Permanence and Periodic Solutions for a Two-Patch Impulsive Migration Periodic N-Species Lotka-Volterra Competitive System

We study a two-patch impulsive migration periodic N-species Lotka-Volterra competitive system. Based on analysis method, inequality estimation, and Lyapunov function method, sufficient conditions for the permanence and existence of a unique globally stable positive periodic solution of the system are established. Some numerical examples are shown to verify our results and discuss the model further.


Introduction
Owing to natural enemy, severe competition, seasonal alternative, or deterioration of the patch environment, species dispersal (or migration) in two or more patches becomes one of the most prevalent phenomena of nature.Generally speaking, species dispersal is mainly concluded as the following three types: (i) dispersal occurs at every time and happens simultaneously between any two patches, that is, continuously bidirectional dispersal; (ii) dispersal occurs at some fixed time and happens simultaneously between any two patches, that is, impulsively bidirectional dispersal; (iii) dispersal shows itself as a total migration form, that is, impulsively unilateral diffusion (or migration).
Many empirical works and monographs on population dispersal system with type (i) have been done (see [1][2][3][4][5][6] and references cited therein).For example, in [3] where   () represents the dispersal rate from patch  to patch  at time  and the dispersal established in this model is continuous and bidirectional; that is, the dispersal occurs at every time and happens simultaneously between any two patches  and .In recent years, some population dynamical models with impulsively bidirectional dispersal have been proposed and studied (see [7][8][9][10] and references cited therein).For instance, in [7], the authors studied the following autonomous impulsive diffusion single species model: where   ( = 1, 2) is the dispersal rate in the th patch.
The pulse diffusion occurs at every  period ( is a positive constant).Obviously, in this model, species  inhabits, respectively, two patches before the pulse appears; when the time at the pulse comes, species  in two patches disperses from one patch to another, that is, impulsively bidirectional dispersal.
However, in all of these investigated dispersal models considered so far, there are few papers to consider the total impulsive migration system, that is, impulsively unilateral diffusion (type (iii)) system.Practically, in the real ecological system, with seasonal alternative, some kinds of birds or vegetarians will migrate from cold patches (or food resource poor patches) to warm patches (or food resource rich patches) in search for a better habitat to inhabit or breed; fish will go back from ocean to their birthplace to spawn and so on.Obviously, this kind of diffusing behavior exists extensively in the real world.Therefore, it is a very basic problem to research this kind of impulsive migration systems.Zhang et al. in [11] studied a single species model with logistic growth and dissymmetric impulse dispersal and obtained some very general, weak conditions for the permanence, extinction of these systems, existence, uniqueness, and global stability of positive periodic solutions by using analysis based on the theory of discrete dynamical systems.In our previous work [12,13], a two-patch impulsive diffusion periodic singlespecies logistic model (see [12]) and a two-patch prey impulsive diffusion periodic predator-prey model (see [13]) have been proposed and studied and some interesting results have been established, respectively.In this paper, we will continue our study on the two-patch impulsive diffusion model to a -species competitive system.Motivated by the above analysis, in this paper, we consider the following two-patch impulsive migration periodic species Lotka-Volterra competitive system: where   is the population density of the th species;  1 () and  2 () represent the intrinsic growth rates of the th species in patch 1 and in patch 2, respectively;  1 () and  2 () denote the intraspecific competition coefficients of the th species in patch 1 and in patch 2, respectively;  1 () and  2 () ( ̸ = ) are the interspecific competition coefficients between the th species and the th species in patch 1 and in patch 2, respectively.The species migration occurs at every pulse time  +1 ( = 0, 1, 2, . ..),where  0 = 0,  1 <  2 < ⋅ ⋅ ⋅ <   < ⋅ ⋅ ⋅ is sequence of positive numbers with lim  → ∞   = +∞.We suppose that the system is composed of two patches.When  ∈ [ 2 ,  2+1 ), all the species live in patch 1; because of the change of the environment, the populations will migrate to patch 2 and the migration loss is  1 ( − 2+1 ) ( = 1, 2, . . ., ); then the populations will live in patch 2 during the period  ∈ [ 2+1 ,  2+2 ).When the environment changes again, all the populations will migrate back to the previous patch; here, the migration loss is  2 ( − 2+2 ).In this paper, we always assume the following: In addition, we assume that the investigated  species always migrate between the two patches almost simultaneously.We will establish some sufficient conditions for the permanence, extinction, and existence of a unique globally asymptotically stable positive periodic solution of the system.The methods used in this paper are inequality estimation and Lyapunov functions which are introduced in work [14] "the permanence and global stability for nonautonomous species Lotka-Volterra competitive system with impulses." The organization of this paper is as follows.In Section 2, as preliminary, an important lemma on the two-patch impulsive migration periodic single-species logistic model is introduced.In Section 3, sufficient conditions on the permanence and extinction of system (3) are established.In Section 4, conditions for the existence and global stability of the unique positive periodic solution are obtained.Finally, some examples and numerical simulations are proposed to illustrate the feasibility of our results and discuss the model further.
() If system (4) satisfies then it has a unique globally attractively positive -periodic solution  * (); that is, () If condition ( 6) is replaced by and condition ( 5) is retained, then Proof.Due to the fact that the population dispersal is only restricted in two patches and shows itself as aggregate migration, we can rewrite system (4) as follows: In order to prove proposition (a), firstly, we prove the permanence of system (4); that is, there exist two positive constants  and  such that for any positive solution () of system (4) we always have From conditions ( 5) and ( 6), there are positive constants  1 ,  2 , and  such that We first of all prove that there is a constant  > 0 such that lim sup for any positive solution of system (4).In fact, for any positive solution of system (4), we only need to consider the following three cases.
Next, we consider Case 3. Obviously, there is  1 ⩾ 0 such that ( 1 ) <  1 .Then we prove that, for all  ⩾  1 , where Furthermore, there exists and integrating this inequality from  3 to  2 we have which contradicts with (17).This proves that (16) holds.Lastly, if Case 2 holds, then we directly have Choose constant  =  1 exp( 0 ) + 1; then we see that ( 14) holds.By a similar argument as in the proof of ( 14) we can prove that there is a constant  > 0 such that lim inf for any positive solution () of system (4).Conclusion ( 11) is proved.Now, we prove proposition (a).Let () and  * () be any two positive solutions of system (4).It follows from ( 11) that there are positive constants  and  such that Choose Lyapunov function as follows: For any  = 0, 1, 2, . .., we have Hence, () is continuous for all  ∈  + and from the Mean-Value Theorem we can obtain Calculating the upper right derivative of (), then from (25) we obtain From this, we further have, for any  =  + , where  ⩾ 0 is an integer and 0 ⩽  <  is a constant, Hence, () → 0 as  → ∞.Further from (25) we obtain lim Lastly, we prove that system (4) has a unique positive -periodic solution.Consider the sequence  * (,  * 0 ).It is obviously bounded in the interval [, ] for all  = 0, 1, . ... Let  * be a limit point of this sequence, and hence  * = x * .The solution  * (,  * ) is the unique periodic solution of system (4).By (28), it is globally attractive.This completes the proof of proposition (a).Now we prove proposition (b).From ( 5) and ( 8), for any constant  > 0, there is a positive constant  0 such that From this, a similar argument as in the proof of ( 14), we can obtain for all  large enough.Finally, from the arbitrariness of , we obtain () → 0 as  → ∞.Lemma 1 is proved.

