On the Dynamics of a Stochastic Ratio-Dependent Predator-Prey System with Infection for the Prey

In this paper, we investigate the dynamics of a stochastic predator-prey model with ratio-dependent functional response and disease in the prey. Firstly, we prove the existence and uniqueness of the positive solution for the stochastic model by using conventional methods. Then we obtain the threshold 0 s R for the infected prey population, that is, the disease will tend to extinction if 0 1 s R < , and it will exist in the long time if 0 1 s R > . Finally, the sufficient condition on the existence of a unique ergodic stationary distribution is obtained, which indicates that all the populations are permanent in the time mean sense. Numerical simulations are conducted to verify our analysis results.


Introduction
The research of eco-epidemiology involving ecological and epidemiological models is a significant field in mathematical biology. To our knowledge, Anderson and May were the first to study the spread and persistence of infectious diseases by formulating an eco-epidemiological prey-predator model [1]. Recently, a large number of researchers have devoted to the study of eco-epidemiological models (see [2]- [7]). For example, Chakraborty et al. [2] have investigated the positivity and boundedness of the solutions for a predator-prey model with disease in prey population. Mondala et al. studied the local and global dynamical behavior of a predator-prey eco-epidemiological model with disease in predator [5].
In predator-prey system, functional response plays an important role in the population dynamics. Holling types functional response functions, namely Holling types I, II, III and IV, have been extensively used and investigated [8] [9]. In recent decades, Beddington-DeAngelis and Crowley-Martin type functional response are also widely chosen to model the predation [10] [11]. Li et al. [10] analyzed a stochastic predator-prey model with disease in the predator and Beddington-DeAngelis functional response. They showed that the stochastic system has a similar property to the corresponding deterministic system when the white noise is small enough. In many cases where the predator has to seek for the prey, the per capita predator growth rate should be a function of the ratio of prey to predator abundance in predator-prey model. Thus, the predator-prey models with ratio-dependent functional responses have been proposed and mathematically studied [12] [13] [14] [15]. Based on the literatures, we propose an eco-epidemiological model with infection in the prey and ratio-dependent functional responses as follows where ( ) S t , ( ) I t and ( ) P t denote the densities of the susceptible prey, infected prey and predator respectively. Here, the susceptible prey is subject to the logistic growth, r is the intrinsic growth rate, and r k denotes the interspecific competition rate. The transmission of the disease in the prey is governed by the bilinear incidence rate bSI , where b represent the incidence rate of infected prey to susceptible prey, ( ) S t and ( ) I t denote the densities of the susceptible prey, infected prey. Moreover, the parameters α and β represent the capturing rates of predator to the susceptible and infected prey, respectively; m is the so-called half saturation constant; 1 d and 2 d are the natural death rates of the infected prey and predator. All coefficients mentioned are positive.
As a matter of a fact, most realistic ecosystems are affected by environmental noise (see [16] [17] [18] [19]. Motivated by the method in [20], we introduce to system (1) Gaussian white noise which are directly proportional to S(t), I(t) and P(t), and obtain In this article, in order to better study the spread of infectious diseases among interacting populations, it is more practical to establish a more accurate random ecological infectious disease model. We will concentrate on the dynamics of the stochastic model (2). The rest of the article is organized as follows. In Section 2, the existence and uniqueness of the positive solution is proved for system (2). In Section 3, we analyze the extinction and persistence of the infected prey. In Section 4, we obtain the conditions on the existence of stationary distribution for model (2). In Section 5, numerical simulations are conducted to support the theoretical results. A conclusion is given in the last section.

Existence and Uniqueness
To begin with, we recall some basic notations in stochastic differential equation. let ( ) X t be a regular time-homogeneous Markov process in d  described by the stochastic differential equation The diffusion matrix of the process Furthermore, the differential operator L is defined by To investigate the dynamical behavior of the model, the first concern is whether the solution is global and positive. In this section, we show that there exists a unique global positive solution of system (2) by constructing an appropriate Lyapunov function.
Define a 2 C -function where a is a positive constant to be determined later. The nonnegativity of the function can be obtained from where N is a positive parameter. Thus, we can obtain Open Journal of Applied Sciences Interacting and taking the expectation of both sides of (5) yield By (6), we can obtain Taking n → ∞ induces ∞ > +∞ , which is a contradiction. Hence, we have τ ∞ = ∞ , a.s.. The conclusion is confirmed.

Extinction and Persistence
According to the theory in [21], the basic reproductive number 0 R is a threshold to control whether the disease will spread. If 0 1 R ≤ , the disease disappear; If 0 1 R > , the infectious population will be persistence in the mean. It is easy to conclude the basic reproductive number 0 and system (1) has the following properties: R > , the infectious prey population will be persistence in the mean.
In this section, we turn to establish sufficient criteria on the extinction and persistence of infected prey population for the stochastic system (2). Before giving our main results, we first recall the following lemma.
be the solution of system (2) integrating Equation (7) integrate Equation (9) from 0 to t and divide it by t yields, we obtain By the Itô's formula, we also have integrating Equation (9) from 0 to t and dividing it by t, one can get integrating Equation (11) integrate Equation (13)

Stationary Distribution
Now we present the following lemma.
We aim to prove that 1 LV ≤ − , consider the bounded set D ( )  1, which follows from (16).
where the time increment By simple computations, The numerical simulation is shown in Figure 1, from which one can see that the susceptible prey is persistent, the infectious prey and predator will die out. By simple computations, The conclusion of Theorem 3.3 holds, and the numerical simulation is shown in Figure 2. We note that the susceptible prey and the infectious prey will persist and the predator is going to die out.  The numerical simulation is shown in Figure 3, which is consistent with our conclusion in Theorem 4.2. The difference between (a) and (b) of Figure 3 is the intensity of white noise. We can conclude that with the noise intensity decreases, the dynamics of stochastic system (2) is getting close to the deterministic system (1). Figure 4 shows the simulation of density functions, where system (2) has a unique stationary distribution.