Research Article | Open Access
Dynamics Analysis of a Stochastic Leslie–Gower Predator-Prey Model with Feedback Controls
This study focuses on the investigation of a stochastic Leslie–Gower predator-prey model with feedback controls and Holling type II functional responses. First, the existence and uniqueness of a global positive solution to the system under white noise interference are proved. Second, the conditions for the existence of the system’s positive recurrence are established by constructing suitable Lyapunov functions. Additionally, the persistence and extinction of prey and predator in the system are discussed, and the impacts of noise interference and feedback controls on the system are revealed. Finally, we validate the theoretical results by numerical simulations.
The relationship between biological populations is usually expressed and analyzed by differential equations [1–5]. Among numerous studies of biological populations, Leslie  presented a classical predator-prey model as follows:with and , where and are the density of the prey and the predator at time t, respectively. K denotes the carrying capacity of prey, and and are the intrinsic growth rates of the prey and the predator. is the functional response of the predator to the prey. h is the measure of food quantity that the prey provides for conversion into one predator birth. The term is the predator’s environmental capacity, and is called the Leslie–Gower term. Leslie accentuated that the decrease in the number of predators was related to the per capita availability of the prey to be hunted, and the environmental carrying capacity of the predator was proportional to the number of preys.
On the basis of model (1), Aziz-Alaoui and Daher Okiye  proposed a modified Leslie–Gower predator-prey model with Holling type II functional responses as follows:where a is the maximum of the average predation rate and b is the maximum digestibility rate of predators. n and m are the environment protecting effects of the prey and the predator, respectively. Combined with the Lyapunov function methods, the global stability of the coexisting interior equilibrium point of model (2) was investigated.
As is known to all, ecosystems are inevitably disturbed by the environmental noise [7–15]. Strong environmental disturbances can cause population fluctuations and even extinction. Based on model (2), Ji et al.  presented the following Leslie–Gower predator-prey model with stochastic perturbation:
To simplify the problem, they chose (assuming the predator and prey receive the same protection from the environment) in model (3). Here, denotes the intensity of white noise, and are the standard Brownian motions independent of each other in the complete probability space . Also, the conditions for the persistence in mean and extinction of the system were provided. In recent years, many research results have been achieved in the study of stochastic ecosystems [17–25]. For instance, Miao et al.  verified the stationary distribution of an n-species stochastic differential equation model. Lahrouz et al.  studied a stochastic Leslie–Gower predator-prey model with regime switching where the effects of white noise and colored noise on the model were considered comprehensively, and the long-term dynamic behavior of the population and sufficient conditions for the existence of stationary distribution were investigated. In , a predator-prey model of stochastic Holling type II schemes with Markov switching and prey harvesting was proposed, and they obtained the conditions for the existence of the stationary distribution and got the optimal harvesting strategy.
How to regulate the ecosystem reasonably to ensure its balance and sustainable development is of importance for issue facing mankind [29–33]. Maltby et al.  considered ecosystem management to be a control of physical, chemical, and biological processes, which linked the regulation of organisms and their nonliving environment and human activities to create an ideal ecosystem state. In recent years, widely research has been conducted on the biological mathematical models with feedback controls [35–40]. By introducing control variables, Gopalsamy and Weng  presented a competition model with time delay and feedback controls following by the discussion on the existence and global attractiveness of the model’s positive equilibrium. In , a Lotka–Volterra two-species competition model was studied, and the influence of feedback controls on the global stability of the system was explored.
Synthesizing the above literatures, we construct the following stochastic Leslie–Gower predator-prey model with feedback controls:where and are the feedback control variables and are all positive constants from the realistic biological significance.
The rest of this paper is structured as follows: in Section 2, we prove the existence and uniqueness of global positive solutions to system (4). In Section 3, we obtain the conditions for the existence of positive recurrence of the solutions to system (4). In Section 4, we analyze the survival conditions of the system and obtain sufficient conditions for persistence in mean and extinction. In Section 5, we use MATLAB to perform some numerical simulations to supplement and illustrate the result.
