THE STABILITY
OF A PREDATOR-PREY LINK

BY

Philip Haney Crowley

Heterogeneity of spatial structure can stabilize a predator-prey
link: Let the stability of an ecosystem or of a predator-prey inter—
action be its ability to absorb perturbations without extinction or
escape of the prey population. Then the presence of saturable refuge
space in the system makes stable, predator-limited prey equilibria
possible by displacing the functional response curve toward higher prey
densities. And if the time scale of prey increase is less than or about
equal to the time scale of the predator's numerical response, then
stability can be operationalized and calculated directly from parameter
values of a mathematical model.

An analysis of the effects of stability of adjusting the model's
parameters shows that reducing the chance of prey extinction and the
chance of escape from predator regulation are often incompatible goals.
For example, increasing the amount of refuge space in a system decreases
the chance of prey extinction but increases the chance of their escape;
similarly, increasing intraspecific interference among predators
decreases the chance of extinction and increases the chance of escape.
These results and those for other parameters can be quantified for

specific environmental conditions and control measures, providing a

Philip Haney Crowley

means of predicting and policing the behavior of spatially-distributed

ecosystems in stochastic environments.

SPATIAL HETEROGENEITY AND THE STABILITY

OF A PREDATOR-PREY LINK

By

Philip Haney Crowley

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Zoology

1975

ACKNOWLEDGMENTS

I thank the members of my Guidance Committee, D. J. Hall,
Chairman, D. L. Beaver, W. E. Cooper, and E. D. Goodman for advice and
encouragement during this project. J. E. Breck and J. A. Van Sickle
contributed several helpfu1 suggestions; D. M. Johnson, supplied some
data and an unpublished graph; and Bodil Burke, who sustained a tender
fingertip, exhibited flawless predatory behavior.

I gratefully acknowledge financial support from NSF-RANN
Grant GI-20 to H. E. Koenig and W. E. Cooper and from a Zoology
Department Teaching Assistantship.

All other essential forms of sustenance came from my wife
Lillie: as typist, bread-winner, mathematical consultant, and counselor,

she is here on every page.

ii

TABLE OF CONTENTS

Page
LIST OF TABLES . . . . . . . . . . . . . . . . . iv
LIST OF FIGURES. . . . . . . . . . . . . . . . . v
LIST OF SYMBOLS. . . . . . . . . . . . . . . . . vii
INTRODUCTION. . . . . . . . . . . . . . . . . . 1
Stability . 2
Spatial Heterogeneity (SH) . . . 3
Responses of Predators to Prey and Predator Densities . . . 5
STRUCTURAL SPATIAL HETEROGENEITY AND STABILITY: DERIVATION
AND ANALYSIS . . . . . . . . . . . . . . . . . 12
The Functional Response in a Universe Containing
Refuge Space . . . . . . . . . . . . . . . 12
Equilibrium Prey Density . . . . . . . . . . . 21
The Escape and Extinction Thresholds . . . . . . . . . 31
Stability of a Predator- -Prey Link . . . . . . . . . . 43
Multiple Refuges. . . . . . . . . . . . . . 47
Interference Among Predators. . . . . . . . . . . . 55
Summary of Theoretical Results . . . . . . . . . . . 60
DISCUSSION . . . . . . . . . . . . . . . . . . 64
Three Predator- -Prey Interactions Revisited . . . . . . . 64
Stabilizing Responses of Predators. . . . . . 74
Application to Biological Control and Game Management . . . 78
LIST OF REFERENCES. . . . . . . . . . . . . . . . 80
APPENDIX A DISC EXPERIMENTS . . . . . . . . . . . . 87

iii

LIST OF TABLES

Table
1. Qualitative effects of increases in each of thirteen
parameters on the stable equilibrium density H, the
extinction and escape thresholds (elastic boundaries)
H and H, the conformable boundaries £75 andf7p, and the
overall chances of extinction and escape.
2. Parameters from the disc experiments.

3. Stability variables from the disc experiments.

iv

Page

62
90

96

Figure

10.

11.

LIST OF FIGURES

The type 2 functional response .

The type 3 functional response .

The type 2 functional reSponse of predators hunting
outside the refuge space in a universe containing

a perfect refuge . .

The type 2 functional response inside and outside
the refuge space .

The type 3 functional response inside and outside
the refuge space .

Equilibrium prey densities with type 2 predators
and with prey population growth rate independent

of prey density in a universe containing refuge Space.

Equilibrium prey densities with type 3 predators
and with prey population growth rate independent

of prey density in a universe containing refuge space.

Equilibrium prey densities with type 2 predators and
logistic population growth of prey in a universe
containing refuge space.

Equilibrium prey densities with type 2 predators and
with prey population growth featuring an Allee effect
in a universe containing refuge space .

The prey density t0pography with type 2 predators and
with prey population growth independent of density
in a universe containing refuge space . .

The prey density t0pography with type 2 predators and
logistic prey population growth in a universe
containing refuge space.

Page

11

14

17

18

25

27

30

33

36

38

Figure

12.

13.

14.

15.

16.

17.

18.

19.

The effect of a prey refuge occupancy quotient (q)
inversely related to prey density on the functional
response of type 2 predators .

The functional response of type 2 predators hunting
outside the refuge space at four different exchange
rates of prey with constant R(H).

The functional response of type 2 predators in a
universe containing three kinds of refuge space.

The prey density topography with interference among
type 2 predators in a universe containing refuge space

Densities of simocephalus in eight wading pools
observed at 4-day intervals

The functional response of ishnura to simocephalus
density in eight wading pools.

The functional response of a lab technician to the
density of sandpaper discs thumbtacked to a bulletin
board (no refuge space).

The functional response of a lab technician to the

density of sandpaper discs thumbtacked to a bulletin
board (with refuge space)

vi

Page

41

51

54

59

68

72

92

94

LIST OF SYMBOLS

Functions and variables are capitalized (except t); parameters are
lower case.

l 93

a*

rate of successful attack (outside), volume)predator—time)_1

rate of successful attack inside, volume(predator-time)"1
rate of encountering predators, volume(predator-time)-1
handling time (outside), predator-time prey"1

handling time inside, predator-time prey"1

interference time, time

conformability, an ordered pair of differences between the
magnitudes of a parameter and each of the boundaries of the
conformable region

conformable region, the interval of parameter magnitudes
within which a stable, predator-limied prey density

equilibrium can exist

Allee coefficient, the prey density above which the per
capita prey increase rate becomes positive, prey volume‘l

radius of the large discs used in the disc experiments, cm
radius of the small discs used in the disc experiments, cm
elasticity, an ordered pair of differences between the
magnitudes of the stable (predator-limited) prey density and
escape and extinction thresholds, prey volume‘

elastic region, the interval of prey densities within which a
stable, predator-limited prey density equilibrium can exist,
prey volume-1

. . . . -1
exchange rate of prey between 1n51de and out51de, time

functional response (outside), prey (predator-time)—1

vii

I’TJ

F9:

III:

I} I( 2!: In

It

functional response inside, prey (predator-time)—1
predators encountered per predator-time, time-l
effective fingertip radius from the disc experiments, cm
dummy variable for prey density, prey volume-1

tap rate in the disc experiments, taps min-l

prey density, prey volume-1

positive prey density at which inside and outside functional
response curves intersect, prey volume‘1

minimum prey density just above which a stable, predator-
1imited prey density equilibrium can exist, prey volume-1

maximum prey density just below which a stable, predator-
limited prey density equilibrium can exist, prey volume-1

prey density inside, prey volume-1

prey density outside, prey volume-l
equilibrium prey density, prey volume-1

stable equilibrium prey density, prey volume—1

extinction threshold, lower unstable prey density
equilibrium, prey volume-1

escape threshold, upper unstable prey density equilibrium,
prey volume"

. -l

refuge capac1ty, prey volume
an index, dimensionless
an index, dimensionless

. . . —l
carrying capac1ty of a universe, prey volume
an index, dimensionless
time scale of the numerical response, time

number of prey populations in the system

viii

”GI

H

r/p

r/p

[m

| <

predator density, predators volume—

number of predators minus one, divided by the universe
volume, predators volume—1

refuge occupancy quotient, a function of the exchange and
per capita increase rates, dimensionless

refuge occupancy quotient (when the exchange rate greatly
exceeds the per capita increase rate), dimensionless

per capita increase rate of prey excluding predation.effects,
a function of prey density, time‘

time scale of prey increase, time
maximum per capita increase rate of prey, time-

minimum r/p just above which a stable, predator- limited
equilibrium prey density can exist, volume (predator- time)1

maximum r/p just above which a stable, predator- limited
equilibrium prey density can exist, volume (predator- time) 1

sigmoid coefficient (outside), prey volume-
sigmoid coefficient inside, prey volume-1

stability topography, a function of prey density,
prey 2volume time-1

time
volume (area) of the universe
universe volume: outside volume, dimensionless

universe volume: inside volume, dimensionless

ix

It may be that, in some way at present unknown, the heterogeneity of
environments in space and time can independently bring about natural
control, but I cannot see a logical basis for this and shall remain
unconvinced unless someone can demonstrate how it works out (or at

least, for a start, how it could do so).
M. E. Solomon, 1957

INTRODUCTION

Densities of predator and prey populations in a flask or on a
tray of oranges fluctuate violently (Gause, 1934; Huffaker, 1958), but
densities of the same populations in natural ecosystems may remain
relatively constant. Such observations have suggested to many ecologists
that trophic complexity is somehow responsible for the stability of
ecological systems, a view currently being challenged by some rather
elegant theory. According to May (1973), Maynard Smith (1974), and
others, biologically complex systems should be lg§§_stable than simpler
ones, often strikingly so. The question has now become, "Where is all
that natural stability coming from?"

Smith (1972) catalogued major stability-related influences on
ecosystems and demonstrated the critical stabilizing role of spatial

heterogeneity (patchiness in space) (also see Holling, 1968). Unfortu-

 

nately, however, little progress has been made in quantifying spatial

 

heterogeneity, though Smith's work and that of Levin and Paine (1974)
appear promising. And even stability itself has yet to be quantified
in an ecologically general and empirically useful way, as attested by
the disparate stability notions of MacArthur (1955), Margalef (1969),
Lewontin (1969), Smith (1972), Holling (1973), and many others.

