A stochastic predator–prey model with Holling II increasing function in the predator

This paper is concerned with a stochastic predator–prey model with Holling II increasing function in the predator. By applying the Lyapunov analysis method, we demonstrate the existence and uniqueness of the global positive solution. Then we show there is a stationary distribution which implies the stochastic persistence of the predator and prey in the model. Moreover, we obtain respectively sufficient conditions for weak persistence in the mean and extinction of the prey and extinction of the predator. Finally, some numerical simulations are given to illustrate our main results and the discussion and conclusion are presented.


Introduction
The dynamical relationship between predators and preys is one of the most important and interesting topics in biomathematics [20]. Some models have been presented, which study a two-dimension predator-prey model [16,29,40], multi-predator model [7,35] or multiprey model [13,33,39]. The dynamic property of a predator-prey model with the disease spreading is also one of the dominant themes in biomathematics. To study the effects of disease on the population, these models with sick prey or sick predators have been studied [10,11,18,34,45,46]. In addition, some models with the functional responses have also been proposed [8,19,29,30]. Many conclusions have been drawn and are expected to become more substantial in the future.
The relationship between pests and their natural enemies is a typical predator-prey relationship. In agriculture, how to control pests is a key point. Among the pest control methods, biological control is a common approach. There has been a lot of research and some good results [14,37,38,44].
Tang [37] proposed a pest management predator-prey model with the prey-dependent consumption and established the following ODE model with Holling II increasing function in the predator: dx dt = x(r − by), where x(t) and y(t) represented the densities of the prey and the predator at time t, respectively; r was the growth rate of x(t); the prey's contribution to the predator's growth rate was λbxy 1+bhx , where b and h respectively denoted the searching rate and handling time, parameter λ was the rate at which ingested prey in excess of what was needed for maintenance was translated into predator population increase; d 1 denoted the mortality of y(t); r, b, h, λ and d 1 were positive constants. It was assumed that predators may consume a progressively smaller proportion of prey when the prey density increased [37]. And Tang proposed that this model had the same dynamical behaviour as the classical model.
To understand the effect of individual competition for a limited amount of food and living space, the environment capacity is taken into account in [17,21,25,41]. Sun et al. [36] studied the following model with Holling II increasing function in the predator: where K was the environment capacity and other parameters were the same as the model (1). If 0 ≤ h ≤ λ d 1 , system (2) has three equilibrium points Furthermore, O(0, 0), A(K, 0) are saddle points and E * (x * , y * ) is a globally asymptotically stable focus [36]. In fact, population dynamics is inevitably affected by environmental white noise which is an important component in an ecosystem [12]. But in the deterministic model, all parameters are not disturbed by the environment. Hence the deterministic model has some limitations in mathematical modelling of ecological systems and is quite difficult to fitting data perfectly and to predict the future dynamics of the system accurately [1]. May [32] pointed out the fact that the birth rate, death rate, carrying capacity and other parameters in the system are affected by random fluctuations. To understand the impacts of randomness and fluctuations, it is convenient and effective to model population dynamics through a stochastic differential equation [17,22-24, 26-28,42].
In order to study the influence of environmental disturbance on the population, we introduce the method of [47]. For model (2), given t > 0 and time instant t = j t, intro- 4 = o( t), where i = 1, 2 and m = 0, 1, 2, . . ., and σ 2 i denote the intensities of stochastic disturbance. In each interval [m t, (m + 1) t), assume that X (j) increases according to model (2) and is also affected by the random amount (x m R t 1 (m), y m R t 1 (m)) T . Hence, for m = 0, 1, . . . we get According to Theorem 7.1 and Lemma 8.2 in [6], as t → 0, X m converges weakly to the solution of the following equation: where B i (t), i = 1, 2 denote the standard independent Brownian motion. The rest of this article is organized as follows. In Section 2, we give some definitions and lemmas to complete the structure of the article. In Section 3, the analytic results of dynamics of the stochastic predator-prey model are given which include the existence and uniqueness of the global positive solution, existence of the stationary distribution and the persistence and extinction of the prey and the extinction of the model (3). We give some numerical simulations to verify our theoretical results in Section 4. Finally, we provide a brief discussion and the summary of the main results in Section 5.

Preliminaries
Throughout this paper, unless otherwise specified, we let ( , F , {F t } t≥0 , P) be a complete probability space with a filtration {F t } t≥0 satisfying the usual conditions (i.e. it is right continuous and F 0 contains all P-null sets).
As a matter of convenience, we define some concepts and introduce some base definitions and symbols. Let R n In addition, for a function z(t) for t ∈ (0, ∞), define First, some definitions and useful lemmas of permanence and extinction will be given. (i) If lim t→∞ x(t) = 0, then x(t) is said to go to extinction almost surely.
(ii) If x(t) * > 0, then x(t) is weakly persistent in the mean almost surely.
Next, the definition of stationary distribution and some assumptions and lemmas will be proved.
Denote E l to be Euclidean l-space. Let X(t) be a homogeneous Markov process in E l denoted by the following equation: The following diffusion matrix [15] is

Definition 2.2 ([2, 3]):
The corresponding probability distribution of an initial distribution γ can be written as P γ which shows the initial state of the system (4) at t = 0. If the distribution of X(t) with initial distribution γ converges in some sense to a distribution π = π γ , satisfy for all measurable G, where a priori π may depend on the initial distribution, then the system (4) has a stationary distribution π(·).

