Abstract
A stochastic prey-predator system in a polluted environment with Beddington-DeAngelis functional response is proposed and analyzed. Firstly, for the system with white noise perturbation, by analyzing the limit system, the existence of boundary periodic solutions and positive periodic solutions is proved and the sufficient conditions for the existence of boundary periodic solutions and positive periodic solutions are derived. And then for the stochastic system, by introducing Markov regime switching, the sufficient conditions for extinction or persistence of such system are obtained. Furthermore, we proved that the system is ergodic and has a stationary distribution when the concentration of toxicant is a positive constant. Finally, two examples with numerical simulations are carried out in order to illustrate the theoretical results.
Highlights
The Lotka-Volterra model [1,2,3] is a classical model in the study of biological mathematics, and the continuous LotkaVolterra model which is modeled by ordinary differential equations and delay differential equations is widely used to characterize the dynamics of biological systems [4,5,6,7,8,9,10,11,12,13]
The functional response functions are important in the population ecological models [14]
Functional responses fall into two categories: one depends only on the density of the prey, such as Holling I–III [15,16,17]; the other depends on the density of both the prey and the predator, such as Beddington-DeAngelis type [18, 19]
Summary
The Lotka-Volterra model [1,2,3] is a classical model in the study of biological mathematics, and the continuous LotkaVolterra model which is modeled by ordinary differential equations and delay differential equations is widely used to characterize the dynamics of biological systems [4,5,6,7,8,9,10,11,12,13]. The Beddington-DeAngelis functional response has been widely used in the modeling of ecosystems in which there is mutual interference among predators [24, 25]. Nx (t) b2y (t)) , where r2 represents the growth rate of y due to omnivorous nature and bi (i = 1, 2) denote the density-dependent coefficient of the prey and the predator, respectively. (3) t ≠kτ, Δx (t) = 0, Δy (t) = 0, Δce (t) = μ, t = kτ, where σ1(t), σ2(t) are positive, nonconstant, and continuous functions of period τ, ce(t) stands for the concentration of the toxicant in the environment, h denotes the loss rate of toxicant at time t, τ is the impulsive input period and μ is the impulsive input amount, and β1 and β2 represent the dose-response of the prey and predator to the environmental toxicant, respectively. Some examples with numerical simulations have been given to illustrate our theoretical results
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