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 sucient conditions for the persistence and extinction of the model. Finally, the correctness of the theoretical derivation is veried by numerical simulations.


Introduction
In nature, time delays exist in many ecosystems [1][2][3][4][5].For example, maturity stage is a common phenomenon in biological population, and many diseases have a long incubation period.e mathematical model describing this phenomenon with time delay is called the delay di erential equation.In 1999, Saito et al. [6] where x(t) and y(t) stand for the population density of prey and predator at time t, respectively.r i (i 1, 2) represent the intrinsic growth rate of corresponding population.τ 1 and τ 2 are discrete time delays.a(a < 0), 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 oras and faunas and maintain the diversity of ecosystems is an important issue that needs to be solved urgently.In the process of marine shery production, over shing often results in the exhaustion of shery 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][8][9][10][11][12][13][14].In 2003, Gopalsamy and Weng [15] studied the following population competition model with feedback control: where u 1 (t) and u 2 (t) are the feedback control variables, b i > 0, a ij > 0, α i > 0, η i > 0, and a i > 0 (i, j � 1, 2).ey also discussed the existence of positive equilibrium point and global attraction of the model.In 2013, Li et al. [16] introduced feedback control variables into the twospecies competition system and discussed the extinction and global attraction of equilibrium points.ey 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. [17] discussed a Lotka-Volterra predator-prey model with discrete delays and feedback control as follows: where u(t) is the feedback control variable, e and f denote the feedback control coefficients, a ii (i � 1, 2) denote the intraspecific competition rates, a ij (i ≠ j, i, j � 1, 2) stand for the capturing rates of the prey and predator populations, τ 1 is the time of catching prey, and τ 2 is maturation delay of predator.Shi et al. [17] show that (i) e solution (x In fact, in nature, ecosystems are inevitably affected by various environmental noises [18][19][20][21][22][23][24][25][26][27][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][30][31]; the other type is the sum of many small, independent random interference, called white noise, which is usually represented by Brownian motion [32][33][34][35].Assume that the population's intrinsic growth rate r i is disturbed by white noise: en, model ( 3) is transformed into and satisfies the initial conditions where B i (t) (i � 1, 2) denote the independent standard Brownian motion, σ 2 i denote the intensity of white noise, τ � max τ 1 , τ 2  , ϕ i (0) > 0, ψ(0) > 0, and ϕ i (θ) and ψ(θ) are both nonnegative continuous functions on [− τ, 0].
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)? is paper mainly studies the dynamical properties of stochastic systems (5) also satisfying initial conditions (6). e second part proves the suitability of the system.e third part discusses the oscillation of the stochastic model near the positive equilibrium point (x * 1 , x * 2 , u * ) of the corresponding 2 Complexity deterministic model.e 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.

Existence and Uniqueness of Global Positive Solutions
e stochastic differential equation is expressed as the stochastic differential equation of V(x, t) along system ( 7) is defined as [36] dV(x(t), t) � V t (x(t), t) + V x (x(t), t)f(x(t), t) where Theorem 1.For any given initial condition (6), model (5) has a unique global positive solution (x 1 (t), x 2 (t), u(t)), and the solution will remain in R 3 + with probability one.
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 (x 1 (t), x 2 (t), u(t)) in interval t ∈ [0, τ e ), where τ e is the explosion time.
To prove that this solution is global, we only need to prove τ e � ∞ a.s.Let k 0 > 0 be a sufficiently large constant for any initial value x 1 (0), x 2 (0), and u(0) lying within the internal [(1/k 0 ), k 0 ].For each integer k ≥ k 0 , define the stopping time Obviously, τ k is increasing as k ⟶ ∞.Let τ ∞ � lim k⟶∞ τ k ; therefore, τ ∞ ≤ τ e a.s.Now, we need to verify τ ∞ � ∞ a.s.Otherwise, there are two constants T > 0 and ϵ ∈ (0, 1) such that P τ ∞ ≤ T   > ϵ.So, there is a positive integer k 1 ≥ k 0 , such that Define a C 2 − functionV: R 3 where e nonnegativity of this function can be obtained from Applying Itô's formula yields where Complexity 3 erefore, where K is a positive constant.So, we get Integrating ( 17) from 0 to τ k ∧T and taking expectation on both sides, we have Set Ω k � τ k ≤ T  , and from inequality (10), we have P(Ω k ) ≥ ϵ.Note that, for every ω ∈ Ω k , there is at least one of x 1 (τ k , ω), x 2 (τ k , ω), or u(τ k , ω) equaling either k or (1/k), and then, we have 4 Complexity It can be obtained by ( 18) where 1 Ω k is the indicator function of Ω k , and letting k ⟶ ∞ yields is is a contradiction; we must have τ ∞ � ∞, and we have completed the proof.