Permanence and Extinction
We first discuss the permanence of all species of system (3).A similar analysis as system (4), system (3) can also be written as follows: For each  = 1, 2, . . ., , we consider the following two-patch impulsive migration systems as the subsystems of system (3): On the permanence of all species   ( = 1, 2, . . ., ) for system (3) we have the following result.
Next, we prove that there is a constant  > 0 such that lim inf We only need to consider the following three cases for each  = 1, 2, . . ., .
For Case 1, since   () ⩽  0 for all  ⩾  4 , then let  =  4 + , where  is any positive integer; integrating system (33) from  4 to , by ( 40) and (43) we have Hence,   () → 0 as  → ∞, which leads to a contradiction.For Case 3, obviously, there is  5 ⩾   0 such that   ( 5 ) >  0 .Then we prove that, for all  ⩾  5 , where If ( 46) is not true, then there is  6 >  5 such that Moreover, there exists  7 ∈ ( 5 ,  6 ] such that If  6 =  7 ,  6 must be an impulsive time.Then there exists a positive integer  such that  6 =  2 or  6 =  2+1 ; thus we have From this we can obtain and integrating this inequality from  7 to  6 we have which contradicts with (48) too.This proves that (46) holds.Lastly, if Case 2 holds, then we directly have Then  is independent of any positive solution of system (3) and we finally have lim inf This completes the proof of Theorem 3.
Next, we study the extinction of all species   for system (3); we have the following result.

Numerical Simulation and Discussion
In this paper, we have investigated a class of two-patch impulsive migration periodic -species Lotka-Volterra competitive system.By means of inequality estimation and Lyapunov functions, we have given the criteria for the permanence, extinction, and existence of the unique globally stable positive periodic solution of system (3).
However, if the survival environment of the two patches is austere, the intrinsic growth rates of the two species will decrease.Hence, if we take  which satisfy condition (57) of Theorem 4 (condition (56) is obvious).Hence, all species of system (73) will go extinct.See Figure 3(a).
In addition, if the environment of the two patches is survivable, but the migration loss of the two species is large, that is, if we take  11 = 0.7,  12 = 0.6,  21 = 0.5, and  22 = 0.6 and all other coefficients are unchanged, then we can verify that   and then we have ( 1 ) ⩽ 0 if 0.6453 ⩽  1 < 1 and ( 2 ) ⩽ 0 if 0.4598 ⩽  2 < 1 (see Figure 4); that is, species  1 and  2 will go extinct if 0.6453 ⩽  1 < 1 and 0.4598 ⩽  2 < 1.This shows that the migration loss during the migration also plays a crucial role on the permanence and extinction of the two species.Remark 6.In the course of the above discussion, we have established conditions that guarantee that the two species are permanent or extinct simultaneously.Hence, an interesting and important open problem is under what conditions one species is permanent and the other is extinct.
Remark 7. In all of the above discussion, we have established that if Moreover, if we extend the two-species competitive system to our investigated -species competitive system, what results can be obtained under the similar cases, which are also interesting open problems.

Figure 3 :
Figure 3: The extinction of the two species  1 and  2 of system (73).The extinction illustrated in (a) is caused by the austere survival environment of the two patches and the extinction illustrated in (b) is caused by the large loss during the migration ( 11 = 0.7,  12 = 0.6,  21 = 0.5, and  22 = 0.6).Here, we take initial values  10 = 1 and  20 = 2.