A stochastic theory of search: Implications for predator-prey situations

MATHEMATICAL

BIOSCIENCES

A Stochastic Theory Situations

105

12 (1971), 105-132

of Search: Implications

for Predator-Prey

JYRI E. PALOHEIMO Department of Mathematics and Zoology, University of Toronto, Toronto, Ontario, Canada Communicated

by K. E. F. Watt

ABSTRACT A stochastic model is developed to study the effect of density and distribution of prey on total accumulated catch by the predator. The model is applicable to search by a predator for prey when the latter are randomly located on a plane with given density, clustered with variable cluster size, or schooling and schools can be represented by disks of variable size. Allowance is made for possible escapement of prey. Capturing or delay time for each sighting or catch made can be taken as variable and dependent on the number of prey caught. The radius of detection of prey by the predator may also be fixed or variable. Examples are given for randomly distributed, clustered, and schooling prey.

1. INTRODUCTION

Stochastic models are developed to study the success of a predator in its search for prey or of a fishing vessel in its search for fish. Earlier studies have primarily been restricted to the special case when the prey species are randomly (Poisson) distributed. Typically they have not allowed for delay time for each prey caught; moreover, the models have been deterministic (e.g., models of Baranov [I], Nicholson and Bailey [2], Lotka [3], and Ricker [4, 51. This leads to a catch with the mean proportional to the exploited population. The probabilistic nature of the Poisson searching process has been considered by Neyman [6] in studying the relationship between the catch of herring and numbers of fishing vessels; by Koopman [7] in connection with searching for enemy vessels or submarines; and by Skellam [S] in the effort to estimate the size of a population by a series of random transects. Neyman extended the basic model by introducing a constant delay time for each successful catch made. Rashevsky [9] considered predation on contagiously distributed prey in essentially nonprobabilistic fashion but was led to the somewhat erroneous conclusion that the catch would be less from a clustered than from a randomly distributed population. Copyright 0 1971 by American Elsevier Publishing Company, Inc.

106

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E. PALOHEIMO

The models introduced here are applicable to search by a predator for prey when the latter are randomly located with a given density, when they are clustered with variable cluster size, or when they are schooling and the schools can be represented by disks of variable size. Allowance is made for possible escapement of prey. Capturing or delay time for each sighting or catch made can be variable and may depend on the numbers of prey caught or on the size of the school. The radius of perception of the predator may be fixed or variable. The distribution of prey animals is considered essentially stationary, and their removal by the predator is ignored. This restriction allows us to reduce the predator-prey relationship to a renewal type of problem. The extension of the study to a nonstationary situation when the predation changes the distribution of prey (i.e., distribution in the statistical sense) is not considered in this article. The models have been developed as if the prey were more or less immobile. When the average speed of travel of prey in relation to that of the predator is appreciable, some modifications in the formulas are required. Search for moving random points has been considered by Koopman [7] and Skellam [8]. The search for mobile prey can be reduced to the case considered here (i.e., immobile prey) by replacing the speed of the predator with the vector sum of the speeds of the prey and the predator. However, we will not consider this extension in detail. A number of empirical studies have been made of predator-prey interactions, notably by Helling [IO]. Holling has summarized his work in a series of computer models. Some of the aspects have been formulated mathematically by Watt [ll]. The present article is seen as a complement to their studies in that the formulation of the predator-prey interaction is stochastic and examines the component that hitherto has not been well understood, namely, the effect of contagiously distributed prey. 2.

SEARCHING

MODEL

In an earlier article [ 121, we formulated a general problem of searching by what we term here a predator for randomly or contagiously distributed points, called prey, on a plane. The process was specified by defining the distribution of the searching time to locate a prey or a school or cluster of prey and the joint distribution of handling time and numbers caught from each cluster sighted. To quote, we define F(t) = the probability that the predator has sighted a prey or school of prey and reached its location in time t given that the predator began or resumed its search at time t = 0; G(t,x) = probability generating function for the number of animals

A STOCHASTIC PREDATOR-PREY MODEL

107

caught in time t from a school or a cluster the location of which was reached at time t = 0; more specifically, the coefficient xk, let us say, gk(r), denotes the probability that k prey were caught from the school and that the predator has resumed its search after elapse of time t. In the foregoing F(t) denotes the recurrent searching time distribution and G(t, x) the joint distribution between numbers caught and the time required for the operation. The catch of the predator depends on the distribution of prey. Prey distributions considered are random (Poisson) distributions, and two general types of contagious distributions, namely, an example of a doubly stochastic Poisson distribution and a general class of clustering distributions. For a more detailed treatment of doubly stochastic Poisson and clustering processes, see [ 121. When the individuals follow a Poisson distribution with the density i,, the probability of an individual in a small square dx is I dx, and the probability of n individuals in an area S is (nS)n (exp -1S)/n!. The distribution between the nearest neighbors is also well known and has the probability density function (exp -h2)2hr dr where r refers to the distance. In an ecological context this function was given previously by Moore [13] and Clark and Evans [14]. The’mean distance betweenlthe nearest neighbors is found to be 1/(2JA). F(t) depends on the distribution of prey and on the searching practices of the predator. In the simplest case we assume that the prey are immobile, and that their distribution is Poisson with a mean density i. The predator is assumed to move at a constant speed, which for convenience we scale to be equal to one; hence, speed = time, which we denote by t. The radius of detection is also assumed to be constant and equal to R. The searching time distribution can now be written as dF(t) = exp { -A[anR* + 2R(t - R)]}2RA dt (1) where a takes the values ranging from 0 to 1 depending on the degree to which the prey avoid the area around the predator when it is making its kill or just resting afterward; a = 1 when there is no avoidance within the radius R, and a = 0 when there is a complete avoidance (cf. derivation in [12]). When the probability of a prey within the distance t < R is small (1) may be approximated by dF(t) = exp (-21Rt)2RA dt. (2) The last formula is a sufficiently good approximation whenever the probability is small that the distance between the nearest neighbors s less than the radius of detection R. If the prey that came within the radiu s of detection of the predator are detected only with a probability, let us say, P, then (2) can be written as dF(t) = exp (- 211PRt)2RAP dt.

(3)

108

JYRI

E. PALOHEIMO

The last formula

implies that if the prey are detected with a probability time distribution is equivalent to the searching calculated assuming a prey density AP instead of the

P < 1, then the searching

time distribution actual density I.. Three important extensions in the applicability of F(t) can be readily made. Suppose in the first place that the prey are distributed in diskshaped schools that are dense enough so that the predator can detect the school by detecting the boundary of it. Let 1. refer to the Poisson density of the numbers of schools, and r to the radius of a school; r is assumed to be random variable with a cumulative distribution function K(r). It can be shown that the previous formulas apply if the radius of detection R is replaced by the sum of the radii of detection and the mean school radius, that is, by R + f and ? is calculated

from r =

J r&(r).

