Analysis of a Delayed Stochastic One-Predator Two-Prey Population Model in a Polluted Environment

This paper is concerned with the dynamics of a delayed stochastic one-predator two-prey population model in a polluted environment. We show that there exists a unique positive solution that is permanent in time average under certain conditions. Moreover, the global attractively of system is studied. Finally, some numerical simulations are given to illustrate the main results.


Introduction
With the rapid development of economy, environmental pollution has gradually become the major social problem today. With a growing number of toxicant and contaminants entering into the ecosystem, the quality of our living environment has declined. Then many species have been extinct, and some of them are on the edge of extinction. Therefore, controlling environment pollution has become a major topic in many countries, which draws researchers to investigate the influence of environment pollution.
In the 1980s, Hallam et al. [1] [2] [3] firstly proposed the deterministic models to study the impact of environment toxicant on the survival of biological population. Their studies have provided useful bases about protecting species for us. However, population system is often affected by environmental noise, and there are many scholars who have studied the dynamics of stochastic models with toxicant [4] [5] [6] [7].
On the other hand, more realistic models of population interactions should take the effects of time delay into account [8] [9]. Further, in the natural world, it is a common phenomenon that a predator feeds on some competing preys [10] [11] [12]. However, there is little research on the delayed stochastic one-predator two-prey model in a polluted environment. Thus we consider a stochastic delayed one-predator two-prey model with toxicant input in this article. The rest is organized as follows. In Section 2, we show some notations and introduce a stochastic delayed one-predator two-prey model in polluted environment. In Section 3, we show that the system (SM) has a unique global positive solution. In Section 4, we give the main theorems and their proof. In Section 5, the attractively global system is investigated. In Section 6, we present numerical simulations to illustrate our mathematical findings.

The Model and Notations
In this section, we will give some notations on stochastic one-predator-two-prey system. The stochastic predator-prey system in a polluted environment takes the following form:

t r a C t c x t c x t t x t x t r a C t c x t c x t t C t k C t g m C t t C t k C t g m C t t C t hC t u t t
The above model does not incorporate the effect of time delay, but for a long time, it has been recognized that delays can have a complex effect on the dynamics of a system [9] [13]. In the same time, the natural growth of many populations is inevitably affected by many random disturbances. Considering the effects of random disturbances, we assume the growth rate of prey and the death rate of predator are perturbed with ( ), 1, 2, 3.   x t In past few decades, delay population systems with one predator and two competing preys have received great attention and have been investigated widely. However, as far as the authors concerned, no one has yet explored the preda-tor-prey system with time delays and toxicant inputs in the same time. Therefore, on the basis of article [14], we establish the following delayed stochastic one-predator two-prey model in a polluted environment: 1  1  1 0  11 1  12 2  12  13 3  13  1 1  1   2  2  2  2 0  21 1  21  22 2  23 3  23  2 2  2   3  3  3  3 0  31 1  31  32 2  32  33 3  3 3  3   0  1 x t is the size of the the prey i, x t is the size of the predator; i r is the growth rate of the i the species, r is the death rate of the predator; ii c is the intra-specific competition rate, 1, 2, 3 i = .  , , 0, 1, 2,3 Although the model is a five-dimensional system, because the explicit solutions of the latter two equations are easy to get, it is actually only necessary to study the first three stochastic differential equations of the model, which is called model ) (SM in this paper.

Existence and Uniqueness of the Global Positive Solution
In order to make the model be sense, we need to show the solution is non-negative and global.
This paper assumes that condition Since the coefficients of (3.2) obey the local Lipstchiz condition, then (3.2) has a unique local positive solution x t is global, i.e., e τ = ∞ . Consider the following system:

y t y t r a C t c y t t y t B t y t y t r a C t c y t t y t B t y t y t r a C t c y t c y t c y t t y t B t
Before we state the main theorem of this paper, we need to introduce several hypotheses. Hypothesis 1.

Permanence in Time Average
In this section, we study the permanent in time average of systems (2.3) and (SM). We firstly do some preparation.

Global Attractivity
Making use of Kirchhoff's Matrix Tree Theorem (See, e.g., [21]), one has 0 It is easy to get that:   , , x t x t x t tend to constants, which is consistent with the results of Theorem 4.1. the solutions of systems (SM) fluctuate around a small zone. Thus, we think the system (SM) is permanent.
Since the parameters given above meet the hypothesis 2: 11 12 13 c c c > + , 22 21 23 c c c > + and 33 31 32 c c c > + . According to Theorem 5.1, we can get the system (SM) is global attractively, see Figure 2. Figure 2 shows the simulations of the solutions of systems (SM), From Figure  2(a) and Figure 2(b), we can see that the solution of the system is globally attractively, whether with or without random perturbations.

Conclusions and Discussions
The dynamic relationship between predator and their preys has been and will continue to be one of the major themes in ecology due to its importance and Journal of Applied Mathematics and Physics  , , x t x t x t . The green line represents one prey population ), ( 1 t x manent in time average are given. Our main result in part 4 reveals the impacts of stochastic perturbations on the persistence and extinction of every species. Finally, our results are confirmed by numerical simulation.
Some questions deserve further explorations. In the first place, it is significant to study the delay population system with other disturbance, such as Lévy jumps, or Markovian switching. Another problem is to consider population models with different functional responses, such as Holling II-IV type and Beddington-DeAngelis functional response. We leave these investigations for future work.