This paper presents an attempt to operationalize the concepts

of stability and spatial heterogeneity in order to help explain and

 

 

measure the behavior of ecosystems distributed in space. Three concepts
central to this analysis are introduced first: stability, spatial
heterogeneity, and the responses of predators to prey and predator
densities. Next, the effects of one kind of spatial heterogeneity on
the stability of a predator-prey link are analyzed theoretically, and
the theory is extended to two more complex cases. The discussion then
focuses briefly on three case studies from the literature, considers
stabilizing responses of predators, and outlines applications to pest
and game management. Finally, an appendix presents some empirical

support for a theoretical result.

Stability

In this paper, stability is taken to be an inherent property of

 

an ecosystem and of its conStituent populations (i.e., a tendency in

response to perturbation), rather than an "emergent property" at the

 

ecosystem level of organization or the observed behavior of the system

 

through time (i.e., persistence or constancy). It is assumed that
ecosystem-level stability simply integrates the characteristics of the
constituent population-level predator-prey interactions (Smith, 1972;
May, 1973; Maynard Smith, 1974). From this perspective, stability can
initially be analyzed within single predator-prey links, with bio-
logically and spatially complex cases elaborated later. Interspecific
competition and multispecies predator-prey interactions are deferred for
consideration elsewhere.

Following a suggestion by May (1974a) and a similar approach in
Anderson (1974), I distinguish two fundamental kinds of stability:

elasticity and conformability. Elasticity is a measure of the tendency

 

for perturbations of state variables (functions of time reflecting the

 

system's history, such as prey density) to be damped (cf. "Lyapunov

stability" in Rosen, 1970). Conformability is a measure of the tendency

 

for elasticity to be maintained in the face of perturbations of

parameters (coefficients independent of the system's dynamics, such as

 

maximum prey per capita increase rate). Quick but drastic changes of
environment can "shock" the state variables directly, whereas more
subtle but longer-lasting changes of environment will "shift" the
parameters of the system (see Takahashi, 1964). If the elasticity and
the conformability of an ecosystem are Operationally defined and
appropriately quantified, then the response of that system to ”shocks”

and "shifts" can be predicted.

Spatial Heterogeneity (SH)

 

Biological populations almost always assume clumped (under-
dispersed) distributions in physical space in response to one or both
of two kinds of spatial heterogeneity:

1. stochastic SH, in which patchiness arises from environmental or

 

demographic stochasticity, and

2. structural SH, in which patchiness is imposed on the organisms

 

by the deterministic structure of the environment (cf. Levin,
1974).
The rocky intertidal zone along the coast of Washington State studied by
Paine (1966, 1969) and his students nicely exemplifies stochastic SH
(though structural SH is certainly also present): barnacles, mussels,
starfish, and other invertebrates occupy the tidal rocks in transient

patterns that slowly shift in response to competition, predation, and

 

the gouging action of logs and waves. At any given place within the
intertidal, the abundance of each population depends much more strongly
on such stochastic events than on local physical structure. Other
systems featuring predominantly stochastic SH include those in Huffaker
(1958), Huffaker et a1. (1963), Pimentel et a1. (1963), Luckinbill
(1973, 1974), Hardman and Turnbull (1974), and several in Levin and
Paine (1974).

The most familiar example of structural SH must be the

Paramecium-Didinium flask systems of Gause (1934), in which the oatmeal

 

sediment at the flask bottoms protected the paramecium from its
predator. Other structural SH-dominated systems are found in Errington
(1946), Smith (1972), van den Ende (1973), Crombie (1946), Huffaker and
Kennett (1956), and Johnson (1973); the last three of these provide the
case studies considered later in this paper. Structural SH, with
physical structure of paramount biological importance, is the major
focus of this analysis; in particular, the implications of including
refuges in a predator-prey universe are examined in detail.

The term "refuge" is used here in a broad sense to designate
structurally definable regions of space within which predators harvest
prey less effectively (cf. "relative refuge" of Huffaker, 1958 and
Smith, 1972). In turn, this reduced effectiveness of predators must
generate a strong selection pressure for prey to occupy such regions
preferentially (Tullock, 1970; Smith, 1972). In Errington's mink-muskrat
system, for example, muskrats with lodges established in ponds or
marshes are nearly (not completely) invulnerable to mink predators

(Errington, 1946). Similarly, in van den Ende's chemostat, bacteria

 

 

l

attached to the vessel walls largely (but not entirely) escaped the

swimming Tetrahymena population and avoided the outflow; moreover, after

 

only a few days the bacteria in the reactor had adapted to the refuge by
failing to synthesize polysaccharide capsules, facilitating adherence to
solid surfaces (van den Ende, 1973). Demonstrating the stability
effects of such refuges in predator-prey interactions depends on an

understanding of the responses of predators.

Re§ponses to Predators to Prey and Predator Densities

 

”Predator" is another term broadly applied here, denoting at
least carnivores, insect parasitoids, and some herbivores (e.g., seed
eaters). Since this analysis emphasizes instantaneous responses and the
prediction of dynamics from the magnitudes of equilibria, many important
distinctions between parasitoids and classical predators (see Royama,
1971) can be ignored to preserve "perspective" (Huffaker, 1971).

Predators can respond to changes in the density of their prey
by changing their 93p density, a numerical response, or? y changi g the
rate at which they kill prey, a functional response (Solomon, 1949).

The numerical response of predators to prey increase may include
reproduction or aggregation or both, but such numerical changes can be
expressed only after a time lag. Although the (behavioral) lag in
aggregation is probably often negligible (e.g., hawks vs. field mice),
the (physiological-developmental) reproductive lag can be quite long
(e.g., fish vs. zooplankton); the stability prOperties of models that
track predator density without a reproductive lag, such as the
"pathologically simple" Lotka-Volterra equations, are probably only

mathematical artifacts (see May, 1973; Nicholson, 1955). Furthermore,

the relation in nature between prey density and the resulting predator
density is invariably quite complex and lacks a truly general form
(Readshaw, 1973). So both mathematical intractability and biological
complexity impede the development of a sufficiently broad, quantitative
model incorporating the reproductive numerical response.

Many predators either have negligibly small or extremely slow
numerical responses relative to the increase rates of their prey (e.g.,
planktivorous fish, insectivorous birds), or they are limited by other
resources (e.g., some territorial birds and mammals). The stability of
these interactions and others considered below depends on the functional
response (Takahashi, 1964; Solomon, 1964), a phenomenon thoroughly
investigated both theoretically and experimentally by Holling (1959a,
1959b, 1961, 1965, 1966). Holling classifies the functional response
to prey density--the killing rate per predator as a function of prey
density--into three main types: type 1, a linear rise of the functional
response with prey density to a plateau; type 2, a negatively accelerated,
monotonically increasing response, rising asymptotically to a maximum;
and type 3, a sigmoid functional response, featuring initial positive
acceleration and an asymptotic rise to a plateau. Type 1 has never been
convincingly demonstrated empirically and is at best uncommon. The vast
majority of laboratory functional response data resemble type 2 and fit
the theoretical curve in Figure l, particularly data from invertebrate
predators and parasitoids and from vertebrates in the absence of
alternate prey (Holling, 1965). There are also considerable data
resembling type 3 for both vertebrates (e.g., Holling, 1959a) and

invertebrates (e.g., Lawton et a1., 1974), usually but not invariably

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with alternate prey present, yet no simple analytical expression is
available in the literature to fit them. An equation derived for this
purpose from the type 2 equation is illustrated in Figure 2; but since
the new equation does not explicitly incorporate the dynamics of
learning or of any of the other behavioral factors commonly supposed to
cause this sigmoid shape (see Krebs, 1973), it will only be used to
suggest the general stability implications of the type 3 functional
response.

Thus, in striking contrast to the numerical response, the
functional response of predators exhibits little or no time lag and
occurs mainly in one or two characteristic forms. Furthermore, all
parameters of the equation describing the most common form can be
obtained independently of the functional response data themselves.
Whenever the functional response dominates predator-prey dynamics,
these important features can greatly facilitate an a_priori stability
analysis of the interaction.

Predators may also exhibit a functional reSponse to predator
density. Though social facilitation may be significant in some cases,
the predominant non—linear consequence of increasing predator densities
is interference. The theoretical derivation below includes a cursory
analysis of the impact of interference on stability in a spatially
heterogeneous universe.

The developmental reponse of predators (Murdoch, 1971), a largely
unexplored phenomenon of possible limited importance for stability, is

ignored in this paper.

10

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STRUCTURAL SPATIAL HETEROGENEITY AND STABILITY:

DERIVATION AND ANALYSIS

The Functional Response in a Universe
Containing.Refuge Space

 

 

Consider a conceptual universe composed of two types of space:
refuge space ("inside") and non—refuge space ("outside"). Ignore
temporarily the question of how these spaces are distributed--as many
small patches of refuge and non-refuge or as a few larger patches. And
suppose initially that the refuge space is a "perfect refuge," that is,
that (1) predators cannot kill prey inside and (2) prey stay inside if
there is room for them. Now if the perfect refuge has a capacity of h
prey density units, then the functional response of predators in this
universe (but outside the refuge space) must be displaced h units to the
right along the prey density axis from its corresponding values in a
refuge-free universe. For type 2 predators in particular, Figure 3
replaces Figure 1.

