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We study a two-patch impulsive migration periodic

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 [

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 [

Motivated by the above analysis, in this paper, we consider the following two-patch impulsive migration periodic

In this paper, we always assume the following:

Functions

Impulsive time sequence

In addition, we assume that the investigated

The organization of this paper is as follows. In Section

In this section, as a preliminary we consider the following two-patch impulsive migration periodic single-species logistic system:

Let

Due to the fact that the population dispersal is only restricted in two patches and shows itself as aggregate migration, we can rewrite system (

In order to prove proposition (a), firstly, we prove the permanence of system (

From conditions (

We first of all prove that there is a constant

We first consider Case

Next, we consider Case

Lastly, if Case

By a similar argument as in the proof of (

Now, we prove proposition (a). Let

Lastly, we prove that system (

Now we prove proposition (b). From (

In [

We first discuss the permanence of all species of system (

Assume that conditions

From condition (

We firstly prove the ultimately upper boundedness of system (

Next, we prove that there is a constant

For Case

For Case

If (

Lastly, if Case

Next, we study the extinction of all species

Assume that conditions

From system (

In this section, we study the existence, uniqueness, and the global stability of the positive periodic solution of system (

Let

Suppose that all the conditions of Theorem

Choose Lyapunov function as follows:

Now let us consider the sequence

In this paper, we have investigated a class of two-patch impulsive migration periodic

In order to testify the validity of our results, we consider the following two-patch impulsive migration periodic

Corresponding to system (

In system (

Dynamical behavior of system (

Further, it is not difficult to verify that

Dynamical behavior of system (

Meanwhile, if we choose

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

The extinction of the two species

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

The trends of

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.

In all of the above discussion, we have established that if

If

If

If

Moreover, if we extend the two-species competitive system to our investigated

The authors declare that there is no conflict of interests regarding the publication of this paper.

This research was supported by the National Natural Science Foundation of China (11401060) and Zhejiang Provincial Natural Science Foundation of China (LQ13A010023).