Comparison of a Deterministic and a Stochastic Model for Interaction between Antagonistic Species

Comparison of a Deterministic and a Stochastic Model for Interaction between AntagonisticSpeciesAuthor(s): George H. WeissSource: Biometrics, Vol. 19, No. 4 (Dec., 1963), pp. 595-602Published by: International Biometric SocietyStable URL: http://www.jstor.org/stable/2527535 .

Accessed: 28/06/2014 12:27

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

International Biometric Society is collaborating with JSTOR to digitize, preserve and extend access toBiometrics.

http://www.jstor.org

This content downloaded from 91.223.28.76 on Sat, 28 Jun 2014 12:27:26 PMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=ibs

http://www.jstor.org/stable/2527535?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


COMPARISON OF A DETERMINISTIC AND A STOCHASTIC MODEL FOR INTERACTION BETWEEN

ANTAGONISTIC SPECIES

GEORGE H. WEISS

The Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, College Park, Maryland, U.S.A.'

SUMMARY

In this paper we present an analysis of the stochastic and deterministic theories for a prey-predator situation in which no births are involved. Both theories can be solved in closed form, and the results of the theories can be compared. It is shown that the stochastic theory yields results which approach those of the deterministic theory as the numbers are increased.

There are two fundamental approaches to prey-predator problems which involve the interaction of two or more species. The first is the deterministic approach in which it is assumed that the interaction between two species is proportional to the product of the numbers of participants in the encounter, and that the numbers vary in a deterministic manner specified by some differential equation. The second approach is a stochastic one in which it is recognized that the interplay of the numbers in the prey-predator engagement is essentially a random process and accordingly, that one may associate probability distributions with these numbers. There are few instances in which one can solve in closed form both the deterministic and the stochastic equations for the same situation for all numbers.

The same problem arises in several different guises. Perhaps the earliest work on deterministic models was that of Ross [1911], who treated the theory of the spread of malaria. Subsequently Lanchester [1917] proposed the same types of equations to describe combat. Next Volterra [1926] used roughly the same approach to prey-predator problems in a discussion of the upset of the balance between different species of fish in the Adriatic Sea due to the first world war. In 1926 McKendrick proposed a stochastic model of epidemics, which however he could not solve. This was followed by a series of papers by Kermack and McKendrick [1927, etc.] on the deterministic theory of epidemics.

'Present address: Rockefeller Institute, New York, N. Y.

595



596 BIOMETRICS, DECEMBER 1963

Following this early work the theory of the prey-predator processes and the theory of epidemics have been widely developed. The latter work is admirably summarized in the book by Bailey [1957]. However most progress has been made in the deterministic theory of these interaction processes since the stochastic theory generally leads to a second order partial differential equation with variable coefficients. Only in a few special cases can this equation be solved. Hence a detailed comparison between stochastic and deterministic treatments of the same problem is hard to make.

It is the purpose of this paper to introduce a simple model of a prey-predator problem in which it is possible, in a sense, to compare the deterministic and the stochastic theories. The model assumes two species, call them A and B, which interact in some manner such that each individual encounter results in the death of a member of one of the populations. We will assume that reproduction can be neglected. This amounts to the assumption that the total encounter takes place over a short period of time. We will study the probability that one side or the other is reduced to zero, and the distribution of the number of survivors in the surviving species. These two problems can be handled both for the deterministic and for the stochastic theory. The method to be used is that of regeneration points, and is analogous to a treatment by Foster [1955] of a problem in epidemiology.

We assume that initially there are M1 individuals in species A and N individuals in species B, that the probability that an A individual is killed in (t, t + dt) is axmndt where there are m A's and n B's alive at time t, and the probability that a B individual is killed in (t, t + dt) is ,Bmndt. In the deterministic theory, the numbers of A's and B's will be denoted by m(t) and n(t) and the differential equations describing the process are

mh(t) = -arm(t)n(t) (1)

n(t) = -3m(t)n(t)

with m(0) = ill and n(0) = N. Let a/3 = o. Then by dividing the first of these equations by the second we find

m(t) - on(t) = M - oN. (2)

Thus, in the deterministic theory the A species will always survive the B species provided

M > oN. (3)

Furthermore, assuming that this condition is satisfied we have the



INTERACTION BETWEEN ANTAGONISTIC SPECIES 597

complete solution

(M - o-N)N exp [(aN - OM)t] n(t) M - o-N exp [(aN - OM) t]

m(t) M - a-N + (M - o-N)o-N exp [(aN - 3M)t] M - a-N exp [(aN - O3M) t]

Thus the number of A's remaining, conditional on the A species wiping out the B species, is

- m(o) = M - o-N. (5)

We shall see how these results have a stochastic analogue. The simplest type of information about the stochastic theory can

be obtained, not by formulating and solving the equations for the state probabilities P(m, n; t) but rather by considering the individual encounters between members of the two species to be regeneration poin.ts. It will be assumed that the result of each encounter will be a single death. We may then solve for the quantities Qk(m, n), which are defined to be the probabilities that there are m A's and n B's left after k encounters (k = 1, 2, *). The parameters k, m, and n are not independent since

k + m + n = M + N. (6)

A simple argument suggested by the referee suffices to establish the formula for Qk(m, n). At any encounter the probability that an A individual dies is v/(1 + a-). Hence M - in, the number of deaths of species A after k encounters, has a binomial distribution:

Qk(m n) =(M -m)(1 m O- + 1- (7)

k O' M-m

M - m) (1 +a-)

We now consider the probability that the A species reduces the B species to zero. This probability will be denoted by PA and is given by

1 M 1 M PA =1 + a E Qk(m, 1) =1 a > QM-m+N-l(m, 1) (8)

1 'M=1 I+ o

since the probability of reducing the B's to zero may be considered in two steps: First reduce the B's to one survivor (assuming that one or more A's survive) and then reduce the remaining survivor. Notice that the problem is equivalent to a two dimensional random walk beginning from the point (M, N) and bounded by two barriers at (0, y) and (x, 0).




