Abstract
In this paper, we investigate the stochastic dynamics of two dispersal predator-prey systems perturbed by white noise, impulsive effect, and regime switching. For the system just interrupted by white noise, we first prove that the stochastic impulsive system has a nontrivial positive periodic solution. Then the sufficient conditions for persistence in mean and extinction of the system are obtained. For the system with Markov regime switching, we verify it is ergodic and has a stationary distribution. And conditions for extinction of the prey species are established. Finally, we provide a series of numerical simulations to illustrate the theoretical analysis.
1. Introduction
Due to the ecological effects of human activities and climate changes, e.g., the pollution of the environment and the construction of manufacturing industries and highways, many complete habitats have been increasingly broken into many smaller patches. In some patches, there is less environmental pollution but more predation, and, in other patches, the situation is quite the opposite. As a consequence, the biological species disperse among different patches to search for a better habitat, which has profound effects on both the permanence and evolution of species. Therefore, the effect of dispersion on the possible persistence of single or multiple species has become an imperative subject in population biology and various dispersal models have been investigated by many authors in recent years.
Levin [1] firstly proposed a single species dispersal system in 1974. Takeuchi in [2] established the sufficient conditions for persistence of the single dispersal system and verified the existence and uniqueness of the globally positive equilibrium. Then two species or multiple species dispersal system have been investigated by many authors [3–7]. Kuang et al. investigated the relationship between the stability of the population and the dispersal rate of the prey species in [3]. When two species interact, one as a predator and the other as prey, then the Lotka-Volterra predator-prey model is frequently used to describe the dynamics of the biological system [8–12]. Cui [4] explored a nonautonomous dispersal predator-prey system and the sufficient conditions for persistence were obtained. In [5], a threshold for the persistence and extinction of a dispersal predator-prey model with time delay was derived.
In view of the environmental pollution, we consider a dispersal predator-prey system with impulsive toxicant input [13–17], which can be described as follows:where are the prey density in patch at time and denotes the predator density at time . The prey species can disperse between two patches and represent the dispersal rates in different patches. , , and . The environments of two patches are polluted, is the toxicant concentration in environment at time and is the toxicant concentration in the organism at time . and . are the intrinsic growth rates of each species. stand for the intraspecific competition rates of each species. The ratio-dependent functional responses are defined by , where represents the predator interference. and are predation rates and ingestion rates, respectively. are dose-response parameters of the prey in different patches to the toxicant. represents the toxicant uptake rate of the prey species from environment, and and denote the organismal net depuration rate and net ingestion rate, respectively. stands for the loss rate of toxicant in environment and is the pulsed input concentration of the toxicant at period , . All the parameters above are positive.
It is acknowledged that deterministic model cannot describe the species activities completely since the species activities are disturbed by environmental noise frequently. Now let us take white noise [18–34] into account. Suppose that parameters are stochastically perturbed by the white noise like where are mutually independent Brownian motions defined on a complete probability space with a filtration satisfying the usual conditions (i.e., it is increasing and right continuous while contains all -null sets). represent the intensities of the white noise. Li et al. in [34] considered a stochastic dispersal population system; then, we perturb system (1) by white noise to the following impulsive stochastic dispersal predator-prey system
Generally, there are two types of noise in environment. One is white noise, and the other is colored noise, namely, telegraph noise. The telegraph noise is always described as a switching in two or more environmental regimes distinguished by factors such as temperature and humidity. According to the experiment done by Aquaculture Institute of Guizhou Province in 2008 [35], the reproductive effects of African catfish in different season were compared by using local African catfish fed by native fisherman with artificial propagation of oxytocin, in which induced spawning groups were 14 in rainy season and 10 in dry season with male and female ratio 1:2. The results show that the average egg number of per kilogram is 29593 in rainy season, which is 89.06 percent more than that in dry season. It is obvious that different seasons can affect the intrinsic growth rates of the species significantly. Recently, the stochastic population models under regime switching have attracted many authors’ attention [36–46]. Usually, the regime switching between environmental regimes is memoryless and the waiting time for the next switching follows exponential distribution. In this paper, we model the telegraph noise by a continuous-time Markov chain with finite-state space , then system (3) becomesFor each , all parameters above that related to regime are positive.
This paper is organized as follows. In Section 2, we introduce some notations and lemmas, which are necessary for later discussion. In Section 3, we verify the existence of a positive -periodic solution of system (3) and establish the sufficient conditions for persistence in mean and extinction of the prey species. In Section 4, we use a class of stochastic Lyapunov functions with regime switching to obtain the ergodic property and positive recurrence of system (4), which account for some recurring events of a population system. We also analyze the extinction of system (4). Furthermore, numerical simulations and examples are introduced and carried out to illustrate the theoretical results in Section 5. Finally, we give some discussions in Section 6.