The probability

of

detecting a school of radius r is now equal to the probability of detecting a school times the probability that the school detected has the radius r; that is, probability of detecting = z$ nF(1) dK(r). a school of radius I These formulas will be applied to a case when the distribution of prey consists of randomly placed disks of variable radius and within each disk the individual prey follows a Poisson distribution with density, say, ,M. Such a distribution is an example of a doubly stochastic Poisson process (cf. [12]). If the area covered by a school is not circular, then it cannot be described by a fixed radius. Any convex figure, however, may be thought of as having a center point (e.g., the point of gravity). For any given orientation of the convex area we define the support of the area as the distance from the center point to the tangent, or to the line touching the extreme corner in the direction of the path of the predator. The support so defined is now the function of the orientation of the convex figure and for any given shape it has now a well-defined distribution H(O) d0/2n where 0 is the angle of orientation. For the purpose of calculating the probability of detection we may take ? = +n

J H(O) c/o

as the mean radius. It is also fairly obvious what additional modifications are called for if the shape or the size of the area is subject to further variations. A second extension of the searching time distribution can be made when the distribution of prey can be described by a clustering process. These have been studied previously by Neyman [15], Neyman and Scott [ 161, Thompson [ 17, I 81 and Bartlett [ 191. Clustering processes are obtained

A STOCHASTIC

PREDATOR-PREY

109

MODEL

by assuming a primary Poisson process giving rise to cluster centers and a secondary process specifying numbers of individuals per cluster and their distribution around the cluster center. An important special case is obtained when the numbers of individuals per cluster follow a geometric series distribution with a parameter p and when the individuals around the cluster center are spread according to the bivariate normal law. Sampling such a spatial distribution by counting numbers of individuals in small randomly located quadrants will give rise to a negative binomial distribution (cf. [ 171). When the prey are distributed in dense disk-shaped schools, F(t) can be modified by replacing the radius of detection R by R + F, the combined radius of detection and mean school radius. The probability of detecting of a school in time dt is now i(R -I- F) dt. A similar procedure can be followed when the prey are clustered, with the exception that the probability of detection of a cluster is based on the probability of detecting any one of the members of the cluster. When the prey are distributed in clusters we put P(/z) = the probability of detecting an individual from a cluster the center of which is at a distance h from the path of travel of the predator. The probability P(h) depends on the distribution of the individuals around the cluster center. If there are n individuals in a cluster, then the probability that at least one of them will be detected is equal to 1 - (1 - P(lz))“; hence, we get P(n) = probability

of detecting

a cluster of size n in time & (5)

=

no [l -(l

-P(h))“]

dh,

s0 since h has a uniform distribution. (P(n) as given here P(n) in Eq. (3.13) of [12], where r, = maximum cluster D. S. Robson (personal communication) pointed out, assume a maximum cluster radius r,.) Assuming now that the distribution of individuals center follows a bivariate, symmetrical normal law then P(h) is given by

is equal to (R + r,J radius. However, as it is not necessary to around the cluster with a variance 02,

(6) where @ refers to the standardized cumulative normal distribution function. Let p(n) = probability of a cluster of size n. When the numbers of prey per cluster follow a geometric distribution with a parameter p, then P(4 8

= (1 - p)cL”.

110

JYRI E. PALOHEIMO

Hence P = probability

of detecting

a ciuster in $

= c p(n)P(n> (7) We note that our definition admits the possibility of an empty cluster (i.e., p(O) = 1 - JL). From (5), however, we see that the probability of detecting an empty cluster is also zero. The third extension in the basic definition of the searching time distribution is arrived at when the speed of the search is assumed to vary. Let a(t) be the speed of the predator. Normally u(t) would be a random function. For any given realization of it the approximation (2) for F(t) would be replaced by (P(r)

= [ exp( - 2AR 11 z,(t) & j]ZiRI;(r)

dr.

(2’)

When z)(t) is a random function, as opposed to being constant, the mean searching time and its variance are different from those given later on. However, the only modification in our model called for when z>(t)is a random function is the substitution of the correct moments of the searching time in place of the ones given for the simpler case. The joint distribution of the delay time and the number of animals caught from a cluster sighted, denoted earlier by G(t, x), depends on both the distribution of prey and the behavior of the predator. Specific examples are considered in later sections. Here we consider the simple case of the predator searching for randomly located prey. We assume that the predator’s behavior can be described by specifying pC = probability

of capturing

~7~= 1 - pC = probability

the prey sighted;

that the prey escapes;

G,(t) = probability that the predator resumes its search in time t after sighting a prey that eventually escapes; the location of the prey at the time of sighting was reached at time t = 0; G,(t) = probability that the predator resumes its search in time t after sighting a prey that is eventually caught; the location of the prey at the time of sighting was reached at time t = 0. Somewhat fact that the from the time the prey was

complicated definitions of G,(t) and G,(t) arise from the searching time distribution F(t) includes the time elapsed of sighting to the time required to reach the location where sighted. This complication is somewhat academic, and by

A STOCHASTIC

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MODEL

subtracting the radius of detection (properly scaled) from t defined by F(t), G,(t) and G,(t) could more simply be defined as the distribution of time from the sighting of the prey to the resumption of the search for a new prey. Let the powers of x stand for the number

of animals

caught and let

G(t, x) = qcG,(t) + ~cGl(t); (8) G(t, X) is then the joint probability generating function for the delay (handling) time and the number of animals caught. We denote by * the convolution of two functions. Then F(t) * G(t, 1) where t F(t) * G(t, 1) = F(t - u) dG(u, 1) s0 gives the total time involved in both searching and catching operations; that is, F(t) * G(t, 1) (i.e., with x = 1) is the probability that the predator, having started its search at time t = 0, has resumed its search for a further prey or school of prey in time t. Moreover, F(t) * G(t, x) is the probability generating function of the predator having caught k prey (in (8), k = 0 or k = 1) and resumed its search in time 1. Assume now that once the predator resumes its search, the process starts from the beginning. Hence, the predation is a renewal-type process or, if our interest is in the total catch by the predator, a cumulative renewal process. Hence, the methods of the renewal theory apply (cf. [20]). The successive renewal points are the times when the predator resumes its search. Besides the total numbers of animals caught in a given time, we may also be interested in the total number of schools or clusters sighted. The exact probabilities for numbers of prey caught or numbers of schools or clusters sighted can be obtained from equations given in 1121. In practical applications the mean catch (numbers or schools) and its variance are of more interest. They can be expressed in terms of the moments of the searching and catching time distributions. The moments of F(t) * G(t, 1) are denoted by pi and those of G(t, x) by vii. More specifically, PI =

