Dynamics of a stochastic density dependent predator–prey system with Beddington–DeAngelis functional response

Abstract

In this paper, we discuss a stochastic density dependent predator-prey system with Beddington-DeAngelis functional response. First, we show that this system has a unique positive solution as this is essential in any population dynamics model. Then, we investigate the asymptotic behavior of this system. When the white noise is small, the stochastic system imitates the corresponding deterministic system. Either there is a stationary distribution, or the predator population will die out. While if the white noise is large, besides the extinction of the predator population, both species in the system may also die out, which does not happen in the deterministic system. Finally, simulations are carried out to conform to our results.

Study of Certain Stochastic Predator-Prey Models

“…It is well known that for any initial value (x(0), y(0)) ∈ R + almost surely (see [9]). To proceed, we first consider the equation on the boundary by setting y = 0…”

“…[11] studied a stochastic predator-prey model with modified Leslie-Gower and Holling type II schemes; see also [10] in which stochastic ratio-dependent predator-prey models were considered. Moreover, several stochastic models with the well-known Beddington-DeAngelsis functional response were also studied in [9,15,21]. In ecology models, an important concept is stochastic permanence, which indicates that the species will survive forever.…”

“…Much effort has been devoted to finding conditions needed for stochastic permanence. In some of the aforementioned papers, using Lyapunov functions, some conditions for extinction or permanence were also provided and ergodicity was investigated; see [9,17]. However, their conditions are restrictive and not close to a necessary condition.…”

A stochastic predator-prey system with Watt-type functional response

“…In fact, a lot of stochastic version of existing deterministic models have been introduced recently by different authors and here we only mention some of them. For example, Khasminskiǐ and Klebaner in [10] gave an analysis of Lotka-Volterra system with small random perturbations, Ji and Jiang in [9] analyzed a stochastic predator-prey system with BeddingtonDeAngelis functional response, Bandyopadhyay and Chattopadhyay in [2] studied the effect of environmental fluctuations on a ratio-dependent predator-prey system, Mandal and Banerjee in [13] investigated stochastic persistence and stationary distribution of a Holling-Tanner type prey-predator model, Dung in [5] provided an explicit solution to delayed logistic equations with fractional noise, etc. proposed by Watt [23], which can be described by the following differential equations…”

Dynamics of a Stochastic Intraguild Predation Model

“…In essence, random factors can lead to complete extinction of populations even if the population size is relatively large. Previous studies have explored the dynamic properties for stochastic single species models [14][15][16], stochastic predator-prey models [17][18][19][20][21][22][23], stochastic competitive models [24][25][26][27], stochastic mutualism model [28][29][30][31]. Specially, Liu and Wang [32] investigated a two-prey one-predator model with random perturbations.…”