Assumption 2.1 ([15]):
There exists a bounded domain U ⊂ E l with regular boundary, which has the following properties: (H1) The smallest eigenvalue of the diffusion matrix A(x) is bounded away from zero in the domain U and some neighbourhood thereof. (H2) If x ∈ E l \ U, the mean time τ is finite at which a path issuing from x reaches the set U and for every compact subset κ ⊂ E l it holds that sup x∈κ E x τ < ∞.

Lemma 2.3 ([3]):
Let f (·) be a functional integrable about the measure μ. If Assumption 2.1 holds, then the Markov process X(t) has a stationary distribution μ(·) and for all x ∈ E l . Moreover, if f (·) is a function integrable with respect to the measure μ, then

Dynamics of the SDE model
In this section, we will analyse the dynamics of model (3). First, the existence and uniqueness of the global positive solution will be proved, which is a prerequisite for analysing the long-term behaviour of model (3).

Existence and uniqueness of the global positive solution
for any initial value (x(0), y(0)) ∈ R 2 + , and the solution will remain in R 2 + with probability 1.
Proof: Consider the following system: where where τ e is the explosion time since the coefficients of model (5) satisfy the local Lipschitz condition. Consequently, by the application of Itô s formula, system (3) has a unique local solution (x(t), y(t)) ∈ R 2 + for any initial value (x(0), y(0)) ∈ R 2 + . Next, we only need to prove that this solution is global, i.e. τ e = ∞ almost surely. Let For each integer k > k 0 , we define the stopping time as follows: Set inf ∅ = ∞ (∅ denotes the empty set). Let τ ∞ = lim k→∞ τ k , then τ ∞ ≤ τ e almost surely. We assume τ ∞ = ∞ almost surely. Otherwise, there is T > 0 and ε ∈ (0, 1) such that P{τ ∞ ≤ T} > ε. Therefore, there exists a constant k 1 > k 0 which satisfies P{τ k ≤ T} ≥ ε Applying Itô s formula, it can be derived that According to Lemma 4.1 of Dalal et al. [5], for x i ∈ R + , Therefore, the following inequalities holds.
Integrating from 0 to τ k ∧ T and taking the expectation by applying Grownwall's inequality, . Then one can be derived that where 1 t (θ ) is an indicator function of k . This contradicts the hypothesis. Consequently, the proof is complete.

Existence of the stationary distribution
The stationary solution means that it is a stationary Markov process, suggesting that the prey x and the predator y are persistent and cannot become extinct. In other words, if the stationary distribution of the solutions of the system exists, we can get the stability in stochastic sense. In this section, we prove the existence of the stationary distribution in model (3).
exists a stationary distribution and it is ergodic. 2 . By Itô s formula to V 1 , it can be derived that An application of Itô s formula to V 2 , it can be given that where It is easy to prove that When ω < min{( r K − by * − σ 2 Let U be a neighbourhood of the ellipsoid which satisfies U ⊆ E 2 \ U, hence there is a positive constant K such that LV ≤ −K for (x, y) ∈ E 2 \ U. In other words, condition (H2) in Assumption 2.1 is satisfied. Moreover, for all (x, y) ∈ U and ξ ∈ R 2 , there exists N = min{σ 1 x 2 , σ 2 y 2 , (x, y) ∈ U} > 0 such that which implies condition (H1) in Assumption 2.1 is satisfied. Therefore, according to Lemma 2.3, the system (3) has a stationary distribution which is ergodic.

Persistence and extinction
Different noise intensities may lead to different behaviours of the population x(t) and y(t) in studying the population long-term behaviour, either extinction or persistence. Therefore, we consider the persistence and extinction of x(t) and extinction of y(t) of this part.

