Abstract
We study a two-patch impulsive migration periodic -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.
1. 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 [1–6] and references cited therein). For example, in [3], Teng and Lu have investigated the following single-species nonautonomous dispersal model with delays: 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 [7–10] and references cited therein). For instance, in [7], the authors studied the following autonomous impulsive diffusion single species model: where 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 single-species 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; and represent the intrinsic growth rates of the th species in patch and in patch , respectively; and denote the intraspecific competition coefficients of the th species in patch and in patch , respectively; and are the interspecific competition coefficients between the th species and the th species in patch and in patch , respectively. The species migration occurs at every pulse time , where is sequence of positive numbers with . We suppose that the system is composed of two patches. When , all the species live in patch ; because of the change of the environment, the populations will migrate to patch and the migration loss is ; then the populations will live in patch during the period . When the environment changes again, all the populations will migrate back to the previous patch; here, the migration loss is .
In this paper, we always assume the following:Functions , and are -periodic continuous defined on and for all and .Impulsive time sequence satisfies for all . Moreover, for all and are constants.
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.
2. Preliminaries
In this section, as a preliminary we consider the following two-patch impulsive migration periodic single-species logistic system: where , and are -periodic continuous functions defined on , , and for all and impulsive time sequence satisfies for all . Moreover, and are constants. We have the following result.
Lemma 1. Let be any positive solution of system (4).
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 , , and such that We first of all prove that there is a constant such that 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.
Case 1. There is such that for all .
Case 2. There is such that for all .
Case 3. is oscillatory about for all .
We first consider Case . Since for all , then for , where is any positive integer, integrating system (10) from to , by (12), we have Hence, as , which leads to a contradiction.
Next, we consider Case . Obviously, there is such that . Then we prove that, for all , where . If (16) is not true, then there is such that Furthermore, there exists such that and for all . Taking an integer such that , then for all we have and integrating this inequality from to we have which contradicts with (17). This proves that (16) holds.
Lastly, if Case holds, then we directly have Choose constant ; then we see that (14) holds.
By a similar argument as in the proof of (14) we can prove that there is a constant such that 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 , 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 is an integer and is a constant, Hence, as . Further from (25) we obtain Lastly, we prove that system (4) has a unique positive -periodic solution. Consider the sequence . It is obviously bounded in the interval for all . Let be a limit point of this sequence, . Then . Indeed, since and as , we get The sequence , has a unique limit point. On the contrary, let the sequence have two limit points and . Then, taking into account (28) and , we have and hence . 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 , there is a positive constant 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 as . Lemma 1 is proved.
Remark 2. In [12], to prove the globally attractively positive -periodic solution and the extinction of system (4), we required conditions and for all besides conditions (6) and (8). However, we improve the conditions and for all to (condition (5)) in Lemma 1, which is superior to conditions given in [12].
3. 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 , we consider the following two-patch impulsive migration systems as the subsystems of system (3): On the permanence of all species for system (3) we have the following result.
Theorem 3. Assume that conditions and hold. Moreover, if then system (3) is permanent; that is, there are constants and such that for any positive solution of system (3), where is the globally attractively positive -periodic solution of system (34).
Proof. From condition (36) we directly have and by Lemma 1(a) we can obtain that defined in Theorem 3 is existent and globally attractive. Therefore, for any positive solution of system (34) and any constant , there exists such that We firstly prove the ultimately upper boundedness of system (3). From conditions (35) and (36), there are constants small enough such that for each . Let be any positive solution of system (3). Since by the comparison theorem of impulsive differential equations, we obtain where is the positive solution of system (34) with initial condition . By taking in (39), we can obtain that Choose a constant ; then is independent of any positive solution of system (3). Obviously, we have for all and .
Next, we prove that there is a constant such that We only need to consider the following three cases for each .
Case 1. There is such that for all .
Case 2. There is such that for all .
Case 3. is oscillatory about for all .
For Case , since for all , then let , where is any positive integer; integrating system (33) from to , by (40) and (43) we have Hence, as , which leads to a contradiction.
