Asymptotic Behavior of Predator-Prey Systems Perturbed by White Noise
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Asymptotic Behavior of Predator-Prey Systems Perturbed by White Noise Nguyen Hai Dang · Nguyen Huu Du · Ta Viet Ton
Received: 8 November 2010 / Accepted: 1 July 2011 / Published online: 21 July 2011 © Springer Science+Business Media B.V. 2011
Abstract In this paper, we develop the results in Rudnicki (Stoch. Process. Appl. 108:93– 107, 2003) to a stochastic predator-prey system where the random factor acts on the coefficients of environment. We show that there exists the density functions of the solutions and then, study the asymptotic behavior of these densities. It is proved that the densities either converges in L1 to an invariant density or converges weakly to a singular measure on the boundary. Keywords Prey-predator model · Diffusion process · Markov semigroups · Asymptotic stability Mathematics Subject Classification (2000) 34C12 · 47D07 · 60H10 · 92D25
1 Introduction The stable predator-prey Lotka-Volterra equation X˙ t = (αXt − βXt Yt − μXt2 ), Y˙t = (−γ Yt + δXt Yt − νYt2 ),
(1.1)
where Xt and Yt represent, respectively, the quantities of prey and the predator populations; α, β, γ , δ, μ and ν are positive constants, has attracted a lot of attention. It has been proved that the solution of (1.1) is asymptotically stable. For the stochastic predator-prey Lotka-Volterra equation, we have to mention one of the first attempts in this direction, the very interesting paper of Arnold et al. [1] where the N.H. Dang · N.H. Du () Faculty of Mathematics, Mechanics and Informatics, Hanoi University of Science, Hanoi, Vietnam e-mail: [email protected] T.V. Ton Department of Applied Physics, Graduate School of Engineering, Osaka University, Osaka, Japan
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N.H. Dang et al.
authors used the theory of Brownian motion and the related white noise models to study the sample paths of the equation
dXt = (αXt − βXt Yt − μXt2 )dt + σ Xt dWt , dYt = (−γ Yt + δXt Yt − νYt2 )dt + ρYt dWt ,
(1.2)
where Wt is the one dimensional Brownian motion defined on the complete probability space ( , F , {Ft }t≥0 , P) with the filtration {Ft }t≥0 satisfying the usual conditions, i.e., it is increasing and right continuous while F0 contains all P−null sets. The positive numbers σ, ρ are the coefficients of the effect of environmental stochastic perturbation on the prey and on the predator population respectively. In this model, the random factor makes influences on the intrinsic growth rates of prey and predator. In continuing this study, in [10–12], the authors showed that the distribution of each solution starting at a point in intR2+ has the density which either converges in L1 to an invariant density or converges weakly to a singular measure on the boundary (0, ∞) × {0}. This paper studies a stochastic predator-prey system where the intrinsic growth rate of the prey, μ, and the one of the predator, ν, are perturbed stochastically μ → μ + σ W˙ t and ν → ν + ρ W˙ t . This means that we consider the following stochastic equation
dXt = (αXt − βXt Yt − μXt2 )dt + σ Xt2 dWt , dYt = (−γ Yt + δXt Yt − νYt2 )dt + ρYt2 dWt ,
(1.3)
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