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On Lotka–Volterra predator prey models

Published online by Cambridge University Press:  14 July 2016

L. Billard
L. Billard*
Affiliation:
Florida State University
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Abstract

Ever since Lotka (1925) and Volterra (1926), (1931) first considered mathematical formulations for prey-predator processes, the resultant equations have resisted attempts to solve them. However, over the intervening 50 years, standard techniques have allowed a few isolated results to be obtained for some simplified versions of the original process, but the classical equations for the stochastic model have remained unsolved. We give here solutions to the classical process for the case in which interactions occur over a sufficiently short period of time that no births occur. The technique used is one recently developed by Severo (1969a), (1969b), (1971). The approach can be easily generalised to allow solution for the case in which births do occur, as well as for the simplified versions of the original process.

Keywords

LOTKA–VOLTERRA PREY-PREDATOR MODELSSTATE PROBABILITIESSTOCHASTIC MODEL
Type
Short Communications
Information
Journal of Applied Probability , Volume 14 , Issue 2 , June 1977 , pp. 375 - 381
Copyright
Copyright © Applied Probability Trust 1977 

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References

Bartlett, M. S. (1957) On theoretical models for competitive and predatory biological systems. Biometrika 44, 27–42.Google Scholar
Bartlett, M. S. (1960) Stochastic Population Models in Ecology and Epidemiology. Methuen, London.Google Scholar
Bartlett, M. S., Gower, J. C. and Leslie, P. H. (1960) A comparison of theoretical and empirical results for some stochastic population models. Biometrika 47, 1–11.Google Scholar
Becker, N. G. (1970) A stochastic model for two interacting populations. J. Appl. Prob. 7, 544–564.CrossRefGoogle Scholar
Billard, L. (1974) Properties of the stochastic prey-predator model. Technical Report No. 57, Stanford University.Google Scholar
Leslie, P. H. (1948) Some further notes on the use of matrices in population mathematics. Biometrika 35, 213–245.Google Scholar
Leslie, P. H. and Gower, J. C. (1958) The properties of a stochastic model for two competing species. Biometrika 45, 316–330.CrossRefGoogle Scholar
Lotka, A. J. (1925) Elements of Physical Biology. Williams and Wilkins, Baltimore.Google Scholar
Mertz, D. B. and Davies, R. B. (1968) Cannibalism of the pupal stage by adult flour beetles: An experiment and a stochastic model. Biometrics 24, 247–275.Google Scholar
Neyman, J., Park, T. and Scott, E. L. (1956) Struggle for existence: the Tribolium models, biological and statistical aspects. Proc. 3rd Berkeley Symp. Math. Statist. Prob. 4, 41–79.Google Scholar
Park, T. (1954) Experimental studies of interspecies competition II. Temperature, humidity and competition in two species of Tribolium. Physical Zool. 27, 177–238.Google Scholar
Puri, P. S. (1975) A linear birth and death process under the influence of another process. J. Appl. Prob. 12, 1–17.CrossRefGoogle Scholar
Severo, N. C. (1967) Two theorems on solutions of differential-difference equations and applications to epidemic theory. J. Appl. Prob. 4, 271–280.CrossRefGoogle Scholar
Severo, N. C. (1969a) A recursion theorem on solving differential-difference equations and applications to some stochastic processes. J. Appl. Prob. 6, 673–681.CrossRefGoogle Scholar
Severo, N. C. (1969b) Right-shift processes. Proc. Nat. Acad. Sci. U.S.A. 64, 1162–1164.Google Scholar
Severo, N. C. (1971) Multidimensional right-shift processes. Adv. Appl. Prob. 3, 200–201.CrossRefGoogle Scholar
Volterra, V. (1926) Variazoni e fluttazioni del numero d'individui in specie animali conviventi. Mem. Acad. Lincei Roma 2, 31–113.Google Scholar
Volterra, V. (1931) Leçons sur la théorie mathématique de la lutte pour la vie. Gauthier–Villars, Paris.Google Scholar
Weiss, G. H. (1963) Comparison of a deterministic and a stochastic model for interaction between antagonistic species. Biometrics 19, 595–602.CrossRefGoogle Scholar