J

tF(t) * G(t, 1) dt = mean total (searching

J

t”F((t) * G(t, 1) dt = second moment

and delay) time per

sighting, p”z =

v0 1 = G:(co, X)~=~ = mean number

of the total time,

of prey caught per cluster,

vo2 = G&(co, x),=1 + vo1 = second moment of the number caught per cluster, VlO =

J t d,G(t,

1) = mean delay time,

of prey

112

JYRL E. PALOHEIMO V 20

=

V 11

=

I

s

d,G(t, 1) = second

t2

t

d,G:(t,

moment

x),=~ = joint

moment caught.

of the delay time, of the delay time and numbers

We denote by M,(t) the mean number of clusters or schools exploited in time t; Var,(t) the variance of the number of clusters or schools exploited in time t; M,.(t) the mean number of prey caught in time t; and Var,(t) the variance of the number of prey caught in time f. It can now be shown that the mean numbe1.s of clusters or prey (i.e., M,(f) and n//,(t)) and their variances can be expressed in terms of the moments k(r, pz, vo,, . . ., vl, : (9) M

_(t) =

x

?!! + & _ !“-:!! I’

Varz(t)

where o(l)+

1

2/d

ill

\‘o1+0(1),

>

(10)

= “‘;;l,l’:t+o(l),

0 as f+

co.

Furthermore, p, and p2 (i.e., the moments of the total time involved in both searching and catching operations) can be expressed in terms of the moments of the component processes F(t) and G(t, x). Hence, once the searching time distribution and the delay or catching time are specified, the foregoing formulas give the mean catch and its variance, in terms of either individual prey or schools of prey. The formulas for the mean catch, (9) and (10): show that the common sense formula, mean catch = [total time/(time required per school)] multiplied by mean number per school (in (IO)), is almost an unbiased estimate of the expected total catch. The bias resulting by using the obvious estimator is independent of total time t; hence. its significance decreases as we consider the catch over a longer time interval. The mean number of prey caught and its variance, expressed in (IO) and (12) respectively, can be interpreted readily in terms of the number of clusters or schools sighted (9) and its variance (11). The mean number caught is the mean number of schools sighted multiplied by the mean number caught per school; the variance of the numbers caught consists of three terms: the first component expresses the variance of the numbers caught per cluster; the second component the variance of the searching time; and the third component what might be considered as the covariance of the capturing time and the numbers caught per school.

A STOCHASTIC

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MODEL

Potentially of more interest are the formulas given for the variances, as they can be thought of as a measure of the uncertainties faced by a predator in a given situation. When comparing different predator species, such measures of uncertainty must obviously be scaled by the predator’s food requirements. When comparing different prey species of a given predator, the uncertainties must be scaled by the food value of each prey species. 3. APPLICATIONS The general formulation given in the preceding section may be employed to study a variety of different aspects of predation. In this section we give explicit formulas for the mean searching time and its variance. These formulas are then used to calculate the mean catch and variance for the random or Poisson search with a delay time. The effect of clustering or schooling is illustrated for clustering and doubly stochastic Poisson distributions of prey. Examples considered show that the catch is very much dependent on the distribution of prey and not only on its mean abundance. Applications to fisheries have been considered elsewhere [21]. a. Calculation of Searching

Time

The distribution of the searching time or distance, these two being equivalent when the speed of travel was assumed constant and equal to 1, is given in (1) for randomly located prey. In practice the distribution may often be approximated by Eq. (2) (i.e., S(t) = (exp -2ARt)2RA dt). Denoting the mean distance by Y1 and its second moment by Yz, that is, Y, =

we

Y’, = m t S(t), s0 get, using the approximate formula (2),

Y’, = 2&

Yy,-YT

= mean searching

1 = ___ = variance 4J2R2

m

tZ

S(t),

s 0

time,

of the searching

(13) time.

(14)

We note that the variance equals the square of the mean, a property of the exponential waiting time distribution. When the more exact form (1) of the searching time distribution F(t) is used, we get y = exp( - RaxR’)+ @[R(2,Ian)*] - 4 (15) 1 (La)+ ’ 2AR Y’, = &+

exp(-AazR2)

a7c-1 1 ~ ___ lan +2A2R2 > .

(16)

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JYRI

E. PALOHElMO

It can be shown that when R(2;la7c)+ is small and < 1, that is, when the distance from a prey to its nearest neighbor is large compared with the radius of perception of the predator, we may take

which in turn may be approximated by (13) and (14). When R(2lan)f ti I (i.e., when the distance between the nearest prey neighbor is on the average

TABLE MEAN

I

SEARCHING DISTANCE 0~ TIME(EQ. (15)) FOR TWO

OF U AND APPROXIMATE

MEAN

Mean

SEARCHING

distance

VALUES

DlSTANCE(EQ.( 17))

(22/xY,)

Eq. (15)

22/hR

(I = 1.0

a z 0.51

Eq. (17)

0.010 0.050 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.000 1.100 1.200 1.300 1.400 1.500 1.600 1.700 1.800 1.900 2.000 3.000 4.000

199.998 39.992 19.984 9.968 6.617 4.93 1 3.911 3.225 2.730 2.356 2.066 I.836 I.652 1.504 1.387 1.293 1.220 1.163 1.119 I .086 1.061 1.043 1 .OOl 1.000

200.005 40.026 20.052 10.102 6.817 5.194 4.234 3.603 3.157 2.825 ?.568 2.363 2.195 2.056 1.938 1.838 1.752 1.679 1.617 1.564 1.519 1.482 1.336 1.325

200.000 40.000 20.000 10.000 6.667 5.000 4.000 3.333 2.857 2.500 2.222 2.000 1.818 1.667 1.538 1.429

1.333 1.250 1.176 1.111 1.053 1.000 0.667 0.500

A STOCHASTIC

PREDATOR-PREY

small compared mean distance

with the radius

of perception),

&la)-++

Y’, =

115

MODEL

it can be shown that the

exp( - Aa~R2)(a~ - 1) .