For Case , obviously, there is such that . Then we prove that, for all , whereIf (46) is not true, then there is such that Moreover, there exists such that If , must be an impulsive time. Then there exists a positive integer such that or ; thus we have From this we can obtain which contradicts with (48). If , we can choose an integer such that ; then we have for all and integrating this inequality from to we have which contradicts with (48) too. This proves that (46) holds.
Lastly, if Case holds, then we directly have Let constant . Then is independent of any positive solution of system (3) and we finally have This completes the proof of Theorem 3.
Next, we study the extinction of all species for system (3); we have the following result.
Theorem 4. Assume that conditions and hold. Moreover, if then all species of system (3) are extinct; that is, for any positive solution of system (3).
Proof. From system (3) we directly have Hence, for each , we have for all , where is the positive solution of system (34) with initial condition . According to conditions (56) and (57), by Lemma 1(b), we finally have for any positive solution of system (3). Theorem 4 is completed.
4. Periodic Solutions
In this section, we study the existence, uniqueness, and the global stability of the positive periodic solution of system (3).
Let and be any two positive solutions of system (3). From Theorem 3, we can obtain that there are constants and such that
Theorem 5. Suppose that all the conditions of Theorem 3 hold and there are constants and a nonnegative continuous function , satisfying such that Then system (3) has a unique positive -periodic solution which is globally attractive; that is, any positive solution of system (3) satisfies
Proof. Choose Lyapunov function as follows: For any impulsive time , we have Hence, is continuous for all . On the other hand, from (61) we can obtain that for each and any and For any and , calculating the upper right derivative of , from (63) and (67) we obtainwhere . From this, we further have for any Hence, it follows from (62) that as . Therefore, from (61) we obtain Now let us consider the sequence , where and . It is compact in the domain since for all and . Let be a limit point of this sequence, . Then . Indeed, since and as , we getThe sequence has a unique limit point. On the contrary, let the sequence have two limit points and . Then, taking into account (70) and , we have and hence . The solution is the unique periodic solution of system (3). By (70), it is globally attractive. This completes the proof of Theorem 5.
5. 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).
In order to testify the validity of our results, we consider the following two-patch impulsive migration periodic -species competitive system:
Corresponding to system (34), two subsystems of system (73) are taken as follows:
In system (73), we take for all . Hence, we have . Moreover, we take , , ,,, , , , , , , , ,. Obviously, which guarantee that system (74) with has a globally attractively positive 2-periodic solution from Lemma 1(a) and . See Figure 1(a). Similarly, we have that is, system (74) with also has a globally attractively positive 2-periodic solution from Lemma 1(a) and . See Figure 1(b).
(a)
(b)
Further, it is not difficult to verify that which satisfy condition (36) of Theorem 3 for each . Therefore, species and are permanent. The numerical simulation is given in Figure 2.
Meanwhile, if we choose in Theorem 5, we can verify that then we can choose . Therefore we have that is nonnegative and continuous and . All conditions of Theorem 5 are satisfied. Hence, system (74) has a unique positive -periodic solution . See also Figure 2.
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 , and and all other parameters are retained, then we obtain which satisfy condition (57) of Theorem 4 (condition (56) is obvious). Hence, all species of system (73) will go extinct. See Figure 3(a).
(a)
(b)
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 , , , and and all other coefficients are unchanged, then we can verify that and condition (57) of Theorem 4 satisfies. Therefore, the two species and of system (73) will also go extinct. See Figure 3(b). Meanwhile, if we fix , and , let and denote and then we have if and if (see Figure 4); that is, species and will go extinct if and . This shows that the migration loss during the migration also plays a crucial role on the permanence and extinction of the two species.
(a)
(b)
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 the two species are permanent. And if then the two species are extinct. Therefore, we have the following open problems.If , and , what trends of all solutions of system (73) are.If , and , what trends of all solutions of system (73) are.If and , what trends of all solutions of system (73) are.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.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This research was supported by the National Natural Science Foundation of China (11401060) and Zhejiang Provincial Natural Science Foundation of China (LQ13A010023).