Now the two restrictions that made the refuge "perfect" can be
relaxed: Distinguish "3?, the rate of successful attack for predators
hunting inside, from ”a", here restricted to the rate of successful
attack for predators hunting outside, such that Ofgfa; and let "q"
represent the refuge occupancy quotient, the fraction of prey occupying
the refuge space when H<<h, with the restriction that Q3931. (Thus for

a perfect refuge, a_= O and q = 1.) Then in the general case, with

12

 

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15

O < a_< a and O < q < 1, functional responses of predators hunting
inside and outside the refuge space may differ strikingly (see

Figures 4 and 5): Predators hunting inside encounter prey at densities
reflecting the prey occupancy quotient q and the volume ratio X2 but

at some critical density of prey in the universe, H, the refuge space is
just saturated and cannot admit additional prey; equating the functional
response when H §_H with the functional response with H :_H and solving
for prey density shows that H equals h/q for both predator types. So
the functional response curves of predators inside rise from the origin--
in the typical hyperbola for type 2 or in the sigmoid-shape for type 3--
bending sharply to a plateau at prey density h/q. Predators hunting
outside exhibit a functional response similar in shape to the inside
curve at prey densities below h/q; the prey density outside depends
directly on v and on l-q, the fraction of prey occupying pppfrefuge
space at low densities. Since the initial curves reach the original
displaced curves (e.g., Figure 3) at h/q--the saturation density, above
which the refuge space is indistinguishable from a perfect refuge--the
displaced curves specify the functional response at prey densities
greater than h/q. In sum, predators inside encounter prey at a density
of qu when 0 :_H f_h/q and at KP when H :_h/q; predators outside
encounter prey at a density of v(l-q)H when 0 :_H :_h/q and at v(H-h)
when H :_h/q. And for both type 2 and type 3 predators, the functional
response curve has a slope discontinuity at h/q, above which it remains
constant for predators inside and rises to an asymptote at 1/b for

predators outside.

16

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ouen onu on :e: .ooemm owsmon ogu opnmpzo pce opnmcn omcommon Heconpocsm m omxp one

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17

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18

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.m ensene

 

19

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menmen

 

 

 

 

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20

The outside curves of Figures 4 and 5 exemplify what Holling
(1959b) has called the type 4 functional response--curves displaced
along the prey density axis by a ”threshold of security." Unlike the
other types of functional response curves, however, these are dominated
by the influence of environmental structure rather than by intrinsic
biological characteristics of the predators and prey; therefore it seems
preferable to view them as modified type 2 and 3 curves rather than as
an additional distinct type.

Notice in Figure 4 at zero prey density that since the slope of
the inside F-curve (azq) exceeds the slope of the outside F-curve
(av(l-q)), the two curves must intersect. The intersection prey density
:Q can be found by equating the two functional response expressions (4)

for H :_h/q and solving for H:

 

3V
_ ll —
=§ - h“av+aa(pfb)xh)+1)
(6)
~ 1165+ 1) when p~b.

Figure 4 suggests that if apq > av(l-q), predators should hunt inside the

refuge space whenever H < H and outside whenever H > §:to maximize their
kill rates. This conclusion implies that the magnitudes of a, a) q, and
other parameters may be viewed as evolutionary "strategies" with
predictable trajectories over evolutionary time in a universe of known
structure. For example, it can be shown that prey facing predators with
a strong tendency to aggregate in regions of high prey density should
reduce their refuge occupancy quotient q when H <:§ and increase q when

H >.H. But evolutionary arguments are peripheral to the main thrust of

21

this paper: exploring the stability implications of spatial heterogeneity

on an ecological time scale.

 

Appendix A presents.a preliminary experimental validation of the
new functional response curves in Figure 4. By modifying and rerunning
the disc experiments of Holling (1959b) with and without refuge space, I
obtained good agreement between data and independently-parameterized
theoretical curves. The appendix also tabulates stability variables
of the observed outside functional response curve that were estimated
following the analysis in later sections of the text.

Much of the remainder of this analysis emphasizes the outside
curves of Figures 4 and 5: hunting within the refuge space is ignored.
This approach lends tractability to the algebra and approximates many
actual predator-prey interactions featuring predator—exclusive refuges.
Moreover, in a preliminary investigation, the non-exclusive refuge case
yielded similar results unless a large proportion of the predator

population hunted inside the refuge space.

Equilibrium.Prey Density

 

If both the functional response of predators (F) and the per
capita increase rate of the prey population excluding predation effects
(R) are known functions of prey density (H), and if the numerical
response by predators is negligible, then equilibrium prey densities

(H) can be calculated by solving equation (7) for H.

R(H)H = PF (H) (7)

22

where R(H) is per capita increase rate of prey excluding
predation effects, time‘l,
H is equilibrium prey density, prey volume‘l,
and p is predator density, predators volume-1.
Equation (7) states the equilibrium condition: the rate at which prey
population density increases from net reproduction (RH) equals the rate
at which density decreases from predation (pF).

Now consider the simple case in which both R and p are relatively
independent of prey density over the density range of interest; that is,
set R = r, a constant. From equations (4) and (7), assuming that
predators are type 2 and that all are hunting outside the refuge space,

rH = pF = av(l—q)pH when 0 :_H :_

l+abv(1-q)H ’ (8)

 

43“?

which has at most two roots within the interval [0, h/q),

 

 

= av(l-q)p -r
abv(l-q)r
(9)
and H = 0,
where r is the constant per capita increase rate of prey excluding
predation effects, time—1.
_ _ av (H-h)p 11
rH - pF - l+abv(H—h) when H i-q , (10)

having two or fewer roots equal to or greater than h/q,

 

H _ abvhr + app - r i_/r2(1 - abvh)2 + avp(an - 2r(1 + abvh))
' _ 2abvr

(11)

23

Now for type 3 predators, by the same argument, equations (5)
and (7) determine a maximum of three roots in the interval [0, h/q),

av(l—q)p-r.1/(r-av(l-q)p)2-4sabv(1-q)r2

H = 2abv(l—q)r

 

(12)

and H = 0, when 0 :_H f_2—.

And (abvr)H3 + (r-2abvrh-avp)H2 + (sr-rh+abvrh2 + 2avhp)H - avth = O

h (13)
when H :_a-;

this equation has at most two positive, real roots greater than or equal

to h/q.

Equation (7) also implies that prey density equilibria must be
the prey densities at intersection points of the functional response
curve with the graph of the "prey increase function," RH/p. Figures 6
and 7 illustrate this graphical technique with R and p independent of
prey density for type 2 and type 3 predators respectively.

Now consider the general case in which the prey Per capita
increase rate is not independent of prey density. Logistic prey
p0pulation growth is the linear representative of the common class of

functions for which R(H) declines monotonically with H:
H
Rm)=M1-p no

where r is the maximum per capita prey increase rate, time"1
and k is the carrying capacity of the universe for prey,
prey volume'l.

For type 2 predators hunting outside the refuge space, equations (4),

(7), and (14) imply that

24

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.Hm\nnANm\NnAmm\mn

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open :pzonw monnensmom xonm nun: pae mnoueponm N omxu can: monunmCop 5on9 Ednnnnnnsom

.o onSMnm

25

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1m
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O

 

 

 

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26

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.n onsmne

27

H

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m

SM?

n onswnm

 

I mcnnnomnea
I oaneumnEom E

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dsax Teuorqoun; '3

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28

when 0 :_H :_h/q,

 

_ r(abv(l—q)k-l) iI/r2(abv(l—q)k+l)2 - 4a2bv2(l—q)2rkp
H _

abv(l—q)r (15)

and H = 0; when H :_h/q,
(abvr)H3 + r(l-abvk-abvh)H2 + k(abvrh+avp—r)H — avhkp = 0, (16)

an implicit solution.
From (5), (7) and (14), comparable equations could also be written for
type 3 predators.

Graphical solutions in Figure 8 emphasize the similarity of
these results to those for density-independent prey reproduction, except
that monotonically decreasing R(H) usually generates an additional
stable equilibrium slightly below the carrying capacity k. Mathematical
and biological implications of this new uppermost equilibrium are
mentioned in latter sections.

Notice the four different kinds of equilibria (H) in Figures 6—8.
An equilibrium prey density is stable if the functional response F
exceeds the prey increase function RH/p at prey densities greater than
H, gpd the prey increase function exceeds the functional response
below H. Thus if H "accidentally" becomes slightly greater than H, then
the functional response exceeds the prey increase rate, and H tends to
return to H; if H becomes slightly smaller than H, then the prey increase
rate exceeds the functional response, pushing H back toward H. At an
unstable equilibrium, however, "accidental" shifts of prey density
away from H tend to become accentuated; if H rises above H, the prey

can increase faster than predators can eliminate them (i.e., RH/p > F),

 

29

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.w onswnm

w onsene

HIoEsHos. xonm .xunmcow xonm .:

.ClU‘

:M .X

 

30

  

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I £836 9

   
 

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r—I

 

 

31

but if H falls below H, predators can reduce prey density faster than
prey can be replaced. Semistable equilibria are stable for prey
density shifts in one direction away from H but unstable in the other;
that is, shifts along the prey density axis are accentuated in one
direction but counteracted in the other. And absorbing equilibria may
locally resemble any of the other three kinds except precisely at H:
if prey density ever reaches H, it remains there permanently. In any
universe closed to immigration, such as those considered here, H = O
(prey extinction) is an absorbing equilibrium.

In addition to those with constant or logistic R, another kind
of prey increase function features an "Allee Effect," in which R is
negative below some critical non-zero prey density. An expression
derived from the logistic to incorporate this effect is drawn with a
refuge—displaced functional response curve in Figure 9, indicating
patterns of equilibria similar to those in Figure 8; but note in
Figure 9 that whenever c > 0, stable equilibria are accompanied by lower

unstable equilibria (Holling, 1973).

The Escape and Extinction Thresholds

Because the magnitudes of state variables spontaneously tend to
approach stable equilibria and avoid unstable equilibria, a stable
equilibrium may be said to have a "domain of attraction” delimited by
the adjacent unstable equilibria (Lewontin, 1969). In particular, any
perturbation of prey density to a new value within its domain of
attraction returns toward the stable H, but perturbations beyond the
boundaries of the domain carry prey density toward a different stable H

in a new domain of attraction.

32

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.m onsmnm

HIoEsHo> monm lhpnmcop honm .m m “anew:

 

33

  
    

U

x Mmogmo Ho 0
_
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AmoL: m
imo-e51e-xve

    

   

Rub: a

              

  

   
 

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34

A function T(H), defined as in equation (17) such that dT/dH =
-dH/dt, can clarify the stability properties of this predator—prey
interaction. In Figures 10 and 11, each r-p pair specifying a prey
increase function on the functional response graph also specifies a
graph of the stability topography function T(H). Imagine that the H-T
plane is a vertical cross—section of a surface on which a small
momentumless marble is rolling, and suppose that the marble spontaneously
rolls downslope such that the horizontal component of its velocity
equals the slope of T at each point. Then the marble's position along
the H—axis represents the prey density varible, and the dimensions of
terrain features delineate and measure the domains of attraction that
constrain the marble's trajectory; for example, when r = r3 and p = p3
in Figures 10 and 11, the stable equilibrium at Hg has as its domain of
attraction the interval (H3,Hg).

