Dynamics of a General Stochastic Nonautonomous Lotka-Volterra Model with Delays and Impulsive Perturbations

A stochastic nonautonomous N-species Lotka-Volterra model with delays and impulsive perturbations is investigated. For this model, sufficient conditions for extinction, nonpersistence in the mean, weak persistence and stochastic permanence are given, respectively. The influences of the stochastic noises, and the impulsive perturbations on the properties of the stochastic model are also discussed. The critical value between weak persistence and extinction is obtained. Finally, numerical simulations are given to support the theoretical analysis results.


Introduction
Population ecology is a major subfield of ecology that deals with the dynamics of species populations and the way these populations interact with the environment.It is concerned with the study of groups of organisms that live together in time and space and compete for the limited resources or in some way inhibit others' growth.Modelling of dynamic interactions in nature allows us to understand better how these complex interactions and processes work.The wellknown model that regards dynamic of population models is the Lotka-Volterra model.The investigation of the Lotka-Volterra model is one of the dominant themes in mathematical ecology due to its importance.The Lotka-Volterra model with delays has received more and more attentions and had lots of nice results [1][2][3][4][5].More details of the Lotka-Volterra model with delays are discussed in the books by Gopalsamy [6] and Kuang [7].
In the real world, the population models are inevitably influenced by the environmental noise which is an important component in an ecosystem [8][9][10].Moreover, May [11] has pointed out the fact that, due to environmental noise, the birth rate, carrying capacity, competition coefficient, and other parameters involved with the system exhibit random fluctuation to a greater or lesser extent [12].
On the other hand, populations may be affected by a variety of factors both naturally and manly, such as earthquake, drought, flooding, fire, crop-dusting, planting, hunting, and harvesting; the inner discipline of species or environment often suffers some dispersed changes over a relatively short time interval at the fixed times, which makes it unsuitable to be considered continually.In mathematics perspective, such sudden changes could be described by impulses.With the development of the theory of impulsive differential equations [13,14], we can establish adequate mathematical models of impulsive differential equations to investigate the dynamic behaviors of such ecosystems with impulsive effects.Consequently the dynamical behaviors of impulsive population dynamical models and stochastic population dynamical models have been extensively studied [15][16][17][18][19].The nonautonomous N-species Lotka-Volterra competitive system with impulsive perturbations was discussed in Hou et al. [15].Nspecies nonautonomous Lotka-Volterra competitive system with delays and impulsive perturbations was considered in Zhang and Teng [18].While these papers did not discuss the persistence and extinction of the stochastic Lotka-Volterra model, from the viewpoint of applications, it is critical to find out when the population will go to extinction or survival.
A major problem in population biology is to understand what determines extinction of a population.Population extinction is often a result of habitat destruction and modification which can be widespread.Moreover, dramatic changes in ecosystem structure or function are often caused by the species additions in the form of invasive species.In addition, the extinction of native populations may be caused by the growth of invasive species [20].Moreover, even large populations may be destroyed by some extraordinary perturbation [21].When the time is sufficiently large the population of some species may not become extinct, but the size of that population may be close to zero so that the species can be endangered.In other words, there exists a critical number between extinction and survival of population.In this sense, Ma and Hallam [22,23] proposed the concepts of nonpersistence in the mean and weak persistence for some deterministic models and Lu and Ding [24] applied these concepts to stochastic logistic models instead of the stochastic Lotka-Volterra model.
Inspired by the works referred above, in this paper, we will investigate the persistence and extinction of a general stochastic nonautonomous Lotka-Volterra model with delays and impulsive perturbations.To our knowledge, there are few results of this aspect for the stochastic nonautonomous Lotka-Volterra model.Moreover, all the publications have not obtained the persistence-extinction threshold for the general stochastic nonautonomous Lotka-Volterra model with delays and impulsive perturbations.The problems above are explored and some main results are given in this paper.The general stochastic nonautonomous Lotka-Volterra model with delays and impulsive perturbations which has a unique positive global solution is investigated.For this model, sufficient conditions for extinction, nonpersistence in the mean, weak persistence, and stochastic permanence are established.The influences of the stochastic noises and impulsive perturbations on the properties of the stochastic model are discussed.Comparing with deterministic results [25][26][27], if the noise is sufficiently small, the property permanence that the related deterministic system possesses is preserved in the stochastic model.However, with the increase of stochastic noise, the properties of the system may be changed greatly.For example, the solution to the associated stochastic model will be extinct with probability one with the increase of stochastic noise being sufficiently large, although the solution to the original deterministic model may be persistent.The properties of the system are not affected by the impulsive perturbations which are bounded; otherwise, the properties may be changed significantly.The critical value between weak persistence and extinction is obtained.
The rest of the paper is arranged as follows.The stochastic nonautonomous Lotka-Volterra model with delays and impulsive perturbations is formulated and some notations and preliminaries are given in Section 2. Section 3 shows that the general nonautonomous Lotka-Volterra model has a unique positive global solution.Then, sufficient conditions for extinction, nonpersistence in the mean, weak persistence, and stochastic permanence are established in Section 4. The simulation results in Section 5 are given to illustrate the main results obtained in this paper.Finally, the conclusions are given in Section 6.
On the other hand, in practice, model ( 1) is affected by impulsive perturbations.As a result, model (1) becomes the following stochastic nonautonomous Lotka-Volterra model with impulsive perturbations: Advances in Mathematical Physics 3 where  denotes the set of positive integers, 0 <  1 <  2 < ⋅ ⋅ ⋅ < lim  → +∞   = +∞.This model will be studied in this paper.
Then () is said to be a solution of ISDE (4).
For the aim of simplicity, we define the following notations: For any sequence {  ()} (1 ≤ ,  ≤ ), define The following definitions are commonly used and we list them here.