Substituting formula (7) into formula (8) we find for PA the result

PA 1 ] (j+N N- 1 (9) (+ Of k /i-o 0.

i.e., a negative binomial distribution. It is known, however, that this sum can be put in the form of an incomplete beta function, (Fieller and Pearson [1933]):

PA =Il/(l+)(N, M) (10)

where

Ip (N, M) = 2V A 1(I t)m-1 dt/ t2-1(j _ t)Af-1 dt. (I11)

We can also ask for the distribution of the number of the A species remaining, conditional on A being the victorious species. If we let Urn be the probability that there are m of species A which survive, then

U,=[Ij(1+,(N, M)V'1?l+~ A +j. ;i-1 (12) (1 + 0.)M+N-m M M m ) (2

The expected number of survivors is NA and we can write for this quantity

NA-= ZmU. = M-oN mn=1

+ M( M - 1) (1 ?fI+N-1 [Ij/(1+,)(N, M)]1. (13)

The formulae given so far are exact. For larger values of Mt and N we may use the normal approximation to the incomplete beta function

I1/(l+,)(N, M) ( + N 1) M+N-1

N ~ ~~~~~)+~

' erfc -erfc [](M+N)o (14)

where

\/2?r - 2/ erfc f xoer du. (15)

If (At + N)o- > 5 then the last term of formula (14) can be neglected




1.0

.8| N=L100/0

.6- / ?

PaX .4

.2 _

.2 .4 .6 .8 1.0 1.2 1.4 .6 .8 2.0

FIGURE 1

PA AS A FUNCTION OF a = MIN FOR f = 1, N = 10, 100.

and we can write, approximately

1 oo PA = j e du + O(e(M+N)f/ ( ) (16)

2/ (Na-/M)/A(M+N-)

It is now evident that the condition 111 > oN plays a different, but an intuitively satisfying role in the stochastic theory. When M > aN the probability that the A species will wipe out the B species will be greater than 2. We may also verify the fact that the variation Of PA with M and N is intuitively reasonable. When

(M - o-N) /M~N)o- >> 1 (17)

PA is approximately

PA-'- 1 - -M N exp {

1 (M- N)

2

(18)

i.e., as one moves away from the critical condition M/N = C-, the chance of a result contrary to the deterministic theory becomes neg- ligibly small. Similarly when

(o-N - M)/ (M +N)c?>> 1 (19)




one has

PA - TexP {2 (M + N)'}(2

Between these two extreme cases one must use the more exact formula (9). When

1.0

.9

.8

.7

.6

PA

5

.4 / o-=1.2a

.2 /

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 8 1?9 M

FIGURE 2 PLOT OF PA AS A FUNCTION OF M FOR N = 10, o- = 1.00, 1.22.




(M-oN)/ (M +NorE (21)

is very small, the expression for PA can be expanded in a Taylor series in En and we have, approximately

1 1E __ _

PA +~~2i +O0(E). (22) 2 27r :1 2r(2

Let us now hold M/N = a,, fixed, fix C-, and inquire as to the behavior Of PA as N, or the total number of members of both species is increased. In the approximation of large numbers we find

PA=erfc [V (N ] (23) (1.+ ao)a-

Hence the effect of increasing N is to decrease the region of uncertain outcome if PA is plotted as a function of aoe. This is illustrated in Figure 1.

We have plotted, in Figures 2 and 3, some exact results using formulae (10) and (13). It is evident both from formula (13) and from the curves, that the number of survivors is always greater than that predicted by the deterministic theory.

9

8-

7-

6- NA

4

3-

I 2- 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 lB 19 20

FIGURE 3 PLOT OF NA AS A FUNCTION OF M FOR N = 10, - = 1.00, 1.22.




In this paper we have examined in some detail the deterministic and stochastic theories for the same situation. For the present case the two treatm-lents agree, at least qualitatively, and in the limit of large numbers the results of the stochastic theory agree with those of the deterministic theory. This is usually assumed to be the case. However, there are significant situations in which the two types of theory give different results. Principal illustrations of this point may be taken from genetical models.

ACKNOWLEDGEMENT

This research was supported in part by the United States Office of Naval Research.

REFERENCES

Bailey, N. T. J. [1957]. The mathematical theory of epidemics. Griffin, London; Hafner, New York.

Fieller, E. C. and Pearson, K. [1933]. On the applications of the double Bessel function to statistical problems. Biometrika 25, 158-78.

Foster, F. G. [1955]. A note on Bailey's and Whittle's treatment of a general stochastic epidemic. Biometrika 42, 123-25.

Kermack, W. 0. and McKendrick, A. G. [1927, 1932, 1933, 1937, 1939]. Contri- butions to the mathematical theory of epidemics. Proc. Roy. Soc. A 115, 700-21; 138, 55-83; 141, 94-122; J. IHyg. Camb. 37, 172-187; 39, 271-288.

Lanchester, F. W. [19171. Aircraft in warfare, the dawn of the Fourth Arm. D. Appleton, New York.

McKendrick, A. G. [1926]. Applications of mathematics to medical problems. Proc. Edin. Math. Soc. 44, 98-130.

Ross, R. [1911]. The prevention of malaria. Murray Publishers, London. Volterra, V. [1926]. La lutte pour la vie. Gauthier, Paris.



Documents

Comparison of a Deterministic and a Stochastic Model for Interaction between Antagonistic Species