”accidentally" rolls beyond Hg, then it can spontaneously roll to zero,

But notice that if the marble

i.e., the prey population becomes extinct; thus for those parameter

values, H

3 is an extinction threshold, the prey density below which the

 

prey population density tends to decline to zero. 0n the other hand,
if the marble rolls beyond Hg, then it can roll toward higher prey
densities, i.e., the prey population escapes regulation by predators;

thus for those parameter values, H

3 is an escape threshold, above which

prey density increases toward a resource-limited maximum (of. Holling,
1973).

When the per capita increase rate of the prey population
(neglecting predation) is approximated by a constant R, this maximum
prey density is not mathematically specified and presumably is at or

near the resource limit k. When R(H) declines monotonically with H,

Figure 10.

35

The prey density topography with type 2 predators and with
prey population growth independent of density in a universe
cpntaining refuge space. r5/p5>r4/p4>r3/p3>r2/p2>rl/p1

"HI!
”Fl/H
"Fr"
"G"
and

is the extinction threshold, prey volume '1;

is the stable equilibrium prey density, prey volume-1;
is the escape threshold, prey volume- ;

is a dummy variable for prey density, prey volume-1;
”T" is the stability topography, preyzvolume'2 time-1.

36

 

 

  
  
  

 

 

L rSH/ r4H/Io4 :3H 3 _:2:I:p3_ r _ l
l/b ——————————
|
o.—I
In I |
8’5 I
o.
7', g : V stable H
§ § I A unstable H
I), E. : E semistable H
I: v .
(a >‘ | | 0 absorbing H
. 3 I
In 0.. | l
' I
l/ I
| _
I! I H,prey density, l prey volume 1
\l I
|
|
|
|
I
l
N: .p )
2 2
43.4
8'.» C
8‘5 .
9"“ T(r rp3)
° 3
“I“
>< (D
E 5
u-I 0
g > dG
fi‘iq T(H) = j]H(dt)dG (17)
I»
~ n 0
B 94

 

)

T(r

5’p5

Figure 10

37

Figure 11. The prey density topography with type 2 predators and
logistic prey population growth in a universe containing
refuge space. r4/p4>r3/p3>r2/p2>r1/p1.

38

 

 

 

 

 

 

 

 

 

o.-I
an
I: A
o o
3‘51
o u 3
n
E 3
a p
3% '. I Vstable H \‘ 2
u M | Aunstable H
o o. l . r H H
5" _ | | Esenustable H "g;— (1-?
M 3' l. I Oabsorbinq H
a? 6‘. ’ I l
I | I
c 1 f l
0 h
#3 HI prey density, prey Volume'l k
I
i T(rZIPZ)
>, I
‘3 l
0-H
S'o I
m (r 9)
a5 3: 3
o
”‘7'
3: w 0
:5.
232 \
um
(I: :n
~ 3’.
B 04
Figure 11

39

as in logistic population growth, then the size of the carrying capacity
largely determines the density of the highest stable prey population
equilibrium. Barring a precipitous drop in R(H) near k, however,
Figures 8 and 11 clearly suggest that pp£h_predators and resources
determine the level of this uppermost stable equilibrium (see May, 1973);
and conversely, owing to one of the much—lamented unrealities of the
logistic, prey are still partly limited by resources even at the lower-
most stable equilibrium. Nevertheless, the lower and upper stable
equilibria may usually be considered "predator—limited" and "resource—
limited" respectively, implying that the unstable equilibrium separating
their domains of attraction represents a true escape threshold.

Figure 10 indicates that the prey density interval in which a
stable equilibrium can exist with constant R(H) is (H, H). In other
words, if rz/p2 < r/p < rz/p4 in the figure, then there exists a stable
equilibrium prey density within (H, H). The limits of this interval can

be found as follows: For type 2 predators,

H = , (18)

..O|:r

the prey density at which the refuge space becomes full. But this
calculation rests on the assumption that the refuge occupancy quotient

q is really independent of prey density, even near h/q; more likely, q
will decrease somewhat as H approaches h/q, smoothing out the transition
between the intersecting functional response curves. In Figure 12, this
effect is pronounced enough to lower H to zero; the functional response
has become positively accelerated at prey densities below h/q,

resembling the type 3 functional response of Figure 2. In some instances,

40

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momnn omaommon Henonuocsm onn .owsmon on» opnmcn npnmaop zonm mcnmeonocn non:
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:2 ease:

41

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(emtl-Ioaepexd) Aexd
dsex {Quotaoung

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Q\H

 

42

the functional response curves of Figure 10 may be more realistic than
the curve in Figure 12, and equation (18) will be applicable. H, in
contrast, does not depend on q, and Figure 12 suggests a means of

calculating its magnitude for type 2 predators: H is uniquely the

tangent point at which F(H)/H dF/dH. From equations (4),

 

 

F(H) _ av(H-h)
H ' (1+abv(H1h))H (19)
and
dF(H) = av(
dH (l+abv(H-h))2
Equating equations (19) and solving for H,
._ 77-h"
H - 5-5; + h (20)

For type 3 predators, H 0, since the positively accelerated
functional response at low prey densities insures stable equilibria in
this range; for all r/p > O, prey densities very near zero tend to

increase spontaneously, moving away from the absorbing equilibrium at

H = O. H for type 3 predators can be found as above, using equations

(5):

F(H) _ av(H—h)2

- (21)
H (s+H-h+abv(H-h)2)H

 

and

dF(H) = av(H-h)2+2avs(H—h)
(s+H-h+abv(H-h)2)2

 

 

43
Equating equations (21)
abv(H-h)3 - (H-h)h — s(H+h) = o. (22)

Notice from (22) that when h = O, H = Vs/abv: type 3 predators can
stabilize prey density even in a homogeneous universe (Holling, 1965;
and many others).

H and H are more difficult to express explicitly when the per
capita increase rate of prey is density-dependent, even when the
logistic equation provides a suitable representation of R(H). In the
density-dependent case, H is usually zero, especially for type 3

predators (or for high q). H is the same or only slightly larger than

for constant R(H).

Stability of a Predator—Prey Link

 

This analysis suggests that stability of a predator-prey link
can be quantified by appropriately operationalizing elasticity and
conformability. (Recall that elasticity is a measure of the tendency for
perturbations of state variables to be damped; conformability is a
measure of the tendency for elasticity to be maintained in the face of
parameter perturbations.) Let the elasticity of a stable predator-
limited equilibrium prey density, E(H), be an ordered pair expressing
the proximities of the escape and extinction thresholds to the stable

equilibrium. The simplest such expression is

‘- v

R(H) = <H-H, H—H> (23)

44
where H'is a lower unstable (or absorbing) equilibrium prey density,

the extinction threshold, prey(volume)'1,

H'is a stable equilibrium prgyfidensity with a domain of
attraction in the interval (H,H) prey(volume)‘1,

and H is an upper unstable equilibrium prey density, the escape
threshold, prey(volume)‘ .
E measures the size of prey density perturbations away from the stable
density tolerable in each direction without forcing H beyond the domain
of attraction. The domain of attraction is itself specified as the

elastic region, E(H), the interval delimited by threshold boundaries:
5(a) = (fi,fi) (24)

Similarly, let the conformability of a stable predator—limited
equilibrium prey density C(H) express the ordered pair of differences
between the sloPe of the prey increase function and the minimum and

maximum conformable slopes, respectively; that is

ccfifi =<r/p - f7fi, f7} - r/p> (25)

where £7b is the minimum initial slope of the prey increase
function yielding a stable or semi-stable, predator-limited
equilibrium prey density, volume(predator-time)'1,
and £73 is the maximum initial slope of the prey increase
function yielding a stable or semi-stable, predator-limited
equilibrium prey density, volume(predator-time)‘1.
Then the conformable region of a stable equilibrium prey density, C(H),
is the range of conformable initial slopes of the prey increase function,

i.e., those slopes allowing the existence of a stable, predator—limited

equilibrium in the prey density interval (H, H):

45
007) = (r712. r733) (26)
In particular, for type 2 predators and prey with constant R(H),
C(F) = <r/p - FQD/E, Fob/H — r/p> (27)
and CW) = (mp/13, Fan/H)
where H is now seen to be the maximum conformable prey density,
and H is now the minimum conformable prey density, both prey

Volume-1.

But if q decreases with H as prey density approaches h/q, then Figure 12
suggests that the slope of the functional response curve at zero prey
density may provide a more accurate estimate of minimum conformable prey

density. In that case

C(FI’) = <r/p — dF(O)/dH, F(H)/H - r/p> (28)
and C(Ff) = (dF(O)/dH, F(H)/H).

And for type 3 predators and prey with constant R(H),

C(H’)

<r/p, F(H)/H — r/p> (29)
(0,F(H)/H)-

and C (H)

E and E always refer to a given state variable and a given stable
equilibrium. With more than one prey population in the system, the

appropriate values of H, H, and H for some population H1 are the

corresponding H coordinates of the multispecies equilibria; in other

1

E . . . BH and El, E

words, E1, 2,

2, , En can be calculated with n

prey species following the form of equations (23)-(26) using the

respective coordinates of each H = H1, H2, . . . Hn . Similarly,

46

C and C always refer to a given stable equilibrium, a given state
variable, and a given parameter or combination of parameters. The
conformability analyses in this paper focus on r/p, the quotient of two
crucial and biologically "noisy" parameters; calculating the upper and
lower limits of r/p that allow a particular n-species stable equilibrium
to exist clearly requires that H and H be found in n-space, from which
F(H) and F(H) can be obtained. C and C are calculated for each state
variable using its coordinate of the multispecies H and H. If
equations (28) are used, dF(O)/dH should be replaced for example by
3F(O)/3H1 to find C1 and C1, the conformability and conformable region
with respect to prey species 1.