2iarcR

(19)

As R(2?.an)*-+co, this approaches t(k)*, which for a = 1 is the mean distance between the neighboring prey. The mean searching time or distance traveled (15) and approximation (17) to it are tabulated in Table I for a range of values of the radius of perception. The radius of perception R and the mean distance traveled have both been expressed in terms of the average distance between the prey (by replacing R with R2J,? and Y’, with Y12J%). b. Poisson Search with a DeIay Time We assume that the prey are randomly distributed with a Poisson density of ?., that each prey sighted by the predator is caught, and that after each sighting of prey some time elapses (the distribution of which is G(t, 1)) before the search is resumed. Total time from the beginning of the search to the resumption of it for the next prey and its variance are obtained from the moments of F(t) * G(t, 1). The moments of F * G can be written in terms of the moments of F and G: Pl

=

Yl

+

Pz

VlO,

=

y2

+

v20

+

2~lVlO

(20)

where v 1o is the mean delay time per prey caught and v20 is its second moment. If the prey population is not dense, so that approximations (13) and (14) are adequate approximations to ‘Pi and Y2, the mean catch and its variance are given (since vol = vo2 = 1) by 2ARt M,(t) Var,(t)

= =

1

+2ARvlo

+.

2A2R2v 20 (1+2;1Rv,,)’ (21)

21R + 8L3R3(v2, - v:~)~ (1 +~ARv,,)~

’

If (21) are used in an attempt to infer from the catches made by the predator or by a fishing vessel the prey population abundance 1,, we note that I appears only in the product AR in the equations for both the mean catch and its variance. Hence, changes in the prey abundance cannot be separated from changes in the radius of perception of the predator by study of the catch statistics and its variance. This was first observed by Neyman [6] by numerical calculations based on the actual probability distribution of the catch.

JYRI

116

E. PALOHEIMO

If the predator fails to catch some of the prey sighted, then the delay time distribution is given in (8) and its moments are vo1 = PO vi0 = q,s’+p,r;

VO2

=

PC

‘I1 1 =

T’20

=

qc4

PCT,

(22) +Pczz>

where pc is the probability of capture of a prey sighted; q, = 1 - p,; r’ and r; are the first and second moments of the delay time when the prey sighted is not caught; and z and r2 are the corresponding moments when the prey sighted is caught. The dominant (first) term of the mean catch and the variance of the mean catch are

Var,(t)

=

2Rp,q,(l ~~--- -~

c. Effect of Clustering;

+21Rv,o)2+2~R+8~3R3(~~20-~,;)

__~

(1 +~I,Rv,,)~

Probability qf Detection

Constant

In this section we assume that the prey are clustered but that the clusters are randomly distributed. The clusters are assumed to be compact so that the detection of a cluster does not depend on the size of the cluster. Let 3. denote the Poisson density of cluster centers and let p(rz) = the probability of a cluster of size II. To study the effect of clustering, we let /1 and the distribution of numbers per cluster vary in such a manner that A times the mean cluster size is constant; that is, i.Cnp(n) = constant

If the predator written

picks up all the members

= d.

(23)

of the school, this may also be

Av01 -d. -

(24)

The delay time is assumed to consist of two terms: a general term specifying the delay per cluster sighted, and a term that increases proportionally with the number of prey caught. Mathematically this is achieved by convoluting a distribution G,(t) having a mean z’ and representing the time loss per cluster with a k-fold convolution of G,(t), where G,(t) has a mean z and represents the time lost by the predator on each prey caught and where k is the number of prey in the school. In other words, G(t, x) would be given in this case by G(t, x) =

c k>l

p(k)G,(t)

* G,(t) * . . * * G1(t)xk. k terms

(25)

A STOCHASTIC PREDATOR-PREY

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MODEL

To calculate the moments of G(t, X) we take transformation. This is defined as

J

G”(s, x) = s

the Laplace-Stieltjes

e-‘JG(t, x) dt.

(26)

The variable s is a dummy variable introduced for convenience of getting the moments of the distribution. The operation is similar to transforming from a distribution to its moment generating function. The result may be written G”(s, x) = G”o($[Q(&(4) - PKN where Q(X) is the probability generating function animals per cluster, that is, Q(x) = C,“x”p(n). Note that the definitions of some of the well-known butions (e.g., the negative binomial) require a concept In calculating the catching time such schools have to the term p(O) is subtracted in (27) from the probability QbG,

of the numbers

(27) of

contagious distriof an empty school. be excluded; hence, generating function

6)).

of G(t, x) may now be obtained

The moments 1’01 =

Q’(l).

V 02

Q”(l)

=

VI0 = TV

+ -

VI1

=

z’vo1

v20

=

z,(l

-

Q’(l),

Pm +

from (27):

+

VOlT,

+

2rr’vo1 + (vo2 - VO~)Z2+ v,,r,

v02t, p(0))

(28)

where z’ and z; are the first and second moments of G,(t) and 7 and 72 those of G,(t). The moments for the combined searching and catching time distribution F(t) * G(t, 1) were given earlier in (20). When these and the vii’s are substituted in the basic equations (9)-(12), the catch, number of schools, and their variance are obtained. For example, assuming the schools to be far enough apart so that the searching time distribution (2) applies, the mean number of prey caught is given by

M,(t) =

2R dt 1+2ARr’(l

-p(O))f2Rzd

2dv,,lR2 + [I +2ARz’(l

-p(O))

+2Rzd12

(29)

where iv, 1 = d = mean density of prey. Considering the dominant (first) term of the mean catch (29) and ignoring for the time being the term 2ARz’(l - p(O)) (i.e., the delay time for each cluster sighted) in the denominator, we may conclude that when the clusters are compact, the clustering has no effect on the mean catch by the predator (cf. (21)). If, however, the delay time per school sighted is appreciable, then as 1 decreases and the clusters become on an average larger, the catch will increase till it reaches its maximum 2R dt/(l + 2Rzd).

118

JYRI E. PALOHEIMO

d. EfSect of Clustering; Noncompact Clusters In a general case, the probability that the predator detects a cluster depends on the size of the cluster and on how widely it is spread. This is demonstrated by evaluating the catch in time t from a prey population the distribution of which is given by a general clustering process. Using the earlier notation, we put p(k) = probability of a cluster of size h-, P(k) = probability of detecting a cluster of size k, and P = Cp(k)P(k). Assuming further, as in the previous section, that the delay time associated with a cluster consists of a general term plus a term the mean of which goes up linearly with the size of the cluster, we may write G(t, x) = C P(k)p(k)G,(t)

* G,(t) * . . 1 * G,(t);‘.

