Application of inequalities technique to dynamics analysis of a stochastic eco-epidemiology model

Mathematical inequalities play a large role in mathematics analysis and its application. Recently, the inequality technique was applied to impulsive differential systems [1, 2] and stochastic differential systems [3–5], thus some new results were obtained.

Predation can have far-reaching effects on biological communities. Thus many scientists have studied the interaction between predator and prey [6–10]. Interaction between predator and prey is hard to avoid being influenced by some factors. One of the most common factors is the disease. Therefore, there are many scholars who have studied the infected predator-prey systems [11–17]. For instance, Hadeler and Freedman [16] considered a predator-prey system with parasitic infection. They proved the epidemic threshold theorem for where there is coexistence of the predator with the uninfected prey. Han and Ma [15] analyzed four modifications of a predator-prey model to include an SIS or SIR parasitic infection. They obtained the thresholds and global stability results of the four systems.

Species may be subject to uncertain environmental disturbances, such as fluctuations of birth rate and death rate, food, habitat and water, etc. These phenomena can be described by stochastic processes. Recently, the stochastic predator-prey systems have received much attention from scholars [18–21]. Zhang and Jiang [18] studied a stochastic three species eco-epidemiological system. They analyzed the stochastic stability and asymptotic behaviors around the equilibrium points of its deterministic model. Liu and Wang [19] considered a two-species non-autonomous predator-prey model with white noise. They obtained the sufficient criteria for extinction, non-persistence in the mean, and weak persistence in the mean.

The functional response of predator is a very important factor of predator-prey system, which reflects the average consumption rate of predator to prey. Therefore, many scholars prefer to study the predator-prey system with functional response [22–25]. For instance, Wang and Wei [22] explored a predator-prey system with strong Allee effect and an Ivlev-type functional response. Liu and Beretta [23] studied a predator-prey model with a Beddington-DeAngelis functional response. Some biologists have argued that in many instances, especially when predators have to hunt for food and, therefore, have to share or compete for food, the functional response in a prey-predator model should be predator-dependent. This view has been supported by some practical observations [26, 27]. Skalski and Gilliam [26] collected observation data from 19 predator-prey communities, they found that three predator-dependent functional responses (Crowley-Martin [28], Hassell-Varley [29] and Beddington-DeAngelis [30, 31]) were in agreement with the observation data, and in many instances, the Beddington-DeAngelis type looked better than the other two.

To the best of our knowledge, the research on global asymptotic behaviors of a stochastic infected predator-prey system with Beddington-DeAngelis has not gone very far yet. Therefore, according to a deterministic predator-prey model, this paper investigates the stationary distribution and ergodic property of a stochastic infected predator-prey with Beddington-DeAngelis and explores the influence of white noise on the persistence in mean and extinction of the predator-prey-disease system.

First of all, a deterministic predator-prey system is described in [32] by
where (X(t)) is the population density of prey at time t, (S(t)) and (I(t)), respectively, stand for the densities of susceptible predator and infected predator at time t, b is the intrinsic growth rate of (X(t)), c is the natural mortality rate of (S(t)), d is the diseased death rate of (I(t)). (a_{11}, a_{22}, a_{33}), respectively, stand for the density coefficients of (X(t), S(t)) and (I(t)). (a_{12}) is the captured rate of (X(t)), (frac{a_{21}}{a_{12}}) is the conversion rate from (X(t)) to (S(t)), ? represents the infection rate from (S(t)) to (I(t)), (p,q0) are constant coefficients.

The rest of this paper is organized as follows. In the next section, we consider the existence of a global positive solution and the stochastically ultimate boundedness of model (2). In Section 3, we study the global asymptotic behaviors of model (2) around the equilibrium points of its deterministic system. In addition, we explore the stationary distribution and ergodic property of model (2). In Section 4, we obtain the conditions for the persistence in mean and extinction of model (2). In the last section, we summarize our main results and give some numerical simulations.

Throughout this paper, let ((Omega,mathcal{F},{mathcal{F}}_{tgeq 0},mathcal{P})) be a complete probability space with a filtration ({mathcal{F}_{t}} _{tgeq0}) satisfying the usual conditions (i.e. it is increasing and right continuous while (mathcal{F}_{0}) contains all (mathcal{P})-null sets). The function (B_{i}(t)) ((i=1,2)) is a Brownian motion defined on the complete probability space ?. For an integrable function (X(t)) on ([0,+infty)), we define (langle X(t)rangle=frac{1}{t}int ^{t}_{0}X(s),ds, langle X(t)rangle_{*}=liminf_{trightarrow+infty}langle X(t)rangle, langle X(t)rangle^{*}=limsup_{trightarrow+infty }langle X(t)rangle).