The preceding analysis, of course, considers only a few simple
mathematical expressions of elasticity and conformability; many others
are possible. Lewontin (1969), for example, presents another elasticity
metric, an index to characterize the t0pography of a domain of
attraction. And the elasticity concept can be extended to other state
variables--in the present analysis to dH/dt: the boundaries of its
elastic region, easily found for known F(H) and R(H), specify the
maximum sustained rates at which prey can be removed (fished) or
introduced (stocked) without triggering extinction or escape (see
Holling, 1973; Smith, 1968). So in these and other mathematical guises,
elasticity and conformability may help bridge a gap between ecological
theory and application (see below).

Elasticity and conformability differ not only in mathematical
expression but also in biological implications. When "shock" pertur-

bations predominate, as in coastal regions and on tropical and temperate

47

oceanic islands, only highly elastic ecosystems can persist; when
"shift" perturbations predominate, as in strongly seasonal environments
such as temperate lakes and deciduous forest, only highly conformable
systems can survive intact. But it may often be possible to enhance
elasticity or conformability (and reduce the chances of prey extinction
or escape) by tuning critical parameter values, a strategy explored
briefly in the Summary of Theoretical Results and in the Discussion

later in this paper.

Multiple Refuges

 

So far in this analysis the spatial heterogeneity of the
predator-prey universe has been the contrast between inside and outside
the refuge space. No distinction has yet been drawn between a universe
with, say, fifty small refuges of total capacity h and a universe with
a single refuge of capacity'fifl'nor have the implications of several
distinct refuge type§_in the universe been considered. Both omissions
will be remedied in this section.

The most ecologically significant difference between a few large
refuges and many small ones is probably the relative amounts of contact
between "inside" and "outside": the surface-volume ratio (or perimeter-
area ratio) of the refuge space increases dramatically as this space is
partitioned into many small sub-spaces. Now if the extent of,contact
between refuge and non-refuge space partly governs the exchange rate of
prey individuals between inside and outside, as expected if prey
movement has any significant random component, then the functional
response and the stability properties of the interaction may depend

critically on the size distribution of refuge spaces.

48

One simple but representative case is considered here--a system
with type 2 predators, predator-exclusive refuge space, and per capita
prey exchange and increase rates independent of prey density. Let ”e"
specify the exchange rate, the fraction of prey moving gut_across the
refuge perimeter per unit time; then weighting by the occupancy
quotients for inside and outside, the exchange rate of prey moving in_
is eq/(qu). In the steady state the decrease rate of prey density
inside (prey crossing to the outside) must equal the increase rate of

prey density inside (prey crossing to the inside plus prey recruitment

inside):

_ eqxfi
eH_— (l-q)v + rH_ (30)

where H_is prey density inside, prey inside (inside volume)_1,

and H is prey density outside, prey outside (outside volume)_1.

From (30) a new refuge occupancy quotient Q(e)-—the fraction of all prey
that is inside the refuge space as a function of the exchange rate--can

be found by rearranging (30):

Q = 3/1 = eq . (31)
E/XTH/V e—r(1-q)

 

Notice that when e >> r, Q = q; in other words the refuge occupancy

quotient is constant (as assumed in previous sections) for high exchange

rates or low rates of reproduction, but it depends on both "e" and "r"

when those parameters are more similar in magnitude. And note from

(30) that "r" cannot exceed "e” in the steady state, i.e., e :_r.
Replacing "q" by Q(e) in functional response expressions

facilitates the comparison of systems featuring different distributions

49

of refuge space as reflected in their respective prey exchange rates.
Figure 13 shows that greater exchange rates (many small refuges) yield
higher functional response curves at prey densities below refuge
saturation than for smaller exchange rates (few large refuges); so the
system containing many small refuges can have larger H and r p and thus
a greater chance of extinction than a system with few large refuges,
though the chance of escape is unaffected.

There are at least two reasons to question the distinctness of
this contrast: (1) Relaxing predator exclusivity and density-independent
exchange should increase the effective exchange rate more for initially
small "e" than for large "e". (2) "e" is often large relative to "r"
anyway. Yet of the seven examples of structural SH given at the
beginning of this paper, rerge size distribution could certainly be
important in at least two of them: the protozoan—flask systems of
Gause (1934) and the chemostat interaction of van den Ende (1973).

Taking a different slant on operationalizing spatial heterogeneity,

Smith (1972) has shown that the average catchability of prey removed in a

 

non—mixing, spatially heterogeneous universe exceeds the average in the
initial population by the square of the coefficient of variation. This
”catchability bias," though useful in quantifying exploitation, is
inadequate as an index of heterogeneity: the bias disappears completely
in a rapidly-mixing system even though structural SH can still be
strongly stabilizing (see Figure 13).

It has recently been suggested that distributing populations and
refuges in space may "spread the risk" of a destabilizing perturbation

among several prey sub-populations, implying that a system with many

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51

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52

smaller refuges may be mgre_stable than one with few large refuges.

This stochastic view certainly deserves thorough theoretical elaboration--
all the more so if the stability effects of stochastic and structural SH
are frequently antagonistic. In its current vague form (den Boer, 1968
and 1971; Reddingus and den Boer, 1970), however, the spreading-of—risk
concept has generated some confusion and valid criticism (May, 1971 and
1974b; Roff, 1974a; Levandowsky, 1974). Other recent approaches to
stochastic SH (e.g., Roff, 1974b; May, 1974b), though difficult or
impossible to operationalize, may eventually help clarify the relation
between stochastic and structural SH.

Now consider the implications for stability of several different
refuge types in the predator-prey universe. Figure 14 shows the
functional response curve for type 2 predators in a 3-refuge universe,
with the refuge capacities hi and the occupancy quotients qi independent
of prey density. The functional response rises from the origin in a
series of four intersecting type 2 curves: below prey density hl/ql’
all three refuge spaces can hold additional prey with increasing H, but
immediately above hl/ql’ only spaces 2 and 3 still have room; above the
next slope discontinuity, only refuge space 3 is still filling with prey,
and above the highest slope break, all refuge spaces are filled with
jprey. ‘Mathematically, this case simply elaborates the analysis of the
outside curve of Figure 4. In fact, if all three refuge types became
saturated with prey at the same prey density, then Figure 14 would have
only one slope discontinuity and would be indistinguishable from the
outside curve of Figure 4.

Notice that the general shape of the curve in Figure 14 is

sigmoid, especially if the breaks in slope at intersection points are

53

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55

smoothed out by a tendency of each occupancy quotient to decrease
slightly near saturation of its refuge space (see Figure 12). In the
limit, in a spatially complex universe partitioned into an infinite

array of refuge types with different capacities and occupancy quotients--

a spatial continuum from "totally protected" to "totally unprotected"--

the functional responses of either type 2 or type 3 predators must

resemble the sigmoid curve in Figure 2 (Smith, 1972, cf. Comins and

Blatt, 1974).

Interference Among Predators
Although its importance in nature remains a source of controversy
(e.g., see Griffiths and Holling, 1969; Hassell and May, 1973), inter-

ference among individual predators has been shown to have a potentially

:tabilizing effect in some predator-prey models (e.g., Hassell and

ogers, 1973). Thus it seems appropriate to briefly consider the

ifluence of interference on the functional reSponse and on stability in
spatially heterogeneous universe.
The most commonly-cited models of interference effects on the

ictional response are the purely descriptive equations of Watt (1959)

of Hassell and Varley (1969), which ignore the underlying behavioral
hanisms. Recognizing this inadequacy, Rogers and Hassell (1974)
:mpted to account for the reduction in available search time resulting

encounters between pairs of randomly searching parasitoids by the
Dwing argument: If it can be assumed that each encounter costs two

sites some constant amount of search time (see Ullyett, 1949, but

see Hassell, 1971b), then the fraction of parasites effectively

:nted from searching at any given time will reach a steady state

56

depending upon the rate of encounter, the parasite density, and the time
lost per parasite per encounter. Unfortunately, however, the rate of
encounter in Rogers and Hassell‘s model depends on host density,
confounding their searching efficiency parameter.

Consider an alternative approach: In his derivation of the disc

equation, Holling writes an equation analogous to
F = aHIl-bF), (33)

which can be rearranged to yield equation (1); in the above form, however,
the type 2 functional response can be extended to include several prey
populations hunted by a single predator population. If predators hunt
prey of n prey populations via simultaneous and independent searching but
with separate handling of individual prey, then the functional response
of predators to the ith of n prey populations may be written
n

Fi = aiviHi(l-jE1bij)° (34)
(The vi terms correct prey densities to allow for refuge space.)
Solving all n equations (34) simultaneously for Fm,

amvam
F = (35)
m n

l + Z a.b.v.H.
jsl J J J J

03f. Marten, 1973; Timin, 1973; Murdoch, 1973; Lawton et al., 1974;
Ierris, 1974). Now if Rogers and Hassell's assumption of a constant
search time cost per encounter between two searching parasites or
1xredators is valid, then such encounters are closely analogous to

capturing prey of another species, imposing another kind of "handling

57

time” on the searchers. For interference, then, equations (34) can

be replaced by

F

avH(l-bF—b*F*) (36)

and F* a*vp(l—bF-b*F*)

where a* is the rate of "encountering" predators, volume per
predator time

b* is interference time, predator time per predator
encountered,

p is the effective predator density to which each individual
predator is exposed, i.e., the density of predators-minus-
one, predators volume-1,

and F* is the number of predators interfered with per predator

time, time-1.

Now solving equations (36) simultaneously for the functional response,

_ avH
F _ l+ava+a*b*vp (37)

 

(cf. Timin, 1973 and see Salt, 1974).

For type 2 predators in a universe with a perfect refuge, prey
densities of equations (37) must be reduced by"h"density units to allow
for prey inSide the refuge space, yielding equation (38) in Figure 15.
The figure illustrates two related but separable effects on the
stability of the system: (1) varying predator density for a given
intensity of interference and (2) varying the intensity of interference
with a given predator density.

