On a Stochastic Lotka-Volterra Competitive System with Distributed Delay and General Lévy Jumps

This paper considers a stochastic competitive systemwith 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 θ-method to discretize the stochastic model, are introduced to support the analytical findings.

In the real world, the intrinsic growth rates of many species are always disturbed by environmental noises (see, e.g., [7][8][9][10]), which was recognized by many scholars in recent years (see, e.g., [11][12][13][14]).In particular, May [7] 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  1 and  2 are stochastic; then by the central limit theorem, we can replace  1 and  2 by where, for  = 1, 2,   () represents a standard Brownian motion defined on a complete probability space (Ω, F, P) and  2   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, 2 Mathematical Problems in Engineering e.g., [15][16][17][18][19]).Bao et al. [15,16] did pioneering work in this field.In addition, Zou et al. [20][21][22] introduce a general Lévy jumps, which is more reasonable and complicated than the compensator jumps from the viewpoint of biomathematics (see [20]), 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: For convenience, we introduce the following notations: Moreover, we impose the following assumptions in this paper.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.

Lemma 3. Let Assumption 1 hold. For any given initial value
+ ); then system (3) has a unique positive solution () = ( 1 (),  2 ()) on  ≥ − a.s. and the solution satisfies Proof.The proof is similar to Han et al. [29] by defining where In addition, applying the inequality, for So we omit it here.Now let us prove inequality (6).
Case 2 ( = 2).The proof is similar to Case 1; we left out it here.The proof is complete.
The proof of (B) similar to (A) by symmetry and hence is left out.
(C) Suppose that Ψ 1 > Ψ1 and Ψ 2 > Ψ2 .Since Ψ 2 > Ψ2 , it then follows from (33) for sufficiently large .In virtue of (ii) in Lemma 2 and the arbitrariness of , we get Similarly, substituting (32)  Remark 6. Theorem 4 implies an important fact that when −1 <   () < 0,  = 1, 2, the jump process can result in extinction of the population   (), for example, earthquakes and hurricanes, and when   () > 0,  = 1, 2, 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.

Conclusions and Remarks
This paper investigates a stochastic competitive system with distributed delay and general Lévy jumps.Under the assumption Ψ > 0, 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][36][37]).We will research these kinds of permanence of model (3) in detail in our following study.

Figure 1 :Figure 1 Figure 1
Figure 1: The horizontal axis and the vertical axis in this and following figures represent the time  and the populations size (step size Δ = 0.001).