Dynamics of a stochastic population model with predation effects in polluted environments

The present paper puts forward and probes a stochastic single-species model with predation effect in a polluted environment. We propose a threshold between extermination and weak persistence of the species and provide sufficient conditions for the stochastic persistence of the species. In addition, we evaluate the growth rates of the solution. Theoretical findings are expounded by some numerical simulations.

to test population models with predation effects in polluted environments. However, little research has been conducted to exploit this problem (even for deterministic models).
The objective of this paper is testing the above problem. We construct a population model with predation effects in polluted environments in Sect. 2 and testify that the model has a unique global positive solution in Sect. 3. Then we offer a threshold between extermination and weak persistence of the species and provide conditions under which the species is stochastically persistent in Sect. 4. In Sect. 5, we estimate the growth rates of the solution of the model. Finally, we give the conclusions of this paper and numerically expound the theoretical findings in Sect. 6.

The model
Without pollution and predation effects, suppose that the growth of the species follows the logistic role: where (t) is the population size at time t, and b > 0 and ξ > 0 stand for the growth rate and the intraspecific competition rate, respectively. We consider the predation effects. In general, the predation effects saturate at high prey density and vanish quadratically as the prey density tends to zero [19]. Therefore it is reasonable to model the predation effects by the following function: where λ is the upper limit of the predation effects, and ρ 2 measures the saturate effect. Then model (1) is replaced by As said before, the evolution of populations often encounters environmental perturbations [18]. In general, we can take advantage of a color noise process to portray the environmental perturbations [20], and it is suitable to utilize a Gaussian white noise process to depict a weakly correlated color noise [21]. Various ways were developed to incorporate the white noise into deterministic population models. A widely accepted way is to suppose that some parameters in the model are influenced by the white noise (see, e.g., [22][23][24][25][26][27][28][29][30][31][32]). Adopting these approaches, where ψ 1 (t), ψ 2 (t), and ψ 3 (t) are independent standard Wiener processes defined on a certain complete probability space ( , F, {F t } t≥0 , P), and β i , i = 1, 2, 3, stand for the intensities. As a result, model (2) is replaced by To characterize the influence of pollution, we hypothesize that the populations suck up the pollutants into their bodies [4][5][6]. Denote by T(t) the concentration of pollutants in the species, which can lead to a descent of the growth rate [4]: where H 1 (T) is a positive continuous increasing function of T. Accordingly, model (3) is replaced by To depict the pollutants in the species, we pay attention to the following equation: where T e (t) represents the concentration of pollutants in the environment, H 2 (T e (t)) > 0 characterizes the suck up of pollutants from the environment, H 3 (T(t)) > 0 measures the loss of pollutants because of excretion and detoxication. Both H 2 ∈ C 1 and H 3 ∈ C 1 are increasing functions. Finally, we portray the changes of T e (t). Denote by u(t) a continuous and bounded function of t, the input of pollutants from the outside of the environment. Suppose that the changes of T e (t) are governed by the following equation: where H 4 ∈ C 1 , measuring the loss of the pollutants from the environment, is an increasing positive function of T e . According to (4)-(6), we derive the following model: The objectives of this paper is probing some dynamical properties of (t). Note that the last two equations in model (7) do not depend on (t), and they have a unique solution (T(t), T e (t)) for certain initial value. As a result, from now on we concentrate on Eq. (4). Proof We first concentrate on the equation
Proof For arbitrary > 0, we can findT > 0 such that for 0 Integrating both sides fromT to t, we get Taking logarithms leads to As a result, Since is arbitrary, we get (16).
Theorem 6 probes the upper-growth rate of (t). Now let us consider the lower-growth rate of (t).

Conclusions and simulations
In this paper, we have constructed a stochastic single-species model with predator effects in polluted environments. We have probed some dynamical properties of the model, including the existence and uniqueness of the solution (Theorem 1), the threshold between extermination and persistence (Theorems 2-4), stochastic permanence (Theorem 5), and upper-and lower-growth rates (Theorems 6 and 7). To our best knowledge, this paper is the first one to probe population models with predation effect in a polluted environment. Now let us numerically expound the theoretical findings by the Milstein method [34].
We choose H(T(t)) = b 1 T(t) and pay attention to the following discretization equation of model (4):