First, suppose that p1 is one predator per universe volume; that
is, p = O, and there can be no interference. Now increasing p not only

reduces the slope of the prey increase line rH/p, but, in effect, it

58

igure 15. The prey density topography with interference among type 2
predators in a universe containing refuge space.
p4 > p3 > p2 > p1. "a*" is the rate of "encountering"
predators, volume (predator-time)‘1, "b*" is the inter-
ference time, predator-time predator-1; and "p" is the
effective predator density to which each predator is

exposed1 i.e., predators-minus-one density, predators
volume'

59

 

 

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60

also lowers the rate of successful attack, depressing the functional
response at all prey densities in the interval (h, 00). Figure 15 shows
that stable equilibrium prey density still decreases, the chance of
extinction increases, and the chance of escape decreases with increasing
predator density just as in the absence of interference, though the
relative magnitudes of those effects are reduced by interference.
Second, suppose that predator density remains constant at p3,
fixing the prey increase function, but the term a*b* of equation (38)
increases from zero (curve F1) to larger values (F2, F3, and F

4

respectively); increasing a*b* amounts to intensifying interference at
a constant density of predators. This shift clearly increases H:
decreases the chance of extinction, and increases the chance of escape.
Therefore, contrary to the results of Williamson (in_Hassell, 1971b),
Hassell and May (1973), and Rosenzweig (1971, 1972), interference
apparently increases stability only in the sense of avoiding extinction;
escape actually becomes more likely as interference among predators is
intensified.

This analysis can also be extended to include effects of inter;
specific interference among predators by simply adding a term of the
form a*b*Vp to the denominator of equations (37) or (38) for each
additional interfering predator population; note that the appropriate

predator density in such terms would be total density, not predator-

minus-one density.

Summary of Theoretical Results

 

The operational definitions of elasticity and conformability in

previous sections can be used to predict the effects of parameter changes

61

on the stability of a predator-prey link. Since changes of parameter
values tend to shift the boundaries of the elastic region and of the
conformable region all in the same direction, qualitative effects of
such changes on elasticity and conformability also depend on the
magnitudes of the other parameters. But the qualitative effects of
parameter shifts on the chances of prey extinction or escape are
unequivocal and can be inferred from the altered proximities of H and
r/p to the elastic and conformable boundaries, respectively. Table 1

summarizes the qualitative effects of increasing each of thirteen

 

parameters on the stable equilibrium, the boundaries of the elastic
and conformable regions, and the chances of extinction and escape.
(Decreasing each parameter invariably reverses its effect.) The

quantitative implications of any actual or potential set of parameter

 

magnitudes can be found by substitution into the elasticity and con-
formability equations (23)—(29). In the discussion and the appendix of
this paper, both qualitative and quantitative parameter analyses are
used to examine experimental results and suggest applications in eco-
system management.

The principal theoretical results of this analysis, with
references to pertinent figures and tables in the text, are enumerated
below:

1. Heterogeneity of spatial structure can stabilize a predator—
prey interaction, and its stability effects can be quantified
(Figures 6 and 10).

2. The functional responses of predators to prey density in a
universe containing refuge space are basically sigmoid

(Figures 6, 7, 12, and 14).

62

Table l.——Qualitative effects of increases in each of thirteen para—

meters on the stable equilibrium density H,‘the extinction

and escape thresholds (elastic boundaries) H and H, the
conformable boundaries r/p and r7p, and the overall chances

of extinction and escape. "+” denotes an increase in the
magnitude of a variable, "-" a decrease, and ”0" no change.
Where the outcome is ambiguous and depends on the other
parameters, all possible outcomes are given. Note particularly
that the stable equilibrium density and the chance of escape
respond identically to all parameters, and that the chances

of extinction and escape usually respond reciprocally.

 

chance of chance of

 

parameter increased H H H £7b £7; extinction escape
a, successful attack rate — +,0 + +,0 + + -
a*, predator encounter rate + —,0 — —,0 — _ +
b, handling time + _,0 - -,0 _ _ +
b*, interference time + _,0 _ _,0 - _ +
c, Allee coefficient — + + +,O + + _
e, exchange rate 0 +,O 0 +,o o + 0
h, refuge capacity + 0 _ _,0 _ _ +
k, carrying capacity + —,O - —,0 — _ +
p, predator density _ +,O + O O + _
q, occupancy quotient O —,0 0 —,O O — O
r, prey increase rate + —,O - O O - +
s, sigmoid coefficient + O — O — — +
v, volume ratio — +,0 + +,0 + + _

 

63

Shifts of parameter values usually cause mutually reciprocal
shifts in the chances of prey extinction and escape; stable
equilibrium prey density responds like the chance of escape
(Table 1).

Increasing the amount of refuge space in a system decreases
the chance of extinction and increases the chance of escape;
reducing the amount of refuge space has the opposite effects
(Table l).

A system containing many small refuges can have a higher chance
of prey extinction than one with few large refuges (Figure 13,
Table l).

Intraspecific interference enhances the chance of prey escape
and reduces the chance of their extinction; but increasing
predator density decreases the chance of escape and increases
the chance of extinction, even with particularly intense

interference (Figure 15, Table l).

DISCUSSION

Three Predator—Prey Interactions Revisited
In this section three predator-prey studies are reconsidered from
the viewpoint emphasized in this paper. Two of them provide specific but
qualitative examples of how this analysis can help interpret the effects
of spatial structure in predation, and the third attempts a more
quantitative (though somewhat speculative) application to data from an

experimental field study.

Tribolium vs. Oryzaephilus
Crombie (1946) observed the interaction between populations of

the flour beetles Tribolium confusum and Oryzaephilis surinamensis in

 

renewed wheat and flour media. In wheat, the populations coexisted, but
in flour I: confusum consistently eliminated Q, surinamensis via
"voracious" predation on its pupae. However, when the flour medium
contained glass tubing of 1 mm. internal diameter-—large enough to

allow orzyaephilus to enter and pupate but small enough to exclude
tribolium adults and large larvae—~both populations survived just as in
wheat.

Apparently the glass tubing in flour and the bran in wheat can
provide refuges for the vulnerable oryzaephilus pupae, allowing co-
existence of the two beetle populations. The functional response of
tribolium to the density of oryzaephilus pupae is displaced to the

right by the capacity of all tubes or bran within the critical size

64

65

range. Since self-limitation of predators by cannabalism allows
relatively high predator densities (low r/p), and since the prey exchange
rate is low and the occupancy quotient is high, then the stable
equilibrium prey density H must approximately equal the refuge capacity
h. In other words, almost all surviving oryzaephilus pupae must be
inside tubes or bran, and pupal density should be quite stable, as
Crombie observed. This interaction exemplified curve 1 of Figure 13

(but with smaller r/p).

Typhhlodromus vs. Tarsonemus

 

 

Huffaker and Kennett (1956) studied the interaction between the

predatory mite Typhlodromus sp, and the cyclamen mite Tarsonemus

 

 

pallidus in an attempt to deve10p a program for control of I, pallidus
in strawberry fields. The strawberry plant provides an unusually
favorable microenvironment for tarsonemus within the many folds and
crevices in the crown of the plant, furnishing food, humidity, and
protection from predators.

Physical barriers, representing a security threshold that results

from the great heterogeneity in the microenvironment, seem to

preclude actual extermination of the prey on any unit as large

as an entire plant, or at least a group of adjacent plants. Hence,

equilibrium is reached and a rough, although disturbed, balance

at very low densities is characteristic (Huffaker and Kennett,

1965, p. 194).
Structural spatial heterogeneity has clearly stabilized this interaction
much as for tribolium and oryzaephilus.

But there is another important facet of this mite predator-prey

link, recognized by Huffaker and Kennett: even though the predators'

reproductive rate at high prey densities is comparable to the prey's,

this reproductive numerical respdnse is "almost meaningless" for

66

controlling the prey population without a high searching capacity and
the ability to survive at low prey densities. In other words, the
functional response plays a much more vital role in determining the
stability properties of this interaction than does the reproductive
runnerical response, despite similar increase rates of the two pOpu—
lations (see also Huffaker and Kennett, 1966; and Stabilizing Responses

of Predators below).

Ishnura vs. Simocephalus

 

Johnson (1973) studied the effects of predation by damselfly
naiads (Ishnura verticalis) on a littoral cladoceran population
(Simocephalus serrulatus) in plastic wading pools. Each of the eight
pools contained 40 artificial "weeds" composed of thin strips of
plastic window screen hanging down from above the surface to near the
bottom. Three weeds in each pool were sampled every fourth day to
estimate prey and predator densities; by comparing observed prey
densities with predictions of a mathematical model from the densities
and egg counts four days before, Johnson could calculate mortalities due
to predation over the interval.

The time-course of simocephalus density in the eight pools,
modified from Johnson's Figure 4, is presented in Figure 16. In my
view, these curves suggest the existence ofan escape threshold at a
density of about 180 prey liter—1; below that level densities tend to
remain relatively constant, but once above 180 prey liter_1 densities
increase in a sharp, irregular pattern, decreasing again only at the
end of the experimental period with falling ambient temperature. Note

particularly the wide variation in the number of days from predator

 

Figure 16.

67

Densities of simocephalus in eight wading pools observed
at 4-day intervals. The left triangle indicates the date
of ishnura introduction; the right triangle marks the date
at which a given simocephalus population exceeded the
postulated threshold density, near 180 animals liter-1
(data from Johnson, 1973).

 

 

Simocephalus density, animals liter‘l

2000—

1000'

68

 

 

 

 

 

 

 

 

 

400-

200- —————————————

 

O , I I it: 1
July Is ura

22-23 introduced

Figure 16

 

sampling dates, 4-day intervals

69

introduction until the hypothetical threshold is crossed: 8,9,13,0,22,l4,
18, and 38 days respectively. It is tempting to suppose that during the
interval following predator introduction, each simocephalus population
was regulated by ishnura until prey density "accidentally" exceeded the
escape threshold, freeing the prey from limitation by predators. But
since ishnura is not a type 3 predator (at least not without significant
densities of alternate prey--see Lawton et al., 1974) and exhibits
negligible numerical or developmental response to simocephalus density
(Johnson, 1973), spatial heterogeneity should be investigated as a
possible critical stabilizing factor in this interaction.

