Research Article | Open Access
Dynamic Analysis of Stochastic Lotka–Volterra Predator-Prey Model with Discrete Delays and Feedback Control
In this paper, a stochastic Lotka–Volterra predator-prey model with discrete delays and feedback control is studied. Firstly, the existence and uniqueness of global positive solution are proved. Further, we investigate the asymptotic property of stochastic system at the positive equilibrium point of the corresponding deterministic model and establish sufficient conditions for the persistence and extinction of the model. Finally, the correctness of the theoretical derivation is verified by numerical simulations.
In nature, time delays exist in many ecosystems [1–5]. For example, maturity stage is a common phenomenon in biological population, and many diseases have a long incubation period. The mathematical model describing this phenomenon with time delay is called the delay differential equation. In 1999, Saito et al.  studied a Lotka–Volterra predator-prey model with discrete delays, which can be defined as follows:where and stand for the population density of prey and predator at time t, respectively. represent the intrinsic growth rate of corresponding population. and are discrete time delays. , α and β are constants.
Due to the environmental changes and increased human activities, many rare species are at risk of extinction. How to protect endangered species of floras and faunas and maintain the diversity of ecosystems is an important issue that needs to be solved urgently. In the process of marine fishery production, overfishing often results in the exhaustion of fishery resources. It is rewarding for humans to develop and utilize the ecological system of the population rationally, which also contributes to the sustainability of the system [7–14]. In 2003, Gopalsamy and Weng  studied the following population competition model with feedback control:where and are the feedback control variables, , , , , and . They also discussed the existence of positive equilibrium point and global attraction of the model. In 2013, Li et al.  introduced feedback control variables into the two-species competition system and discussed the extinction and global attraction of equilibrium points. They found that if the two-species competition model is globally stable, the system retains the stable property after adding feedback controls and the position of equilibrium point is changed. If the two-species competition model is extinct, by choosing the suitable values of feedback control variables, they can make extinct species become globally stable, or still keep the property of extinction. In 2017, Shi et al.  discussed a Lotka–Volterra predator-prey model with discrete delays and feedback control as follows:where is the feedback control variable, e and f denote the feedback control coefficients, denote the intraspecific competition rates, stand for the capturing rates of the prey and predator populations, is the time of catching prey, and is maturation delay of predator. Shi et al.  show that(i)The solution of system (3) is ultimately bounded(ii)When the conditions are established, system (3) has a unique globally asymptotically stable positive equilibrium point , where , , and
In fact, in nature, ecosystems are inevitably affected by various environmental noises [18–28]. Mathematical models with environmental disturbances can usually be described by stochastic differential equations. Stochastic noise can generally be divided into two categories: one type is a small number of strong interference, usually called colored noise or electrical noise, which can be described by the Markov chain [29–31]; the other type is the sum of many small, independent random interference, called white noise, which is usually represented by Brownian motion [32–35]. Assume that the population’s intrinsic growth rate is disturbed by white noise:
Then, model (3) is transformed intoand satisfies the initial conditionswhere denote the independent standard Brownian motion, denote the intensity of white noise, , and and are both nonnegative continuous functions on .
Due to the interference of stochastic noise, system (5) does not possess an equilibrium point. An interesting question is: Does model (5) still have stability? What is the influence of white noise on system (5)? This paper mainly studies the dynamical properties of stochastic systems (5) also satisfying initial conditions (6). The second part proves the suitability of the system. The third part discusses the oscillation of the stochastic model near the positive equilibrium point of the corresponding deterministic model. The fourth and fifth parts, respectively, obtain the conditions for the persistence and extinction of the stochastic system. Finally, the correctness of the theoretical derivation is verified by numerical simulation.
2. Existence and Uniqueness of Global Positive Solutions
The stochastic differential equation is expressed as
Proof. Since the coefficients of system (5) satisfy the locally Lipschitz condition, for any given initial condition (6), model (5) has a unique local positive solution in interval , where is the explosion time.
To prove that this solution is global, we only need to prove a.s. Let be a sufficiently large constant for any initial value , and lying within the internal . For each integer , define the stopping timeObviously, is increasing as . Let ; therefore, a.s. Now, we need to verify a.s. Otherwise, there are two constants and such that So, there is a positive integer , such thatDefine a : bywhereThe nonnegativity of this function can be obtained fromApplying Itô’s formula yieldswhereTherefore, where K is a positive constant. So, we getIntegrating (17) from 0 to and taking expectation on both sides, we haveSet , and from inequality (10), we have . Note that, for every , there is at least one of , or equaling either k or , and then, we haveIt can be obtained by (18)where is the indicator function of , and letting yieldsThis is a contradiction; we must have , and we have completed the proof.
3. Asymptotic Property
Due to the interference of white noise, the solution of system (5) will have stochastic oscillation. Next, we discuss the asymptotic property of stochastic system at the positive equilibrium point of the corresponding deterministic model. In order to study the problem conveniently, the hypothesis is
Theorem 2. For any given initial condition (6), if hypothesis is established, the solution of system (5) has the property thatwherewhere is the positive equilibrium point of the corresponding deterministic model (3).
Proof. Define the functionwhere and are positive constants. DefineBy Itô’s formula, we obtainwhereSimilarly,whereIn the same way,Therefore, we haveLet , we obtainTherefore,Integrate both sides of (18) from 0 to t and take the expectation, and then we getDivide both sides by t and take the limit superior, and then we haveObviously,whereWe have completed the proof.
Theorem 2 shows that if the condition holds, the solution oscillates around the equilibrium point , and the amplitude of oscillation is positively correlated with the intensity of environmental noise. In particular, if , the influence of environmental noise is not taken into account:The equilibrium point is globally asymptotically stable. This is the conclusion of reference .
In nature, whether ecosystems can survive or not is our main concern. Before discussing the persistence of stochastic system, we give the following assumption:
Proof. According to (37), we haveAs we know, and , from , one can getBy the condition we haveSimilarly, when ,
For the extinction of system (5), we have the following conclusions.
Theorem 4. For any given initial condition (6), the solution of system (5) has the following properties:(i)when , the population is extinct(ii)when , and , population is persistent and and u are extinct(iii)when , , and , population is extinct and and u are persistent
Before proving Theorem 4, consider the following auxiliary system :and it satisfies the initial condition: .
Lemma 1. If , the solution of system (47) has the following properties:(i)(ii)if (iii)if
Proof. By Itô’s formula, we obtainDividing both sides of (48) and (49) by t, we haveBy using Lemma 2 of , from equation (48), it follows thatand therefore,Substitute (53) into (50), and from , we getOn the other hand, computing (50) × a21 + (51) × a11, we haveBy Lemma 2 in literature , when ,If Proof of lemma is completed.
This is where we prove Theorem 4.
Proof. By using Itô’s formula for system (5), we haveWe first prove (i): from equation (58), we can getBy the condition and Lemma 2 in literature ,From (57), (61), and the condition , we haveFurther, from the third equation of model (5), we can get