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Deterministic and stochastic analysis of a nonlinear prey-predator system. (English) Zbl 1054.92044

From the introduction: The object of the present paper is to develop a stochastic dynamic model of a nonlinear predator-prey ecosystem and to examine the stability and extinction of the system under random perturbations. We also want to make a critical comparative analysis of stability of the model ecosystem under both deterministic and stochastic perturbations. For this we have first considered a deterministic analysis of stability about the nontrivial steady-state and bifurcations.
The second part of the work consists of stochastic dynamic modeling of the prey-predator system by addition of random perturbations leading to nonlinear stochastic differential equations. The system of nonlinear stochastic differential equations has then been reduced to a system of deterministic moment equations by the method of statistical linearization. The solution of the system of moment equations leads to an expression of nonequilibrium fluctuation and criteria of stability of the stationary states of the original prey-predator systems. A comparative study of the stability of the system under both deterministic and stochastic perturbations leads to the characteristic behavior of the stochastic model of the system.

MSC:

92D40 Ecology
37N25 Dynamical systems in biology
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
92D25 Population dynamics (general)
34C23 Bifurcation theory for ordinary differential equations
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