Research Article | Open Access
On a Stochastic Lotka-Volterra Competitive System with Distributed Delay and General Lévy Jumps
This paper considers a stochastic competitive system with distributed delay and general Lévy jumps. Almost sufficient and necessary conditions for stability in time average and extinction of each population are established under some assumptions. And two facts are revealed: both stability in time average and extinction have closer relationships with the general Lévy jumps, firstly; and secondly, the distributed delay has no effect on the stability in time average and extinction of the stochastic system. Some simulation figures, which are obtained by the split-step -method to discretize the stochastic model, are introduced to support the analytical findings.
In recent years, delay differential equations has been used in the study of population dynamics. A famous competitive system with distributed delay can be expressed bywhere denotes the size of the th population, , , and are all positive constants, and is a probability measure on . There is an extensive literature concerned with the dynamics of (1) and we here only mention Kuang and Smith , Faria , Freedman and Wu , Bereketoglu and Győri , and Gopalsamy  among many others. In particular, Kuang (see [6, p. 231]) claimed that if and , then model (1) has a positive equilibrium which is globally asymptotically stable, where , , and . It is important to point out that if and , then .
In the real world, the intrinsic growth rates of many species are always disturbed by environmental noises (see, e.g., [7–10]), which was recognized by many scholars in recent years (see, e.g., [11–14]). In particular, May  has pointed out that, due to environmental noises, the birth rates, carrying capacity, and other parameters involved in the system should be stochastic. In this paper, we assume that the parameters and are stochastic; then by the central limit theorem, we can replace and by where, for , represents a standard Brownian motion defined on a complete probability space and is the intensity of the noise.
On the other hand, the population systems may suffer sudden environmental perturbations, that is, some jump type stochastic perturbations, for example, earthquakes, hurricanes, and epidemics. Some scholars have concentrated on the population systems with compensator jumps, and some significant and interesting results have been obtained (see, e.g., [15–19]). Bao et al. [15, 16] did pioneering work in this field. In addition, Zou et al. [20–22] introduce a general Lévy jumps, which is more reasonable and complicated than the compensator jumps from the viewpoint of biomathematics (see ), into population models for the first time. However, there are no articles introducing the general Lévy jumps into population models with distributed delay, to the best of our knowledge. Motivated by these, we consider the famous stochastic competitive system with distributed delay and general Lévy jumps:where , is a real-valued Poisson counting measure with characteristic measure on a measurable subset of with , , is bounded function, and , ; furthermore, we assume that is independent of . Let the initial data , where represents the family of continuous functions from to with the norm , , . Other parameters are defined and required as before.
For convenience, we introduce the following notations:Moreover, we impose the following assumptions in this paper.
Assumption 1. There exists a positive constant such that for .
In this paper, we consider a stochastic competitive system with distributed delay and general Lévy jumps. Unlike the deterministic system, the stochastic system does not have an interior equilibrium. Therefore, we cannot investigate the stability of the stochastic system. In Section 2, we show that the solution to system (3) will tend to a point in time average. Furthermore, we establish almost sufficient and necessary conditions for stability in time average and extinction of each population. In Section 3, we present an example to illustrate our mathematical findings. Section 4 gives the conclusions and future directions of the research.
2. Main Content
Lemma 2 (see Liu et al. ). Suppose that (i)If there exist two positive constants and such that for all , where , , are constants, then(ii)If there exist three positive constants , , and such that for all , then a.s.
In order for the model to be significant, we shall show that the solution is global and nonnegative. However, theorem of existence and uniqueness ([24–28]) is not satisfied in system (3). By using method established by Mao et al. , we will show existence and uniqueness of the global positive solution of system (3).
Proof. The proof is similar to Han et al.  by definingwhereIn addition, applying the inequality, for ,So we omit it here. Now let us prove inequality (6).
Case 1 (). For any , applying the generalized Itô’s formula  to (3) results in ThusThe rest of proof is analogous with Lemma 4.4 in ; we omitted it here.
Case 2 (). The proof is similar to Case 1; we left out it here. The proof is complete.