Johnson’s experimental universe contained two kinds of refuge
space:

1. The sides of the pools, representing about 30% of the vertical
surface area (weeds accounted for the other 70%), were essentially
inaccessible to hunting damselflies; about 30% of the simocephalus were
associated with the sides at all prey densities.

2. The weed surfaces themselves provided a partial refuge for
simocephalus. Because ishnura hunts mainly by ambush, grasping passing
prey that contact its antennae with a quick extention of its labium,
simocephalus attached to weeds (via cervical glands located dorsally on
the carapace) must have been much less vulnerable to damselflies than
those actively swimming within a weed. The pool-side population, not
counted in Johnson's density estimates, increased the effective per
capita increase rate of the prey exposed to predators by a factor of
1.43 (100% e 70%), assuming fairly rapid exchange of prey between the
sides and the weeds, but could not displace the type 2 functional

response curve to stabilize the interaction. The weed surfaces, however,

7O

Innjbably Egu1d_have such a stabilizing effect. As the density of
sxhnocephalus attached to a surface increases, they become much more
active, and the percent of attached animals decreases; if this occurred
art relatively low densities within the artificial weeds, the functional
:response of the damselflies could assume a potentially stabilizing,
sigmoid shape (cf. Haynes and Sisojevic, 1966).

Now examine Figure 17. The data points were calculated from the
positive predation mortality estimates of Johnson (1973). The dashed
line is his unpublished fit to the circled points, for which estimated
and.observed prey density differed significantly. Though this curve was
obtained using standard techniques—-linear transformation and least
squares analysis (Holling, 1959b)--it clearly fails to fit or describe
the data satisfactorily, especially near the origin. By displacing this
same type 2 functional response curve 30 animals liter.1 to the right,
however, a much more plausible fit for low prey densities is found.
(Such a displacement is equivalent to postulating the existence of about
200 highly-sought, essentially invulnerable hiding places for
simocephalus per weed, an interpretation which could only be adequately
evaluated by directly observing simocephalus activity in the weeds as a
function of density.) A high refuge occupancy quotient that decreased
with increasing prey density would imply the very low extinction
threshold and the initial positively-accelerated functional response
illustrated in Figure 17 (cf. Figure 12).

Using the observed mean predator density "p", and estimates of
per capita prey increase rate "r” mathematically predicted by Johnson's
growth model for each interval, the prey increase function rH/p has

been calculated. The lower straight line in Figure 17 incorporates the

71

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.mHoom wanna: unmwo :H spawned msHmnmoooEHm on «Manama mo omcommon Hmaowpocsm one

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72

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73

Inean "r" for the predator-regulated sampling intervals (0.217 day-1);
:Erom.equation 11, the stable equilibrium at the intersection of this
function with the functional response curve (45.5 prey liter—1) is
reasonably close to mean prey density during predator-regulated intervals
(66.7 prey liter—1). The uppe£_straight line in the figure uses the
Inean "r” for sampling intervals in which the prey presumably escaped
regulation by predators (0.480 day-1); since this prey increase function
exceeds the functional response at all positive prey densities, such ”r"
values do appear consistent with prey escape during the corresponding
sampling intervals.

Johnson (1973) emphasizes another source of fluctuating predator
influence on prey. He argues that predation rates of individual
damselflies probably change considerably during each instar such that
any significant degree of molting synchrony could produce important
shifts in effective predator density during the experimental period
(also see Johnson et al., 1975). Thus bgth_parameters of the prey
increase function, "r" and ”p", seem relatively "noisy"; with the prey
increase function bobbing up and down erratically, the effective escape
threshold must be near or somewhat above H, the maximum conformable
prey density. And in fact, the apparent threshold in Figure 16, about
180 prey liter‘l, does only slightly exceed the H found from equation 17,
135.4 simocephalus liter-l.

Fortunately, neither the magnitude of H nor the possibility of
regulation and escape of the simocephalus population depends very
strongly on the exact shape 0f the functional response curve through

the scattered data points at high prey densities. Other parameter

74

'values may alter the H estimate slightly, but for many cases with refuge
capacities near 30 prey liter-1, the observed prey density sequences

seem to agree with the postulated stability properties of the system.

Stabilizing Responses of Predators

Errington (1946, 1956) believed that most predators, especially
Inammals, merely skim off the "excess” from the prey population and exert
no real stabilizing effect; he labelled such predation "compensatory."
If he had attempted to define "excess" objectively by relating prey
density to environmental structure, Errington might have recognized the
stabilizing potential of "compensatory" predators (Huffaker and Watson
in_Huffaker, 1971; Maynard Smith, 1974).

In fact, Errington's view implies that predators in a universe
with limited refuge space should tend to have sigmoid functional responses
to prey density, since capture rates increase sharply as prey density
exceeds his "threshold of security" (cf. Figure 12). Holling (1959b),
Solomon (1964), and Murdoch (1973) recognized this effect but did not
emphasize its significance for stability in spatially complex environ-
ments; they and many others have concentrated on behaviorallyégenerated
sigmoid curves that can be analyzed in simple laboratory systems,
largely ignoring the ubiquitous but conceptually elusive spatial effects
(Tullock, 1970). But the stability implications of the sigmoid
functional response itself (whatever its cause) now seem to be widely
appreciated: wherever the curve increases with prey density faster than
linearly, a stable, predator-limited equilibrium prey density could

occur (e.g., Holling, 1959b; Huffaker et al., 1968; May, 1973).

 

75

Of course, this criterion for a stabilizing functional response

does not strictly require a smooth sigmoid curve, as Figures 6-11
clearly demonstrate (cf. Hassell, 1966; Steele, 1974); nor does it
necessitate learning by the predator, or search-image hunting, or the
1xresence of alternate prey. The following three general conditions,
implicitly quantified in the foregoing derivation, are sufficient to
generate such a curve for ordinary type 2 predators:

1. limited (saturable) refuge space,

2. some tendency for prey to occupy this space, and

3. some tendency for predators to avoid it or be excluded.
Refuge space for pey, at least "relative" refuge space, must be present
in virtually all natural ecosystems. Furthermore, in systems with an
evolutionary history, natural selection must enforce a strong preference
by prey for safe hiding places, as noted previously. And though
adapting to safer surroundings may be a comparatively minor behavioral
or evolutionary feat, evolving the ability to hunt effectively in new
terrane must usually imply such relatively drastic modification that
relative ineffectiveness of predators inside the refuge space is
expected (see Slobodkin, 1974). But Smith's marble game suggests that
"prey populations with predation present will tend to be found in the

more hidden places of their environment, even if_they dg_not select

 

such places" whenever exploitation can reduce catchability (Smith, 1972).

 

lnThe,least generally recognized of the three conditions above is
perhaps the most restrictive: that the average vulnerability of prey must
increase with density over some range below the carrying capacity--i.e.,

that the refuge space can become saturated with prey (Smith, 1972).

 

76

For type 2 predators with a negligible numerical response, protecting a
constant number of their prey yields the stabilizing positive displace-
Inent of the fUnctional response curve along the prey density axis (see

"Figures 2 and 12): per capita protection of prey decreases as the refuge

space fills up. In contrast, protecting a constant fraction of prey at

 

all densities is equivalent to proportionally shrinking the rate of
successful attack "a" in equation (1) (see Griffith's 1969 analysis of
"spatial coincidence"); since the shape of a type 2 functional response
remains convex upward after any such shift of "a", any positive,
predator-limited equilibrium must remain unstable.

The stability effects of these two kinds of protection can be
compared fer several published models: In Rosenzweig and MacArthur's
classic graphical model (1963) both kinds of protection are stabilizing,
whereas Maynard Smith (1974), using modified Lotka-Volterra equations,
shows that only the "constant number" case lends stability to the
interaction. And for Nicholson and Bailey's (1935) model, Varley (1947)
claims that protecting a constant fraction of prey is stabilizing, but
Bailey et a1. (1962) reply that neither kind of protection can be a
common source of stability. Yet because in each of these models the
functional response is over-simplified and masked by the numerical
response, their lack of predictive agreement is not surprising.

In nature, if not always in mathematical models, the relative
impacts of the functional and the numerical responses on stability
reflect the reproductive rates and lags of predators and prey:

1. If the time scale of the predator's numerical response (inverse

per capita increase rate plus lag time) N is small relative to the

77

thne scale of prey increase (inverse per capita increase rate of prey)
‘R, then the numerical response determines stability, yielding a tightly
(humped oscillatory response to density perturbations (e.g., the host—
helnunth parasite interactions of Anderson, 1974).

2. If N~R, then a stabilizing functional response is necessary but
insufficient for system stability, and pOpulation densities oscillate
‘with large amplitude in response to perturbations (e.g., the
'Typholodromus-Tarsonemus interaction of Huffaker and Kennett, 1956;
see Solomon, 1964; Huffaker and Kennett, 1969).

3. If N>>R, then a stabilizing fUnctional response is necessary and
sufficient for system stability, and population densities respond
asymptotically to perturbations (e.g., the marine copepod-phytoplankton
interactions of Steele, 1974; see Takahashi, 1964).

Since a numerical response in nature can be expressed only after
a time lag, the elasticity of an ecosystem must be independent of the
numerical response. Conformability, however, can be increased con—
siderably by the capability for relatively rapid changes in predator
density. For example, a seasonal increase in "r" may increase
reproduction by predators, counteracting the rise in initial slope of
the prey increase function with higher "r". But as implied by the
three categories above, this enhanced conformability depends critically
on the magnitude of time lags in the numerical response: any negative
feedback expressed only after a lag that is long relative to the

natural time scale of the system will be destabilizing (May, 1973).

78

Application to Biological Control
and Game Management

 

The purpose of biological control is to confine.a pest population
below an ”economic threshold” density, whereas in game management a
relatively high, constant density of the game population is usually
desirable. Yet both can require stabilizing a predator—prey interaction
in the face of environmental perturbations, often in structurally
heterogeneous environments. The theroetical analysis in this paper shows
that with some knowledge of the frequency distribution of environmental
perturbations as a function of type (i.e., shocks or shifts) and
amplitude, system parameters can be adjusted to reduce the risk of
extinction or escape or alter stable equilibrium density (see Summary of
Theoretical Results).