Advances in Mathematical Physics
Proof.Consider the following SDE without impulses: with the same initial condition as model ( 2).Now for model (7), there is a unique solution () on  ∈  and the solution will remain in  + with probability 1.
Since the coefficients of model (7) do not fulfil the linear growth condition, the general theorems of existence and uniqueness cannot be implemented for this equation.However, they are locally Lipschitz continuous; hence for any given positive initial condition () = ( 1 (),  2 (), . ..,   ())  ∈ C  , there is a unique local solution () on  ∈ (−∞,   ), where   is the explosion time.To show this solution () is global solution, which means that the explosion time   = +∞, a.s.Let us choose  0 > 0 which is sufficiently large such that For each integer  ≥  0 , define the stopping time where throughout this paper we set inf Ø = +∞ (as usual Ø denotes the empty set).Clearly,   is increasing as  → +∞.

Advances in Mathematical Physics
In addition, So the proof is completed.

The Persistence and Extinction Analysis
In this section, the extinction, nonpersistence in the mean, weak persistence, and stochastic permanence of model ( 2) are discussed.2) goes to extinction a.s.
Proof.Now applying Itô's formula to (7), we can have Then we have )   () ] where The quadratic form of Making use of the strong law of large numbers for martingales [30], The quadratic form of By virtue of the exponential martingale inequality [30], for any positive constants  0 ,  and , we have Advances in Mathematical Physics 7 Choose  0 = ,  = 1,  = 2 ln .Then it follows that Making use of the Borel-Cantelli lemma [30], one gets that, for almost all  ∈ Ω, there is a random integer  0 =  0 () such that, for  ≥  0 , sup This implies that for all 0 ≤  ≤ ,  ≥  0 a.s.Substituting this inequality, (24), and ( 25) into ( 23), we can obtain that for all 0 ≤  ≤ ,  ≥  0 a.s.Therefore, for all 0 ≤  ≤ ,  ≥  0 a.s.In other words, we can get that for 0 <  − 1 ≤  ≤ ,  ≥  0 , Taking superior limit on both sides of (32) and using (25), we have lim sup  → +∞ (ln   ()/) ≤  *  .That is to say, if  *  < 0, one can see that lim  → +∞   () = 0 a.s.So the proof is completed.2) is nonpersistent in the mean a.s.
When it comes to the study of population model, the role of stochastic permanence indicating the eternal existence of the population can never be ignorant with its theoretical and practical significance.So now let us show that the population   () is stochastic permanence in some cases.
Remark 9. Generally speaking, as the biology has implied, in Theorem 4, on one hand, if the species in the process of planting, that is, ℎ  > 0, or harvesting, that is, ℎ  < 0, is affected by stochastic environmental noises which plays a dominant role, then the species will be extinct a.s.In a word, population probably will come to an end in the worst cases which is revealed in Theorem 4, while if the growth rate and the influences of the stochastic noises and impulsive perturbations cancel each other out, then the effects of interspecific (for  ̸ = ) and intraspecific (for  = ) interaction at time , that is,   (), are the dominant factor.So the living chances are considerably rare which is shown in Theorem 5.In Theorem 6, even though the growth rate is larger than the influences of the stochastic noises and impulsive perturbations,   () plays the dominant role; then the population size is limited to zero with the time permitted; however, the opportunity of the survival of it still exists.In Theorem 7, if the growth rate is large enough, then the species will be stochastic permanence.This can well