The P(k)‘s are given in (5) and P in (7). The Laplace-Stieltjes tion of G(t, x) may now be written symbolically

s

-P(h))&(s)))

G”(s,x) = G”,(s) (Q(xC&))-Q(x(1

(30

.

transformadh p

B (30,)

and where Q(x) denotes the probability generating function of numbers of prey per cluster. The moments of G(s, x) may be obtained from (30’) by taking the appropriate derivatives. For the moments with respect to the numbers caught we get P” vo2 = - +v,, P

P’ vo1 = -7 P where we have used the symbolically P’ =

suggestive

(31)

notation

; [Q’(l)-Q’(l-P(h))(l-P(h))]

dh,

s P” =

s

; [Q”(l)-Q”(l-P(h))(l

-P(k))2]

dk,

(32) (33)

and Q’(1) and Q”(1) are the derivatives of the probability generating function Q(x) at x = 1. The moments with respect to time (i.e., vi0 and vZo) and the mixed moment vrr may now be written in terms of \rol and voZ. We have V ,o

=

7’

VI1

=

Volt’

v20

=

5;

+

VOIT, +

+

vo2,

2v,,zr’

+ (voz - v&’

To determine the effect of clustering stant and put, analogously to (23),

+ voiz*.

(34)

we keep the overall density

con-

I.Q’( I) = constant

= d.

(35)

A STOCHASTIC PREDATOR-PREY

The dominant

119

MODEL

term of the mean catch may now be calculated 2dP’t

Mx(t) N [(2~P)-1v~:‘+v,I~3 Example.

For illustration

= Q’(l)+2d(Pz’+P’z)’

we consider

the distribution

from (36)

of prey defined

by (1-p)

Qb) = (l-px) and

Q(x) being the normal (error) integral. Thompson [17] has shown that for small sampling plots the definition above gives rise to a negative binomial distribution of the numbers of individuals. Using Eq. (7) we get for P’ and P” in (33)

(37)

The integration in the foregoing must be done numerically. Although the range is from 0 to co, sufficient accuracy is obtained by integrating from 0 to 20. In substituting any numerical values in the formulas, two conditions must be taken into account. In the first place we recall that in the model the predator exploits the prey species one cluster at a time; hence when the clusters overlap the situation is somewhat artificial. An occasional overlap cannot be avoided, but we may minimize it to the point that it becomes negligible. To this end we think of schools as disks of radius r,,, = 2a and require that no more than 10 % of the disks overlap. This condition is met when Prob

no cluster centers within 4 radii from a given cluster center = exp(-16&r’)

> 0.90

or

0.09 u < -, 2Jn

that is, when the school radii are less than 9% of the mean distance between the school centers. Second, the predator’s radius of detection R should be large enough n relation to the radius of the prey clusters for all of the members of the

120

JYRI

cluster to be picked up by the predator. assumed, somewhat arbitrarily, that R32a where 20142

is somewhat

greater

To ensure

E. PALOHEIMO

this we have simply

2+ (39)

0n than

the mean distance

between

two

prey animals (aJrc/2) and n is the mean number of prey per cluster. The treatment of a more general case where R also takes values less than 20(2/n)+ has not been considered. This would involve calculating the probabilities rk,n of picking k members from a school of size n by use of Monte Carlo methods. The probability generating function for the numbers caught in a given time G(t, x) in (30) would have to be replaced by G(t,

x) = 1 P(n)p(n)r,,,G,(t)

* G,(t) * * . * * G,(I)$.

(30”)

k,n

This modification would also apply when the predator picks up only k prey out of the total n in the cluster. A simple example of this is given in the next section where a predator with what might be called a limited storage capacity preying upon compact clusters of prey is considered. With these comments in mind, Table II has been prepared. It lists the mean catch per cluster (i.e., vol, Eqs. (31) and (32)) and mean distance traveled by the predator between successive clusters detected (Eq. (15) with 1, replaced by iP for a series of values of A, ,n/(l - ,Y), R, and CTsatisfying (38) and (39). Table II can be extended for other combinations of values of R and G by noting that the catch per cluster (for a given value of p) is invariant for any constant ratio of R/o. Similarly, the figures for the mean distance may also be extended, but only when the approximate formula for the mean searching time (2) is applicable; that is, when 2(IP)+(R

+ 20) < 0.4,

(40)

say (cf. Table I), in which case the error occurring in the mean searching time is less than 5 “/o. Then the mean distance traveled for any multiples of b of R and 0 (i.e., for bR and bo), is obtained by multiplying the entry corresponding to R and 0 in Table II by l/b. Catch per searching time is determined by dividing the mean catch per cluster in Table II by the corresponding mean distance. The catch per searching time is plotted in Fig. 1 against the radius of detection and in Fig. 2 against the cluster radius for different combinations of the cluster radius (Fig. I), the radius of detection (Fig. Z), and the mean school size. Not shown in Fig. 1 and 2 is the plot of the catch per searching time against the mean cluster size. Inspection of the catch per searching time ratios obtained from Table II shows, however, that this is more or less independent of the mean cluster size when the cluster radius is fixed except

A STOCHASTIC

PREDATOR-PREY

121

MODEL

1.58 6t

.831 t

,,, .251. a F

.

I

:

:

.125

!

!

.5 RADIUS

:

$

,

OF

,

8

2 DETECTION

!

,

128

32 (I?)

FIG. 1. Catch per searching time plotted against radius of detection (R)for various combinations of school radius (20) and mcan school size (m = p/(1 - p)).

a z

.063-

z s

R

,” ,026 UY

1

M2

.044

.088 SCHOOL

.177 RADIUS

.354

,707

1.41

2.83

(2~)

2. Catch per searching time plotted against school radius (20) for various combinations of radius of detection (R)and mean school size (m = p/(1 - p)). FIG.

II

CATCH

1.00 1.41 2.00 2.83 4.00 5.66 8.00 11.31

R/Za

COMBINATIONS

MEAN

TABLE

SCHOOL

AND

3.40 3.31 3.23 3.17 3.12 3.09 3.06 3.05

Mean catch

MEAN

2373.8 1798.1 1339.0 983.8 715.4 516.3 370.5 264.8

0.03 1

Mean

SCHOOL

BETWEEN

2 h

p)),

1186.9 899.1 669.6 492.0 357.8 258.3 185.4 132.7 593.6 449.7 334.9 246.1 179.1 129.4 93.1 66.8

296.9 225.0 167.7 123.4 90.0 65.2 47.2 34.3

WHEN

0.71 1.00 1.41 2.00 2.83 4.00 5.66 8.00 11.31 16.00

R:~G

AND RADIUS

CENTERS

(20),

SCHOOL RADIUS

with school radius _______ 0.062 0.125 0.250

distance

size :

-

DISTANCE

SIZE (p/(1

A. Mean school

OF MEAN SCHOOL

PER

PREY

5.87 5.68 5.53 5.40 5.29 5.22 5.16 5.12 5.08 5.06

Mean catch

a

--__

3205.1 2542.7 1969.9 1493.7 1078.9 818.5 595.5 429.9 308.7 220.8

801.4 635.8 492.6 373.6 269.9 205.0 149.3 108.1 78.0 56.4

0.177

with school

1602.6 1371.4 984.9 746.9 539.5 409.4 297.9 215.2 154.7 110.9

0.088

Mean distance 0.044

BINOMIAL

size = 4 c

IS A NEGATIVE

B. Mean school

(R)