2. Preliminaries
Throughout this paper, define . is a right-continuous Markov chain taking values in a finite-state space with generator given by where . are the transition rates from to if while . In this paper, we assume that and are independent and the Markov chain is irreducible, which means that the system can switch from any regime to any other one. This is equivalent to the condition that for any , there are finite numbers such that and so implies the ergodicity property according to Markov theory for finite states. Note that always has a trivial eigenvalue. The algebraic interpretation of irreducibility is that rank. Under these conditions, the Markov chain has a unique stationary distribution which can be determined by solving the linear equation subject to .
For convenience, we define , , and .
Firstly, we consider a nonautonomous stochastic differential equation
Lemma 1 (see [42]). Suppose that the coefficients of (6) are -periodic in and there exists a function which is -periodic in , and satisfies the following conditions:
(i) ;
(ii) outside some compact set.
Then there exists a solution for system (6) which is a -periodic Markov process.
Then we introduce a lemma which gives a criterion for positive recurrence in terms of Lyapunov functions. Consider the diffusion process described by the equation:where , satisfying . For each and for any twice continuously differentiable function , has a generator given as follows: where
Lemma 2 (see [43]). If the following conditions are satisfied:
(i) for ;
(ii) for each , is symmetric and satisfies with some constants for all ;
(iii) there exists a bounded open subset of with a regular (i.e., smooth) boundary satisfying that for each there exists a nonnegative function such that is twice continuously differentiable and that for some , , for any , the model (7) is ergodic and positive recurrent. That is to say, there exists a unique stationary density and for any Borel measurable function such that , we have .
Now, we consider the subsystem of systems (3) and (4),
Lemma 3 (see [13]). System (11) has a unique positive solution -periodic solution and for each solution of (11), , as , where for and .
By Lemma 2.2 of [14], we obtain that
Following the same argument as in [13, 22, 36, 47], we can prove the existence and uniqueness of the global positive solution for systems (3) and (4). The proof is omitted here.
3. Analysis of System (3)
By Lemma 3, we know that the limit system of (3) iswith initial value , where is positive and continuous function of period .
3.1. Existence of Periodic Solution of System (3)
Now, we discuss the existence of periodic solution of system (14). Define , .
Theorem 4. Suppose that ; then, there exists a positive -periodic solution for system (14).
Proof. Obviously, is continuously bounded positive periodic functions in . and . We need to show that conditions (i) and (ii) in Lemma 1 hold. Define a nonnegative -function where , and . And are functions defined on satisfying and . Obviously, are two -periodic functions on . Therefore, is -periodic in and satisfies where . Therefore, condition (i) of Lemma 1 holds. Then we prove that condition (ii) of Lemma 1 holds.
Applying Itô’s formula to system (14), we haveSimilarly,andTherefore,Define a bounded closed set where is a sufficiently small number such thatwhere ;where ;where ;wherewherewhere Denote Note that . Now, we prove that , .
Case 1. If , from (20), we obtain thatCase 2. If , from (20), one hasCase 3. If , from (20), we haveCase 4. If , from (20), we haveCase 5. If , from (20), we haveCase 6. If , from (20), we haveHence, Therefore, the proof of Theorem 4 is completed.
3.2. Extinction of System (3)
Theorem 5. For any initial value of system (3), if , then the prey is extinct, where , .
Proof. Define . Applying Itô’s formula to the limit system (14) of (3) yieldsApplying Itô’s formula again, we obtain thatIntegrating both sides from 0 to , one haswhere are local martingales defined respectively by By strong law of large number, we have Dividing (39) by on both sides and then letting , we haveTherefore, , which means the prey species is extinct.
3.3. Persistence in Mean of System (3)
Theorem 6. For any initial value of system (3), if , then the prey species is persistent in mean, where , , .
Proof. Define . Applying Itô’s formula yieldswhere .
Integrating (43) from 0 to , we getwhere
Dividing (44) by on both sides and then letting , we havewhich implies thatAccording to Corollary 3.1 in [13], we have where is a constant, which means the prey species is persistent in mean.
4. Analysis of System (4)
According to (13), we consider the following system:with initial value .
4.1. Extinction of System (4)
Theorem 7. For any initial value of system (4), if , then the prey species will be extinct, where and .
Proof. Similar to the proof of Theorem 4, we define . Applying Itô’s formula to (48), one hasApplying Itô’s formula again, we can obtain thatThen,By the ergodic theory of the Markov chain and the strong law of large number, we haveTherefore,
4.2. Unique Stationary Distribution of System (4)
Theorem 8. Assume that for , , , and , then the stochastic process of model (4) is ergodic and has a unique stationary distribution in .
Proof. By the assumption for , condition (i) in Lemma 2 is satisfied. Let , then Define a bounded open subset as follows: where is a constant. Let , where and . For , we have for all . Thus condition (ii) in Lemma 2 holds. Therefore, it remains for us to verify condition (iii) in Lemma 2.