Suppose, for example, that a game bird population is under such
heavy predation pressure from hunters and other mammalian populations
that a hard winter could cause local extinction. A program calling for
enrichment (supplementary food to increase ”r" and ”k"), barrier con-
struction (fences to reduce l'e" and exclude predators from refuges),
and additional refuge space (increasing "h”) could be designed/to;.
greatly reduce the chance of extinction. Similarly, for an insect pest
with a tendency for seasonal outbreaks, direct predator stocking
(increasing "p”), control of alternate prey, and limiting refuge space
(a special crop variety to decrease "h”) may prevent outbreaks or at
least reduce their frequency. Note, however, that overcompensation
could also be disastrous; for the pest example, extinction or near
extinction of the pest may drastically reduce or eliminate any highly

specific predator, facilitating escape by the pest after a rebound or

79

reintroduction. Thus the stability effects of projected parameter
adjustments must be quantified, perhaps using techniques like those
outlined here, if ecosystem management is to progress much beyond

ecological trial-and-error.

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APPENDIX A

DISC EXPERIMENTS

APPENDIX A

DISC EXPERIMENTS

When Holling (1959) derived equation (1), he attempted to
validate it experimentally with his blindfolded secretary "preying"
on sandpaper discs thumbtacked to the tope of a small table. Using
estimates of handling time and successful search rate obtained from
earlier results, he drew a theoretical curve like Figure l which the
data from his disc experiments fit very well.

Rogers (1972) performed similar experiments with polyethylene
discs that were either removed by the predator (to simulate predation)
or marked and replaced (to stimulate parasitism). He showed
theoretically that when exploitation is taken into account——as it
must be if prey are removed by the predator faster than they are
replaced during the experiment——the randomly searching predator must
actually have a higher successful search rate than Holling had
supposed to fit data from this type of experiment. Since his predator
apparently removed discs without replacing them, Holling's estimate of
the successful search rate must have been automatically biased downward
by this exploitation effect.

In order to obtain experimental support for the equations and
curves of Figure 4, I have repeated and modified the disc experiments

of Holling (1959). But to avoid the exploitation effects considered

87

88

by Rogers (1972) and to obtain a truly instantaneous response, I
replaced the discs as the predator removed them. The predator, a
blindfolded lab technician, searched a circular bulletin board one
meter in diameter by probing with a fingertip.

Three separate sets of experiments were performed:

1. Holling's (1959) experiments were repeated, using discs of
coarse emergy paper 5 cm in diameter thumbtacked to the board. The
number of discs picked up in three minutes was observed at six
densities presented to the predator in random order: 10, 20, 40, 70,
110, and 160 discs per 7854 cm2. Three replicates were run at each
density.

2. Additional experiments were run with 5 cm discs as prey, but in
a universe containing a single refuge having a capacity of 20 prey.
The refuge space, a 90° wedge containing a fourth of the total area,
was bounded by a thin aluminum strip taped to the board, and the
predator was restricted to hunting outside the refuge only. The refuge
occupancy quotient ”q” was arbitrarily set at 0.667; dice were thrown
to determine the actual distribution of prey inside and outside the
refuge. Six replicates were run at each of the six densities.

3. In the third set of experiments, the universe again contained
the refuge of capacity 20, but the predator was told to hunt only
inside the refuge. Prey were 2.5 cm discs, for which the rate of
successful attack was about one third that for 5 cm discs. As in
experiment 2, a ”q" of 0.667 was simulated using dice, and six

replicates were performed at each of the six densities.

89

At various times between experimental runs, handling times for
both large and small discs, tap rate, and effective fingertip radius
were measured independently of the experimental procedure. Disc
handling, clocked to the nearest 0.1 sec with a stopwatch, averaged
2.37 :_0.05 sec for the 5 cm discs and 2.49 i 0.06 sec for the 2.5 cm
discs; the difference between these means is not statistically
significant (t-test, 58 d.f., 0.1 < P < 0.2). Tap rate, counted for one
minute on three separate occasions, averaged 62 :_2.0 taps per minute.
And effective tap radius, obtained by inking the predator's finger
before she tapped around the perimter of a disc, measured 0.5 i about
0.05 cm.

From these independent measurements of handling times, tap rate,
and fingertip radius, the parameters of equations (4) were evaluated
(see Table 2) and the curves of Figures 18 and 19 were drawn. Data
points :_2 S.E. from experiment 1 are plotted in Figure 18; data points
:_2 S.E. from experiments 2 and 3 are plotted in Figure 19. Note that
the error intervals include the theoretical curves in all cases except
two data points from experiment 2. Only one of these points (density
160) has a 95% confidence interval which does not contain the curve.
Careful observation of the predator in action suggests that handling
time may actually have decreased slightly at this highest density, at
which most taps touched a disc-~see also density 160 in Figure 18.

These results provide an initial validation of equations (4) in
a simple, generalized predator-prey system. Since derivation of
equations (4), parameterization of these equations, and gathering of

eXperimental data were all accomplished independently of each other,

90

Table 2.--Parameters from the disc experiments.

 

parameter

magnitude units

calculation

 

a, rate of
successful
attack
outside

a, rate of
successful
attack
inSide;

b, handling
time for
5 cm discs

2, handling
time for
2.5 cm discs

h, refuge
carrying
capacity

q, refuge
occupancy
quotient

u, area of
experimental
universe

v, area .
ratio
outside

3, area
ratio
inside

0.670

0.228

0.0132

0.0138

20

0.667

7854

1.333

7854 cm2

(3min)-1
7854 cm2
(3min)'1

. . -l
3 min disc
. . -1
3 min disc
discs
(7854 cm2)-1
dimensionless
cm

dimensionless

dimensionless

(effective area of ta )(taps
per 3 min) = (n (d+f) )(3g)
where d is large disc radius,
cm.f is effective finger-
tip radius, cm,
and g is tap rate, taps
min“

(effective area of tap)(taps

per 3 min) = (n(d'+f) )(3g)

where d' is small disc radius,
cm

direct measurement
[l/b = 75.8 discs (3 min)-1]

direct measurement
[l/b_= 72.5 discs
(3 min)‘1]

arbitrarily predetermined

die rolled for each disc:
1,2,3, or 4 inside--5 or 6
outside, with a maximum of
20 discs inside

arbitrarily predetermined

total area of experimental
universe e area outside
refuge

total area of experimental
universe % area inside refuge

 

 

 

 

Figure 18.

91

The functional response of a lab technician to the density
of sandpaper discs thumbtacked to a bulletin board (no
refuge space). Data points are means of three replicates
bracketed + 2 standard errors; parameters of the curve
drawn throfigh these points were evaluated independently

of experimental data.

 

92

80

l/b _____________________________

70—

60"

50‘

40“

30—

_ aH _ 0.67H
‘ l+abH ‘ 1+o.ooee4n

 

F, functional response
discs picked up by predator in 3 min

20—

IU—

 

 

1 I I I J
0 O 20 4O 70 110 160

H, prey density, discs per 7854 cm2

Figure 18

Figure 19.

93

The functional response of a lab technician to the density
of sandpaper discs thumbtacked to a bulletin board (with
refuge space). The open-circle points are means of six
replicates in which the predator hunted only inside a
wedge-shaped refuge; the solid points are means of six
other replicates in which the predator hunted only outside.
Error intervals bracket i.2 standard errors. Parameters
of the curves drawn through the points were evaluated
independently of experimental data.

94

 

 

 

 

 

eo-
l/b - ———————————————————————————
1/3 - -- — — — ————————————————————————
7
04 inside outside
v H
E? a q _ 0.608H F: av(i-q)a = 0.2973
1+3!)qu 1+0.00839H 1+abv(1-q)H 1+0.0039zn When 091330
60"
avh av(H-h) 0 893(3-20)
= —--—-— = 14.57 F: —-————-— = '
c 5 l+abvh l+abv(H-h) l+0.0118(H-20) "he“ "330
’2
m
3 c 50‘
8'”
O. H
m 0
0 JJ
H ‘6
1:. E
'3 .S‘
O 04
5 :1
'H
38
mhx 30‘
O
p-a
I: 0:
III
0‘)
In 0
U)
-.-I
'0 zo-
inside

   

 

 

 

l I l I I
0 10 2.0 1‘0 48 I 70 110 160

3‘
:L‘

Hm
31

H, prey density, discs per 7854 cm2

Figure 19

95

there is no cause to suspect accident or artifact in the reasonable
fits of curves and data in Figures 18 and 19.

The predator was not allowed to choose whether to hunt inside,
outside, or both during an experimental run. Such "free choice"
experiments, though they may more realistically reflect options open to
predators in nature, would in this case yield results confounded by
such extraneous factors as the duration of experimental runs and the
motives of the human predator. An alternative approach to the question
of spatial allocation of hunting effort--the approach assumed in these
experiments and in the theoretical deve10pment--is to measure the
functional response inside and outside separately; an overall functional
response can then be obtained as an average weighted by the proportions
of predators in each area or by the proportion of the exposure time
spent per area by each predator.

If the disc population were increasing at a per capita rate
related to prey density by a known function, then some of the stability
properties of this predator-prey system could be calculated from
equations (24)-(27) or equations (24), (25), and (29). Even without
knowing the function R(H), however, g, H, H, and C can be found for
separate cases in which the predator hunts only outside or the predator

hunts wherever the functional response is highest. See Table 3.

96

Table 3.--Stability variables from the disc experiments.

 

 

variable magnitude units calculation
H, lower boundary of the 30 discs equation (18)
Eonformable interval for a (7854 cm2)'1
predator restricted to the
outside; also h/q, the slope
discontinuity
g, the intersection density 40.3 discs equation (6)

(7854 cm2)-1

H, upper boundary of the 61.1 discs equation (20)
conformable interval (7854 cmz)‘1
C, conformable interval (0.226, 7854 cm2 equation (26)
for a predator restricted to 0.405) (predator-
the outside 3min)-1
C, conformable interval (0,297 7854 cm2 equation (28)
for a predator hunting prey 0.405) (predator-
with q that decreases 3min)“1

significantly with density

 

 

 

 

"mmmmm