DISTRIBUTION

OF PERCEPTION

400.9 318.1 246.6 187.2 135.4 103.0 75.4 55.0 40.3 29.8

0.354

VARIOUS

radius

FOR

2 Z m

10.70 10.39 10.13 9.89 9.69 9.52 9.39 9.28 9.21 9.15 9.11 9.08

0.50 0.71 1.00 1.41 2.00 2.83 4.00 5.66 8.00 11.31 16.00 22.63

* The first coIumu b Range of values c Range of values 4 Range of values e Range of values

%+iY

R/20

2184.4 1818.5 1476.2 1166.6 899.8 680.0 505.4 370.9 269.6 194.7 140.0 100.6

4368.7 3636.8 2952.2 2333.0 1799.4 1359.6 1010.5 741.3 538.5 388.4 278.7 199.4

546.4 455.0 369.5 292.1 225.6 170.8 127.3 94.0 69.1 50.9 37.9 28.7

1092,3 909.4 738.3 583.5 450.2 340.3 253.1 186.0 135.5 98.3 71.3 52.0

to the corresponding 0.43 Q P Q 0.63. 0.53 Q P S: 0.77. 0.62 Q P Q 0.87. 0.71 < P < 0.93.

0.500

0.250

gives the ratio R/20 applicable for the probabiijty of detection, for the probability of detection, for the probability of detection, for the probability of detection,

0.125

0.062

--

_,...~._l__l_

Mean distance with school radius

C. Mean school size = 8 J

---R,lZa 20.01 19.53 19.12 18.74 18.40 18.10 17.&l 17.63 17.46 17.34 17.25 17.18 17.13 17.09

Mean catch

line in the table.

0.35 0.50 0.71 1.00 1.41 2.00 2.83 4.00 5.66 8.00 11.31 16.00 22.63 32.00

~-

I

5985.3 5190.3 4407.9 3642.4 2925.3 2288.3 1749.5 1312.6 970.0 708.6 513.2 369.4 264.8 189.5

0.088

2595.2 2204.0 1821.3 1462.8 1144.3 875.0 656.6 485.4 354.8 257.3 185.7 133.7 96.5

2992.7

0‘177 1496.5 1297.8 1102.2 910.9 731.7 572.5 437.9 328.8 243.4 178.4 130.0 94.7 69.3 51.4

0.354

0.707 748.6 649.2 551.5 455.9 366.4 286.9 219.8 165.5 123.2 91.1 67.5 so.5 38.4 29.9

--

Mean distance with school radius

I). Mean school size = 16 c

.-----~_~__~

F3 w

5

8 $ z

3

Lz

@

t? n

Mean catch

37.95 37.15 36.53 36.01 35.51 35.04 34.61 34.25 33.94 33.70 33.51 33.37 33.27 33.19 33.13 33.10

0.25 0.35 0.50 0.71 1.00 1.41 2.00 2.83 4.00 5.66 8.00 11.31 16.00 22.63 32.00 45.25

8267.5 7398.6 6521 .O 5614.7 4699.1 3819.2 3019.6 2329.9 1761.1 1309.2 960.7 698.2 503.8 361.9 259.3 185.9

0.125

Mean

E. Mean school

II (continued)

R/20

TABLE

4133.9 3699.4 3260.6 2807.5 2349.7 1909.8 1510.0 1165.2 880.9 655.1 481.1 350.1 253.3 182.8 132.2 96.3

0.250

distance 1.ooo

1034.0 925.4 815.8 702.6 588.2 478.4 378.7 292.8 222.2 166.4 123.7 92.0 69.0 52.7 41.2 33.9

2067. I 1849.9 1630.6 1404.0 J175.2 955.3 755.5 583.2 441.3 328.6 242.0 177.0 129.2 94.8 70.4 53.3

radius

0.500

with school

size = 32 S

Mean catch

72.64 71.29 70.30 69.56 68.92 68.31 67.71 67.16 66.67 66.27 65.95 65.70 65.51 65.37 65.26 65.19 65.14 65.12

Rf2n

0.18 0.25 0.35 0.50 0.71 1 .oo 1.41 2.00 2.83 4.00 5.66 8.00 11.31 16.00 22.63 32.00 45.25 64.00 11532.2 10578.7 9595.1 8544.3 7422.3 6264.3 5132.5 4088.0 3174.4 2412.0 1800.9 1325.9 965.9 698.4 502.3 360.3 258.5 186. I

0.177

Mean

F. Mean School

2883.4 2645.0 2399.2 2136.5 1856.1 1566.6 1283.8 1022.8 794.7 604.4 452.1 334.0 244.9 179.2 131.7 98.0 74.3 57.9

0.707

with school

64 rl

5766.3 5289.5 4797.7 4272.3 3711.3 3132.4 2566.5 2044.3 1587.6 1206.6 901.2 664.0 484.4 351.1 253.8 183.7 134.0 99. I

0.354

distance

size :

1442.2 1323.1 1200.2 1068 9 928.8 784.2 643.0 512.8 399.1 304.4 229.0 170.8 127.4 95.9 73.3 57.4 47.4 43.3

1.414

___-

radius

Q

0.13 0.18 0.25 0.35 0.50 0.71 1.00 1.41 2.00 2.83 4.00 5.66 8.00 11.31 16.00 22.63 32.00 45.25 64.00 90.51

16214.2 15179.5 14084.9 12879.9 11539.6 10076.7 8548.3 7039.0 5632.7 4391.8 3348.7 2507.1 1850.2 1350.7 978.3 704.8 506.3 363.5 261.9 190.3

0.250

2.000

-i

0.09

R/20

0.13 0.18 0.25 0.35 0.50 0.71 1.00 1.41 2.00 2.83 4.00 5.66 8.00 11.31 16.00 22.63 32.00 45.25 64.00 90.51 128.00

P < 0.96. P f 0.95. P < 0.99. P < 0.99.

