Abstract
This paper is devoted to the dynamics of a stochastic modified Bazykin predator-prey system with Lévy jumps. First, we show that the system has a unique global positive solution and give some properties of solutions. Then, some sufficient conditions for persistence and extinction are derived by Itô formula and some inequalities on stochastic analysis. At last, some simulations are provided to check the main results.
1. Introduction
In population biology, construction of models and relevant qualitative analysis are popular fields [1, 2]. In the last few decades, many predator-prey models with functional and numerical responses have been proposed and studied extensively. Particularly, ratio-dependent functional response is common in some classical literature [3–5].
Alexeev and Bazykin firstly proposed a Bazykin system [1] where and are the size of prey and predator at time . , and are positive constants (some details refer to [1]).
Considering the ratio effect in hunting process of predation, Haque built a modified Bazykin system [2] of which dynamical behavior near equilibrium point and bifurcation are observed. System (2) is more reasonable and many researchers began to pay more attention on it. Its generalized forms have been investigated and a lot of results were obtained [6–8].
However, in the natural world, environmental noise is everywhere, and the growth rate of the populations is not constants. In this case, many systems are described by stochastic differential equations driven by Brownian motion of the actual research. For example, [9] studied the following stochastic modified Bazykin system: where and are two independent Wiener processes defined on a complete probability space and and stand for the level of the white noises. Some sufficient conditions for persistence and extinction are obtained.
Recently, the stochastic differential equation driven by jump has drawn more and more researchers’ attention [10–17]. This is mainly according to sudden perturbation of environment, such as toxic contamination of water, torrential flood, and hurricane. Motivated by the above, this paper considers the following stochastic modified Bazykin system with Lévy jumps: where , represent the left limit of , . , is a Poisson random measure, and is the characteristic measure of on a measurable subset with .
The rest of this paper is organized as follows. In Section 2, some properties of positive solutions to system (4) are discussed. In Section 3, the main results for persistence and extinction are given. Finally, the simulation results show the validity of our results.
2. Properties of Positive Solutions
Throughout this paper, we require that , , and are independent andwith .
First, we present the global existence of positive solutions.
Lemma 1. For any given value , system (4) has a unique positive solution on and the solution will be in a.s. (almost surely).
Proof. Let , then we consider the following system: with initial date on . Since the coefficients of system (6) are locally Lipschitz continuous, then there is a unique local solution on a.s., where is blow-up time. Then is the unique positive local solution to system (4) with initial data . We will show that , and this mean the solution is global.
Consider the following equations:By the comparison theorem of stochastic differential equation, we conclude that According to [10], we can givewhere Because is the existence range of solutions , that means .
Next, we will show the asymptotic property of the solution to system (4).
Theorem 2. The solutions of system (4) are bounded in mean.
Proof. Let . Direct computation, by the formula [18], now leads towhere From actual meanings of parameters and assumption , we get thatis bounded. There exists a constant , such that when ; otherwise . Let ; we have . Denote , then . Similarly, .
3. Persistence and Extinction
In this section, some properties of the solutions of system (4) are investigated. Some sufficient conditions for persistence and extinction are shown. To proceed, some definitions and lemma are as follows.
Definition 3 (see [19]). (1) The population is called to be extinct if
(2) The population is called to be persistent if a.s.
(3) System (4) is called to be persistent if populations and are all persistent a.s.
Lemma 4 (see [20]). Under and , suppose .
(1) If there exist three positive such that for all , where are constants, thenIf and other conditions are the same, then a.s.
(2) If there exist three positive such that for all , where are constants, then Now, the main results about persistence and extinction of system (4) are as follows.
Theorem 5. (1) The populations and are extinct if a.s.
(2) The population is persistent and is extinct if a.s.
(3) The populations and are persistent if a.s.
Proof. (1) By using the Itô formula, the following formulas are hold: Calculated by integral, we have the following form: Then by Lemma 4, note that , and therefore (population is extinct) a.s. Then, for arbitrarily small , there exist a sufficiently large , when , and we have and where . It is clear that (population is extinct) a.s. Thus (1) is correct.
(2) From (1), the following forms are clear. Then by Lemma 4, we have Therefore the population is persistent in mean. For population , we have From Lemma 4 and condition , we know that (i.e., population is extinct) a.s.
(3) In view of above, we can conclude the following when , andThen, we have where is the minimum of .
It is clear that population is persistent.
Remark 6. For (1) in Theorem 5, it implies that the population of extinction a.s. leads to the extinction of population a.s. As shown in Figure 2, the simulations also affirm this point. In Figure 2, we can see that population becomes extinct, and after a while, population becomes extinct.
4. Numerical Simulations
We will demonstrate our results with the help of numerical simulations by using the Euler-Maruyama scheme [21, 22]. In numerical simulation, randomly selected parameters are as follows , and , with simulation time span and step size , where . The initial data is in Figure 2, and others are .
(1) As illustrated in Figure 1, we choose , then , , and . By Theorem 5, the populations and are extinct a.s. Numerical experiments verify the correctness of (1) in Theorem 5.
(2) As illustrated in Figure 2, we choose , then , , and . By Theorem 5, the populations and are extinct a.s. Numerical experiments also verify the correctness of (1) in Theorem 5.
(3) As shown in Figure 3, we choose , then , , and . By Theorem 5, the population is persistent a.s. and population is extinct a.s. The correctness of (2) in Theorem 5 is verified.
(4) As shown in Figure 4, we choose , then , , and . By Theorem 5, the populations and are persistent a.s. The validity of correctness of (3) in Theorem 5 is verified.
5. Conclusions
In this paper, a stochastic modified Bazykin system with Lévy jumps is discussed. Firstly, the existence of global positive solutions is investigated, and boundedness in mean is also given. Further, under related assumption, we obtain the sufficient conditions for the stochastic permanence and extinction of system (4). There are some interesting topics which deserve further discussion. For example, many authors consider the stationary distribution for stochastic predator-prey model with harvesting and delays [23–25], epidemic model with regime switching [26], impulsive stochastic model [27], and budworm growth model with Markovian switching [28]. The above investigations are left for future work.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this manuscript.
Acknowledgments
This work is supported by National Natural Science Foundation of China (nos. 11771001, 11471015, and 11401002), Nature Science Foundation of Anhui Province (no. 1508085QA01 and no. 1508085MA10), Natural Science Foundation of Anhui Colleges (no. KJ2014A010), Program for Excellent Young Talents in University of Anhui Province (no. gxyq2017092), and Anhui Province Workshop of Prestigious Teacher (no. 2016msgzs006).