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August 2001 Long term behavior of solutions of the Lotka-Volterra system under small random perturbations
R. Z. Khasminskii, F. C. Klebaner
Ann. Appl. Probab. 11(3): 952-963 (August 2001). DOI: 10.1214/aoap/1015345354


A stochastic analogue of the Lotka –Volterra model for predator–prey relationshipis obtained when the birth rate of the prey and the death rate of the predator are perturbed by independent white noises with intensities of order $\varepsilon^2$, where $\varepsilon>0$ is a small parameter.The evolution of this system is studied on large time intervals of $O(1/\varepsilon^2)$. It is shown that for small initial population sizes the stochastic model is adequate, whereas for large initial population sizes it is not as suitable, because it leads to ever-increasing fluctuations in population sizes, although it still precludes extinction. New results for the classical deterministic Lotka–Volterra model are obtained by a probabilistic method; we show in particular that large population sizes of predator and prey coexist only for a very short time, and most of the time one of the populations is exponentially small.


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R. Z. Khasminskii. F. C. Klebaner. "Long term behavior of solutions of the Lotka-Volterra system under small random perturbations." Ann. Appl. Probab. 11 (3) 952 - 963, August 2001.


Published: August 2001
First available in Project Euclid: 5 March 2002

zbMATH: 1061.34513
MathSciNet: MR1865029
Digital Object Identifier: 10.1214/aoap/1015345354

Primary: 34E10 , 34F05 , 60H10 , 92D25

Keywords: predator-prey coexistence , Stochastic differential equations

Rights: Copyright © 2001 Institute of Mathematical Statistics


Vol.11 • No. 3 • August 2001
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