2027.7 1898.4 1761.7 1611.1 1443.7 1261.0 1070.2 881.9 706.6 552.2 422.7 318.8 238.5 178.3 134.4 102.9 80.7 66.7 61.1 60.3

0.79 i 0.85 < 0.90 G 0.93 G

4054.0 3795.4 3521.7 3220.5 2885.5 2519.9 2137.9 1760.7 1409.4 1099.5 839.2 629.4 466.1 342.5 251.1 184.9 137.7 104.6 81.5 67.0

1.000

of detection, of detection, of detection, of detection,

8107.3 7590.0 7042.7 6440.2 5770.1 5038.6 4274.5 3519.9 2816.8 2196.5 1675.2 1254.6 926.5 677.3 491.9 356.1 258.2 188.5 139.5 105.5

0.500

Mean distance with school radius

of values for the probability of values for the probability of values for the probability of values for the probability

140.33 138.04 136.41 135.27 134.42 133.69 132.97 132.27 131.62 131.04 130.56 130.17 129.90 129.72 129.62 129.59 129.58 129.59 129.60 129.60

R/2a

f Range 9 Range h Range i Range

Mean catch

G. Mean school size = 128 h

0.354 22905.5 21803.0 20607.7 19250.8 17685.9 15899.7 13925.4 11847.9 9784.2 7850.9 6137.0 4690.4 3518.9 2601.4 1901.5 1378.4 993.5 713.7 512.5 369.3 268.4 198.1

Mean catch 273.40 269.70 267.05 265.20 263.94 263.03 262.23 261.45 260.68 259.98 259.38 258.93 258.63 258.46 258.39 258.38 258.39 258.39 258.40 258.40 258.40 258.40

11453.0 10901.7 10304.1 9625.7 8843.2 7950.2 6963.1 5924.4 4892.7 3926.2 3069.4 2346.3 1761.0 1302.7 953.5 693.0 502.0 364.0 265.7 196.8 148.7 115.6

0.707

2.828 2864.5 2726.7 25714 2407.8 2212.4 1989.2 1742.7 1483.4 1225.9 984.9 771.7 592.2 447.6 335.4 251.1 189.4 145.0 113.8 94.2 86.3 85.2 85.2

1.414 5727.0 5451.4 5152.6 4813.4 4422.2 3975.8 3482.3 2963.1 2447.4 1964.4 1536.4 1175.4 883.5 655.4 482.2 353.8 260.6 194.2 147.5 115.0 94.6 86.4

Mean distance with school radius

H. Mean school size = 256 i

126

JYRI E. PALOHEIMO

when the radius of detection R is about the same size as or larger than the mean distance between the nearest cluster centers. When R is comparatively large, the predator may proceed from one cluster to its nearest neighbor and while the size of the schools increases, the mean distance between them, as well as the traveling time, increases only proportionally to the square root of the size of the cluster when the overall prey density is kept constant. Hence the catch increases up to a point but then when the distance between the schools (and the mean school size) becomes large enough and the predator can less frequently proceed from a school to its nearest neighbor, the catch will start decreasing and finally reach a stable value independent of the mean school size. This conclusion must be modified if there is a substantial loss of time connected with exploiting a school independent of the size of the school (i.e., 5’ of Section 3~). When the delay time t’ is appreciable, then the catch increases with the mean school size. Figures 1 and 2 show the relative influence of R and o on the predator’s catch. In general, though, especially when the mean cluster size is small, the radius of detection R appears more important than the cluster radius 20. However, this is mainly due to the ranges over which cr and R have been varied to satisfy the inequalities (38) and (39). The catch is primarily dependent on the size of R + 20; hence, an equivalent change in R or 0 relative to R + 20 has more or less the same effect. These results may be compared with the experimental observations on feeding of fishes by Ivlev [22]. He found that the food intake of a fish goes up as the variance of the distribution of prey increases. The variance (of a typical contagious distribution) increases if school radius or Q’( 1) = p/( I - p) (mean school size) increases. An increase in ,u/(l - p) was found to have little influence on catch provided that schools were not close together relative to the size of the radius of detection R and that t’ (delay per school of prey) was small. If these restrictions did apply, we could therefore conclude that since the food intake did increase, the increased contagiousness in Ivlev’s experiments was obtained by increasing the school radius cr and not only the mean school size ,u/(l - ~1). e. ,Efect of Limited

Storage

When the predator has a limited storage capacity and can take, for instance, a maximum of m prey animals from each cluster, a further modification in the formulas of the previous sections is required. First, if the detection of a cluster is independent of the number of prey in it, formulas (28) and (29) apply, with the exception that the mean catch per cluster and its variance should be calculated conditional to the maximum number of prey the predator can take. In this case it is clear that when the

A STOCHASTIC

PREDATOR-PREY

MODEL

127

overall prey density is constant, the predator’s catch decreases as the mean cluster size increases. In more general cases the increased contagiousness or mean cluster size increases the probability of the cluster’s being detected and hence increases the predator’s chances. The ceiling imposed on the predator’s ability to exploit a cluster will, on the other hand, counteract its increased chances. Hence, we expect that as the mean cluster size increases, the predator’s catch increases and then, as the cluster size continues to increase, begins to decrease again. This is further illustrated by a numerical example in the next section. J Effect of Variable School Radius Assume that the distribution of prey is described by a doubly stochastic Poisson distribution of a simple kind in which prey are schooling, the schools are randomly distributed and circular in shape, and prey within each school follow a Poisson distribution. The joint distribution between the catching time and numbers of prey caught has a general form GrJt, x), where r refers to the radius of the school and p to the density within the school. The predator is considered to search and to exploit schools rather than individual prey. The alternative modes of exploitation are now perhaps more varied and less easily specifiable. For instance, in the case of fisheries, a fishing vessel can exploit the whole school, as herring seiners do by surrounding the school with a net, or can sample a linear transect of the school, as most trawling vessels do. In the latter instance, however, the vessel often keeps turning back over the same school or piece of ground as long as there is a worthwhile quantity of fish to be caught. The searching models considered have been two dimensional, although if the prey distributions are isotropic, no essential changes are required in three dimensions. Isotropicity, a reasonable assumption in two-dimensional situations, is seldom met in three-dimensional cases. However, many prey-predator interactions that occur in three dimensions can be more reasonably considered by projecting them into a plane. A somewhat special case can occur in fisheries where the searching by the fishing vessels is essentially two dimensional, yet the schools of fish are three dimensional. In considering their two-dimensional projections we have to distinguish between distributions in which the schools change only in length and width and those that change in depth as well. We assume for simplicity that the within-school density p is constant and write G,(t, X) in place of Gr.&(f, x). Our earlier study of the effect of clustering on catch (Section 3d) may be considered as an example of the dependence of catch on searching and schooling when the two-dimensional

128

JYRI E. PALOHEIMO

within-school density is variable. If the schools are three dimensional (i.e., increase in depth with the increase in the school radius), a constant withinschool density implies that for their two-dimensional projection we must take the density p proportional to the school radius. Assuming the mean catch to be a constant fraction of the numbers of prey in a school, and denoting it by 9/7r to avoid carrying constant rt in all our equations, we may thus put [&(rJ)1_*,_.