explain why the conditions are gradually stronger from Theorems 4-6.)], on one hand, we are conscious of the fact that the stochastic noise on   () is detrimental to the survival of the population but the stochastic noise on   () has hardly impressed on the persistence or extinction of the population.Thus, in true ecological modelling, the stochastic noise on   () should be realized but the stochastic noise on   () could be overlooked in some cases. ()/2)], we can find that the properties including extinction, nonpersistence in the mean, weak persistence, and stochastic permanence are not affected by the impulsive perturbations which are bounded and if the impulsive perturbations are unbounded the properties will change significantly.

Examples and Numerical Simulations
In this section, we explore system behavior numerical solutions of model (2).For convenience, consider the case  = 2 are satisfied.In view of Theorem 4, the population () and the population () will go to extinction a.s.In Figure 1(b), we consider   () = 1.12505 + 0.4sin ; the effects of interspecific (for  ̸ = ) and intraspecific (for  = ) interaction at time , that is,   (), are the dominant factor; then the conditions of Theorem 5 hold.By virtue of Theorem 5, the population () and the population () are nonpersistent in the mean a.s.In Figure 1(c), we choose   () = 1.6 + 0.4sin ;   () plays the dominant role; then the conditions of Theorem 6 are satisfied.That is to say, the population () and the population () are weak persistence a.s.This means that even though the species () and () are in the process of harvesting,   () plays the dominant role; then the opportunity of the survival of it still exists.In Figure 1(d), we consider   () = 4.9; the growth rate is large enough; then the conditions of Theorem 7 hold.Making use of Theorem 7, the population () and the population () are stochastic permanence.By the numerical simulations, we can find that stochastic noise on   () (1 ≤  ≤ ) can change the properties of the population models significantly.

Conclusions
In this paper, the persistence and extinction of a general stochastic nonautonomous N-species Lotka-Volterra model with time-varying, infinite delays and impulsive perturbations are investigated.Sufficient conditions for extinction, nonpersistence in the mean, weak persistence, and stochastic permanence are established in Theorems 4-7.The influences of the stochastic noises and impulsive perturbations on the properties of the stochastic model are discussed.On one hand, if the noise is small enough, the property permanence that the related deterministic system possesses is preserved in the stochastic model.On the other hand, with the increase of noise, the solution of the considered model (2) that will become extinct with probability one, nonpersistent in the mean, or weakly persistent has also been shown in this paper.According to  *  = lim sup  → +∞  −1 [∑ 0<  < ln(1 + ℎ  ) + ∫  0 (  () −  2  ()/2)], we can obtain the result that if the impulsive perturbations are bounded, the properties including extinction, nonpersistence in the mean, weak persistence, and stochastic permanence are not affected by the impulsive perturbations and if the impulsive perturbations are unbounded the properties will be affected by the impulsive perturbations and change greatly.Moreover, the critical number between extinction and weak persistence is obtained.Through the observation of Theorems 4-7, there is a very interesting phenomenon that the stochastic noise → +∞ P{|()| ≥ } ≥ 1 −  and lim inf  → +∞ P{|()| ≤ } ≥ 1 − , where | ⋅ | denotes the Euclidian norm in   + .