Define a -function on , where and . is a sufficiently small positive constant satisfying .
An application of the operator to yieldsDefine the vectors , and with , and . As the generator matrix is irreducible, for each , , and , there exists , , and , respectively, which is a solution of the Poisson system Therefore we haveCombining (60), (61) and (62) with (58), we obtainApplying Itô’s formula to yieldsThus can be estimated as follows: It is easy to see that Consequently, we know that, for a sufficiently small , By Lemma 2, we get that is ergodic and positive recurrent; that is, system (4) has a unique stationary distribution.
This completes the proof of Theorem 8.
5. Examples and Simulations
In this section, we introduce some examples and numerical simulations to illustrate our main results.
Example 1. We choose the parameter values in system (3) as follows: , , , , , , , , , , , , , , , , . In addition, we let , , , , for the subsystem. By computation, we obtain that and , satisfying the conditions in Theorem 4.
Figure 1 verifies Theorem 4 from two-dimension and three-dimension. Therefore, there exists a -periodic solution for system (3).
(a)
(b)
Example 2. Firstly, we choose the same parameter values in Example 1, satisfying the conditions in Theorem 6. From the left figure of Figure 2, we find that the prey species and the predator are persistent in mean for system (3).
For the extinction of system (3), decreases from 0.1 to 0.003 and decreases from 0.2 to 0.002. Additionally, we choose , , , , , and . The other parameter values are the same as the values in Example 1. The results are shown in the right figure of Figure 2, in which the prey species both in patch 1 and 2 are extinct.
We suppose that the limit system of system (4) has regimes, i.e., . Then system (48) has subsystems where .
Example 3. Let be a right-continuous Markov chain taking values in (i.e., ) and the generator of the Markov chain is
Hence, . The Markov chain is illustrated in Figure 3.
In regime 1, we choose , , , , , , , , , , , , , , , , , .
In regime 2, we choose , , , , , , , , , , , , , , , , , .
When , we can observe that . The conditions in Theorem 7 hold.
From Figure 4, the prey species in two patches are extinct, which is consistent with Theorem 7.
Example 4. The Markov chain is identical with that in Example 3.
In regime 1, we choose , , , , , , , , , , , , , , , , , .
In regime 2, we choose , , , , , , , , , , , , , , , , , .
When , we obtain that and . The conditions in Theorem 8 hold.
The analytical results are consistent with the simulation results as illustrated in Figures 5 and 6.
From Figures 5 and 6, we know that there exist two density functions which are displayed in Figure 7. Therefore, the prey species has a stationary distribution.
6. Discussion
It is theoretically and practically valuable to investigate the dispersal population system. In this paper, we explore a predator-prey Lotka-Volterra system in two patches. Taking white noise, regime switching and impulsive toxicant input into account, two stochastic impulsive patchy systems are established. We discuss some properties of the two systems and the results are shown as follows.
(i) Suppose that ; then, there exists a positive -periodic solution for system (3).
(ii) For any initial value of system (3), if , then the prey will be extinct.
(iii) For any initial value of system (3), if , then the prey species will be persistent in mean.
(iv) For system (4), given initial value , if , then the prey species will be extinct.
(v) Assume , and ; then, the stochastic process of model (4) is ergodic and has a unique stationary distribution in .
From (ii) and (iii), we can confirm that the extinction and persistence of the dispersal prey species are partly determined by and , where and . When is sufficiently small such that , the prey species is destined to be extinct. In contrary, if is large enough such that , then the prey species will be persistent almost surely. Example 2 also further illustrates the conclusions.
When the dispersal predator-prey system is perturbed by telegraph noise, from (iv), the condition for extinction of the prey species is similar to (ii). However, the parameters are relevant to the regime . That is to say, the variations of dispersal rates and are also subject to the regimes. Different from (iii), the conditions for persistence of the prey in patch are mainly affected by the dispersal rate in the -th patch in (v). If the dispersal rate in patch is not large enough such that , then the prey in patch will be persistent. Examples 3 and 4 simulate the results (iv) and (v).
With the construction of manufacturing industries and highways, refuge can be introduced into the Lotka-Volterra system to enrich the population models, which can simulate the natural reserves for animals. On the other hand, this also contributes to the evolution of the species, which can be found in [48, 49]. Also, Song et al. have explored the spatiotemporal dynamics of diffusive predator-prey models, such as steady-state bifurcations and Turing patterns (see [50–52]). Moreover, in predator-prey systems, time delay [53–56] also should be considered since the digestion process of the predator consumes time. These interesting issues deserve further investigation.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the Research Fund for the Taishan Scholar Project of Shandong Province of China, the SDUST Research Fund (2014TDJH102), and National Natural Science Foundation of China (11561004).