X=, = 9pr2

(41)

for two-dimensional schools. When the schools are essentially three dimensional, the term Y’ in the foregoing is replaced with r3. However, this case is not considered in sequel. Some consequences of the threedimensional schools are considered in [21]. On account of the Poisson distribution within schools, the left side of (41) also gives the variance of the catch from a school of radius r. It is perhaps reasonable to make the further assumption that the time to exploit a school is proportional to its two-dimensional area. We thus put t d, G,(t, 1) = z’ + rr2 (42) s where 7’ is a fixed (mean) delay time per school irrespective of its size representing, say, the time it takes to get a net ready, and T/TCis the time required to exploit a unit area. The joint distribution between the catching time and numbers caught G(t, x) can now be expressed in terms of G,(t, x), which is the joint distribution of the catching time and numbers caught for schools of radius r. Let P(r) = r + R/(r, + R), the probability of detecting a school of radius r; P = I- + Rl(r, + R), the probability of detecting a school; and K(r) the distribution of school radius. Then

G(t, x>= =

Let the variance and denote

s

G,(t, xY(r)

s

of the catching

dK(r) F

rt-R G,(t, x)F+R

dK(r).

time be zero for a fixed school of radius r

mk =

I’

rk dK(r).

(43)

We may now calculate the moments vij of G(t, x) and substitute them in (10) and (12) to arrive at the mean catch and its variance. For the dominant term of the mean catch we get M,(t) Recall that m, = r.

= __

2/?pq(Rm,

+m,)t

1+21(R+m,)z’+2Ar(Rm,+m,)’

(44)

A STOCHASTIC

PREDATOR-PREY

129

MODEL

Example. The mean catch given in (44) depends on the distribution of prey and on catching time (i.e., on 2, p, K(r), and r’ and z) in a manner that is not very readily detectable. Hence we consider a simple example where the following conditions are met. (a) R = 0 (the radius of detection is zero; in fisheries occur when the vessel detects a school of fish directly beneath

(6)

?.pm, = d = constant

this would it).

= mean density/z

(c) K(r) is a step function; radius, say, r.

that is, all the schools

have the same

From the fact that all the schools have the same radius (i.e., r) it follows that m, = F = r, mz = r2, m3 = r3, and so on. The expression for the mean catch (44) can now be written M,(t)

To illustrate (d)

2qdr’pt = rp+2dt’+2dr2z’

the relationship

(45)

(45) we take

q = 1, T’ = 10, and z = 10.

In addition,

the condition

(38) specifying

SCHOOL

RADIUS

an upper bound for school radius

r

FIG. 3. Mean catch (Eq. (45)) plotted against school radius for three different densities of school centers. Note that the within-school density p = 0.1/hr2.

130

JYRI E. PALOHEIMO

in terms of the density of school centers should be met to avoid overlapping among schools. Since r = 2a, it follows from (38) that

In Fig. 3 the mean catch for two-dimensional schools has been plotted against the schools’ radius Y for three different values of I. (i.e., for three different levels of numbers of schools per unit area-in other words, for three different school center densities). We note that at each level of I there is a school radius that results in a maximum catch per unit time. As r increases, the catching time z’ + Tr2 increases. This is at first more than compensated by the decrease in the searching time but later, as r continues to increase, the catching time is proportionally higher, resulting in a decrease in mean catch. The classical assumption (cf. Section 1) that the catch is linearly related to the abundance seems to be valid only when the catching time (i.e., T’ + u’) is negligible. In Fig. 3 this occurs when the school radius is small. In this case the catch per unit time is equivalent to catch per searching time; hence, the mean catch would be given by M,(t)

29 dm,

= -___

m2

or

29 dm4

&f,(t) = ___

112?j

(47)

depending on whether the school is two or three dimensional. We note that although the catch is proportional to the density d, even in these simple cases it is equally dependent on the distribution of prey, namely, on the ratio of its moments. SUMMARY

The results obtained are based on a stochastic model of predator-prey interactions [12]. The main contribution of [12] and of this article is the study of the effect of the spatial distribution of prey on the catch or attack frequency of the predator. We have shown how the searching time distribution can be evaluated when prey follow a distribution of a type belonging to a general class of contagious distributions. By including the probabilities of the numbers to be caught from a school or cluster and the delay times occurring with each capture, the mean catch in a given time by the predator and its variance are obtained, dependent on the parameters of the prey distribution. The delay time associated with the pursuit, capturing, and digestion of prey have been treated as one total. This delay time, which is

A STOCHASTIC

PREDATOR-PREY

131

MODEL

the time from the sighting of prey to the resumption of the search, can also be taken as a stochastic variable. Although we have achieved a fair degree of generality, important limitations have been introduced: the prey organisms have been treated as essentially immobile; the speed of the travel by the predator has been assumed to be constant; the delay time following the capture has been taken as independent of the previous catches. The first two assumptions (i.e., immobile prey and constant searching speed) can readily be removed (cf. [S]). The removal of the last restriction by introducing a delay time that is dependent on the capture history (i.e., on the degree to which the predator is satiated) is not so readily accomplished. It would imply, for one thing. that the stochastic process describing the predation could no longer be treated as a renewal process. ACKNOWLEDGMENT The author thanks Dr. D. S. Robson for his critical review of and comments on the article. Publication of this article was facilitated by financial support from the National Research Council of Canada. REFERENCES 1 F. 1. Baranov, On the question of the biological basis of fisheries, Naurhn. Issled. Ikhtiol Inst. Izv. 1(1918), 81-128. 2 A. J. Nicholson and V. A. Bailey, The balance of animal populations, Proc. Zool. Sot. London 3(1935), 551-598. 3 A. J. Lotka, Elements of mathematical biology. Dover, New York, 1956.

4 W. E. Ricker, Relation of “catch per unit effort” to abundance and rate of exploration, J. Fish. Res. Board Can. 5(1940), 43-70. 5 W. E. Ricker, Further notes on fishing mortality and effort, Copeia 1(1944), 2344. 6 J. Neyman, On the problem of estimating the number of schools of fish, Univ. of California Publ. Statist. 1(1949), 21-36. 7 B. 0. Koopman, The theory of search, I: Kinematic bases, Operations Res. 4(1956), 324-346. 8 J. G. Skellam, The mathematical foundations underlying the use of line transects in animal ecology, Biometrics 14(1958), 385400. 9 N. Rashevsky, Some remarks on the mathematical theory of nutrition fishes, Bull. Math. Biophys. 21(1959), 161-183.

10 H. C. Holling, The functional response of invertebrate Mem. Entomol.

Sot.

Can. 48(1966),

predators

to prey density,

l-85.

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