2. Existence and Uniqueness of Global Positive Solutions
In order to study the dynamic behaviors of system (4), it is necessary to first determine whether there is a global positive solution for any given initial conditions.
Theorem 1. For any given initial value , the model (4) has a unique positive solution on and the solution will remain in with probability one.
Proof. Since the coefficients of the model (4) satisfy the local Lipschitz condition, for any initial value , there exists a unique local solution on , where is the explosion time. We want to prove this solution is global, i.e., to show the a.s. Define a number which is large enough to make . For each integer , we define the stopping time as follows:where the sequence is increasing with respect to h. Let , by the definition of stopping time, a.s. Next, we need to prove so that . If this statement is false, then there exist constants and such thatThus, there is an integer such thatFor the solution of model (4), we define a function V: bywhereBy using the generalized Itô formula, we getwherewhereApplying the Itô formula to (4), one can getTherefore,Integrating both sides of (14) from 0 to and taking the expectation, we obtainFor all , let , we have . Note that, for any , at least one of , or equals or h, and thenwhereCombining inequality (15), which implies thatwhere is the indicator function of . As , there isObviously, this is contradictory. The existence and uniqueness of global positive solutions are proved.
3. Existence of Positive Recurrence
For the stochastic model with feedback controls (4), we are primarily concerned with the long-term dynamic behavior of the population. For example, under what conditions can the path from one point return to a given region for a limited time? This requires us to discuss the positive recurrence of the solution of model (4).
Let be a regular time-homogeneous Markov process in (the l-dimensional Euclidean space) of the stochastic equation:
The diffusion matrix is defined as follows:
Definition 1. (see ). The Markov process is said to be positive recurrent with a bounded domain , if for any , here is the time when the trajectory from x arrives at U, namely, .
Lemma 1. (see ). Suppose that the process almost surely exits from each bounded domain in a finite time. Then, a sufficient condition for positive recurrence is that there exists a nonnegative function in the domain such that
First, we consider the equilibrium point of the corresponding deterministic system. From model (4), we set
By calculating (23), the following equation is obtained:
Proof. Let be a solution to (4) for any given initial value.
. Define a functionwhereObviously, as where . By using the Itô formula, we getTherefore,From the hypotheses (H1) and (H2), we getIt can be written asObviously, the ellipsoidlies entirely in . For any , where U is a neighborhood of the ellipsoid, we have (M is a positive constant).
From Lemma 1, we obtain the existence of positive recurrence to model (4). Because the diffusion matrix of system (4) does not satisfy the uniformly elliptical condition in U, we only prove that the system is positive recurrent but does not obtain the ergodic property.
4. Persistence and Extinction
In the above section, we obtained the existence conditions for the positive recurrence of system (4). In this section, we discuss the persistence and extinction of species of the stochastic model (4) with feedback controls.
For convenience, let us define some notations as follows:
Theorem 3. For any given initial condition , the solution to model (4) have the following properties:(i)When and , and are extinct, and and are also extinct(ii)When , , and , is extinct, and and are persistent in mean(iii)When , , and , is extinct, and and are persistent in mean.
Case 1. According to formula (39), we gainBy Lemma 2.3 in , when ,According to the third equation of system (4), we can acquireSimilarly, by equation (40), when , there isFurthermore, from the fourth equation of model (4), we can obtain
Case 2. We suppose that is a solution to the following equation:with . According to the comparison theorem for the stochastic equation, for , we getFrom the condition , we can get . For arbitrary small , we have for (T is a constant) andBy , we haveFor the arbitrariness of ε, we getAccording to the third equation of system (4), we can deriveSubstituting (53) into equation (39) leads toWhenwe acquireConsidering the third equation of (4) again, we can seeThat is, and are persistent in mean.
Case 3. Let be a solution of the following equation:with . Using the comparison theorem for the stochastic equation, we get that, for ,By the sure reason as the proof of Case (i), we know that holds when . Hence for arbitrary small , there is for . Thus, we can obtainSince , then we haveFrom the arbitrariness of ε, one can getThen, we have from the fourth equation in system (4)Combining (40) with (63) leads to