Theorem 4. For system (3), we suppose that Assumption 1, and , holds. (I)If and , then both and are extinctive almost surely (a.s.); that is, a.s., .(II)If and , then is extinctive a.s. and is stable in time average a.s.; that is, (III)If and , then is extinctive a.s. and is stable in time average a.s.; that is, (IV)If , ,(A)If and , then is extinctive a.s. and is stable in time average a.s.: (B)If and , then is extinctive a.s. and is stable in time average a.s.: (C)If and , then both and are stable in time average a.s.:
Proof. Applying Itô’s formula  to the first equality in (3), we get Making use of the Fubini theorem and a substitution technique, we have Therefore, we deriveSimilarly,(I) Assume that and From (9), By the strong law of large numbers for martingales, we therefore have a.s., ; meanwhile, we define, for , , . Under Assumption 1,, making use of the strong law of large numbers for local martingales (see, e.g., ), we then derive and that Consequently, , a.s. Similarly, by (19), we can show that if , then , a.s.
(II) Suppose that and Since , then, by (I), , a.s. Hence, making use of (9), for arbitrary , there is such that, for , Substituting the above inequalities into (19), we can see that, for ,Since , we can choose sufficiently small such that Applying (i) and (ii) in Lemma 2 to (25) and (26), respectively, we deriveLet . Then, we have , a.s.
The proof of (III) is homogeneous with (II) by symmetry; hence it is omitted.
Now let us prove (IV). For , consider the following equation:In virtue of the classic stochastic comparison theorem , we can find thatSince , , similar to the proof of (II), we can show that Thuswhich, together with (29), implies thatOn the other hand, calculating deducesBy virtue of (6) and (32), for arbitrary , there is such that, for , Substituting the above inequalities into (33) results infor . Meanwhile, calculating yieldsUsing the same way, by (36) we can have that, for ,(A) Suppose and Note that , and then let be sufficiently small such that Applying (i) in Lemma 2 to (35) gives , a.s. The proof of , a.s., is similar to (II) and hence is omitted.
The proof of (B) is similar to (A) by symmetry and hence is left out.
(C) Suppose that and Since , it then follows from (33) and Lemma 2 thatMaking use of the arbitrariness of , we can see thatIt follows from (37), Lemma 2, and the arbitrariness of thatlikewise. Let be sufficiently small such that When (32) and (39) are used in (19), we getfor sufficiently large . In virtue of (ii) in Lemma 2 and the arbitrariness of , we getSimilarly, substituting (32) and (40) into (20) brings about , a.s. This, together with (39), (40), and (42), means and , a.s.
Remark 5. It is important to designate that if , , and , then and cannot hold simultaneously.
Remark 6. Theorem 4 implies an important fact that when , , the jump process can result in extinction of the population , for example, earthquakes and hurricanes, and when , , the jump process is always advantage for the population , for example, ocean red tide.
Remark 7. From the perspective of the condition in Theorem 4, the distributed delay does not influence some the properties including extinction and stability in time average.
3. Numerical Simulations
In this section, we employ the split-step -method, whose approximate solution is mean-square convergent with order (see [32, 33]), to discretize (3). Here, we choose the initial data , , , , , , , , , and . Then . The main difference between the conditions of the following Case 1–Case 4 is that the values of and are different. In Case 1, we let and . Then by virtue of Kuangs work , we have that the positive equilibrium is globally asymptotically stable. Figure 1(a) verifies this. In Case 2, we set , , and . Then and . In view of (A) in Theorem 4, goes to extinction andFigure 1(b) confirms this. In Case 3, we choose , , , and . Then and . It follows from (B) in Theorem 4 that goes to extinction and See Figure 1(c). In Case 4, we let , , , and , that is, and . According to (C) in Theorem 4, we obtain Figure 1(d) validates this.
4. Conclusions and Remarks
This paper investigates a stochastic competitive system with distributed delay and general Lévy jumps. Under the assumption , the almost complete parameter analysis is fulfilled in detail. Our results imply that the general Lévy jumps can significantly change the properties of population models.
Some interesting and significant topics deserve our further engagement. One may put forward a more realistic and sophisticated model to integrate the colored noise into the model [10, 11, 34]. Another significant problem is devoted to stochastic model with infinite delays and general Lévy jumps. We will leave these for future investigation.
It should also be mentioned that “stability in time average” is not a good definition of persistence for stochastic population models. Some papers have introduced more appropriate definitions of permanence for stochastic population models, that is, stochastically persistent in probability or stochastic permanence (see, e.g., [35–37]). We will research these kinds of permanence of model (3) in detail in our following study.
The authors declare that they have no competing interests.
This work is supported by the National Natural Science Foundation of China (no. 11501150), a Project of Shandong Province Higher Educational Science and Technology Program of China (J16LI09), the National Social Science Foundation of China (16BJL087), and the instructional reform item of Higher Education of Shandong Province (2015M091).
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