Lemma 3.1:
For any initial value x(0) ∈ R + , the population x(t) in the system (3) has the following inequalities: Proof: According to the first equation of system (3), by the application of Itô s formula, it can be obtained that Construct a comparison system: Define V 1 = e t ln w. Applying Itô s formula, it is obtained that where Integrating from 0 to t, we can get that Applying the similar method as Zhu et al. [48], we let Applying Borel-Cantalli Lemma, there is i ∈ such that for any constant ∈ i , there exists a constant λ i = λ i ( ), then for all λ 0 > λ i , we derive Hence, ln w(t) + r − r K w(t) + 1 2 σ 2 1 (εe t−λ 0 v − 1) has the supremum for all t ∈ [0, λ 0 v]. In other words, there exists M 1 such that Therefore, lim sup t→∞ 1 t ln w(t) ≤ 0 almost surely (the rest of the proof is the same as Theorem 3.3 and Corollary 3.3 of Zhu et al. [48]). According to the comparison theorem for stochastic differential equations, we get x(t) ≤ w(t). As a result, lim sup t→∞ we structure a comparison system: By the Itô's formula, it can be given that Integrating both sides from 0 to t, where M 1 (t) = t 0 σ 1 dB 1 (s). According to strong law of large numbers, we get Consequently, lim sup t→∞ 1 t ln X(t) ≤ r − σ 2 1 2 < 0 almost surely. According to the comparison theorem for stochastic differential equations, we get lim sup t→∞ 1 t ln x(t) < 0, then lim t→∞ x(t) = 0.
(ii) To prove that the population x(t) is weakly persistent in the mean almost surely, just prove that there is a constant u > 0 that any solution of the system (3) satisfies x(t) * ≥ u > 0. Assume the conclusion is false. Let ε 1 be sufficiently small such that Then for all ε 1 > 0, there exists the solution ( Integrating both sides from 0 to t and divide by t, where M 2 (t) = t 0 σ 2 dB 2 (s). According to strong law of large numbers, lim sup t→∞ Then lim sup t→∞ 1 t ln y(t) < 0. As a result, lim t→∞ y(t) = 0.

Numerical results
In order to make our conclusion more reasonable, we make numerical simulations in this part to verify our conclusion. By application of Milstein's higher order model [9], we simulate the result of the model (3) by giving the positive initial value and parameters. The corresponding discretization equations are where t is time increment and ξ k , ς k (i = 1, 2, . . . , n) is independent Gaussian random variables. For the model (3), choose the initial value (x(0), y(0)) = (0.9, 0.8) and parameters are chosen as follows: Due to 0 ≤ h ≤ λ d 1 , the system (2) exists the positive equilibrium E * = (x * , y * ), where x * ≈ 0.6667, y * ≈ 0.7795. In order to show the effect of white noise on population x(t) and y(t), we respectively take σ 1 = σ 2 = 0 and σ 1 = 0.05, σ 2 = 0.05, as shown in Figure 1(a,b).
In addition, let σ 1 = σ 2 = 0.1 and other values are the same as (10). The calculation pre- 2 ) = −0.0111 < 0, which satisfies the condition of Theorems 3.3(ii) and 3.4. Therefore, the population x(t) is persistent and y(t) tent to extinct  almost surely. The result is shown in Figure 4(a). By increasing the value of σ 1 so that σ 1 = 0.9, we give r − σ 2 1 2 = −0.005 < 0. So the condition of Theorem 3.3(i) holds. That is to say, the population x(t) will go to extinct almost surely. Therefore, we choose σ 2 = 0.2 and other parameters keep consistent with (10), then λbK(r − 2 ) ≈ −0.9011 < 0 where the population x(t) is extinct. Consequently, y(t) will go to extinct such as Figure 4(b).

Discussion and conclusion
We have considered the influence of the white noise on the model (2) in this article. The innovation of the system (2) is that it has taken into account the relationship between  , when x(t) is persistent almost surely and parameters satisfy the condition of Theorem 3.4, the predator y(t) go to extinct almost surely, where σ 1 = 0.1, σ 2 = 0.7 and other values as (10). In (b), when x(t) tend to extinct almost surely and parameters satisfy the condition of Theorem 3.4, the predator y(t) go to extinct almost surely, where σ 1 = 0.9, σ 2 = 0.2 and other values are the same as (10).
predation rate of the predator and the density of the prey and consider the effect of the environment capital of the population x(t). On this basis, due to the disturbance of the environment to the population, we have considered the effect of the noise on predators and prey, which has made our research model (3) more consistent with the ecological significance.
We have first proved the existence and uniqueness of the global positive solution of the model (3), which is the prerequisite for studying the long-term behaviour of predators and prey. Under the condition that the positive equilibrium point of system (2) exists, we have proved the existence of the stationary distribution and its ergodic property which means the predator and the prey are both permanent.
By the comparison of Figures 1, 3 and 4, we have access to the following conclusion: (a) With the increase of σ 1 and σ 2 , the dynamic properties of the system (3) will also change.
(b) White noise has no effect on the system (3) when σ 1 = σ 2 = 0. But when the values of σ 1 and σ 2 become larger, the perturbation effect of white noise will be more obvious.
(c) The population x(t) will be persistent almost surely if r − σ 2 1 2 > 0. Under the premise, the population y(t) will tend to become extinct almost surely if σ 2 is sufficiently large. (d) When σ 1 is sufficiently large, the population x(t) and y(t) tend to become extinct almost surely.
Therefore, we make the population extinct by controlling the size of σ 1 and σ 2 . From the numerical simulation, under the same conditions, a small white noise will make the system persist. And the larger white noise will make species become extinct. It is also possible to control the size of the disturbance so that the prey lasts and the predator becomes extinct. From our model, when the prey is extinct and the predator has no other source of food, the predator must be extinct.