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 (A 1 ) is established, the solution (x 1 (t), x 2 (t), u(t)) of system (5) has the property that lim sup where where (x * 1 , x * 2 , u * ) is the positive equilibrium point of the corresponding deterministic model (3).
Proof.Define the function where ω 1 and ω 2 are positive constants.Define By Itô's formula, we obtain where Similarly, where In the same way, 6 Complexity erefore, we have Let ( Integrate both sides of (18) from 0 to t and take the expectation, and then we get (36) Obviously, lim sup where ( We have completed the proof.eorem 2 shows that if the condition (A 1 ) holds, the solution oscillates around the equilibrium point x * , and the amplitude of oscillation is positively correlated with the intensity σ 2 1 and σ 2 2 of environmental noise.In particular, if σ 2 1 � σ 2 2 � 0, the influence of environmental noise is not taken into account: e equilibrium point x * is globally asymptotically stable.is is the conclusion of reference [17].

Persistence
In nature, whether ecosystems can survive or not is our main concern.Before discussing the persistence of stochastic system, we give the following assumption: Theorem 3.For any given initial condition (6), if assumptions (A 1 ) and (A 2 ) hold at the same time, the solution (x 1 (t), x 2 (t), u(t)) of system ( 5) is persistent that Proof.According to (37), we have lim sup As we know, x 1 (t) ≥ 0 and (44) □

Extinction
Define ( For the extinction of system (5), we have the following conclusions.

□
is is where we prove eorem 4.
en, the limit system of model ( 5) is as follows: From Lemma 1, we can conclude that (67) erefore,

Conclusions and Numerical Simulations
is paper proposes a stochastic Lotka-Volterra predatorprey model with discrete delays and feedback control.We firstly study the existence and uniqueness of global positive solution.By constructing appropriate Lyapunov functions and applying Itô's formula, we discuss the asymptotic behavior of stochastic system at the positive equilibrium point of the corresponding deterministic model.Finally, this paper gives the conditions for the persistence and extinction of stochastic system.eorem 3 shows that the system is persistent if the intensity σ i (i � 1, 2) of random disturbance and the coefficient c of feedback control variable satisfy the condition (A 2 ).eorem 4 indicates that (1) If the coefficient c of the feedback control cu(t)x 2 (t) remains unchanged and the intensity σ i (i � 1, 2) of random disturbance increases, the population cannot resist the disturbance of the external environment and extinct (2) If the intensity of random disturbance σ i (i � 1, 2) remains unchanged and the coefficient c of feedback control cu(t)x 2 (t) is small, Δ 1 > 0 and Δ 2 > 0 are satisfied, which will cause the continuous increase of predator number for a period of time, thus leading to the extinction of the prey population.
erefore, utilizing the feedback control measures to limit the predator quantity within a certain range is beneficial for the sustained existence of the population.
In order to verify the correctness of the theoretical analysis, we carry out the following numerical simulations.Choose the parameters in system (5) as follows: (76) Let Δt � 0.01 and the initial value x 1 (0) � 7, x 2 (0) � 6, and u(0) � 5.