Abstract
In most models of population dynamics, diffusion between two patches is assumed to be either continuous or discrete, but in the real natural ecosystem, impulsive diffusion provides a more suitable manner to model the actual dispersal (or migration) behavior for many ecological species. In addition, the species not only requires some time to disperse or migrate among the patches but also has some possibility of loss during dispersal. In view of these facts, a single species model with dissymmetric bidirectional impulsive diffusion and dispersal delay is formulated. Criteria on the permanence and extinction of species are established. Furthermore, the realistic conditions for the existence, uniqueness, and the global stability of the positive periodic solution are obtained. Finally, numerical simulations and discussion are presented to illustrate our theoretical results.
1. Introduction
In the last few years, mathematicians and ecologists have been actively investigating the dispersal of populations, a ubiquitous phenomenon in population dynamics. Levin [1] showed that both spatial dispersal of populations and population dynamics are much affected by spatial heterogeneity. In real life, dispersal often occurs among patches in ecological environments; because of the ecological effects of human activities and industries, such as the location of manufacturing industries and the pollution of the atmosphere, soil, and rivers, reproduction- and population-based territories and other habitats have been broken into patches. Thus, realistic models should include dispersal processes that take into consideration the effects of spatial heterogeneity.
In recent years, increasing attention has been paid to the dynamics of a large number of mathematical models with diffusion, and many nice results have been obtained. The persistence and extinction for ordinary differential equation and delayed differential equation models were investigated in [2–6]. Global stability of equilibrium and periodic solution for diffusing model were studied in [7–12]. However, in all of above population dispersing systems, it is always assumed that the dispersal occurs at every time. For example, in [7], Beretta and Takeuchi proposed the following single-species diffusion Volterra models with continuous time delays: where is the number of patches, and is the population density in the th patch. The form of the dispersal established in this model is continuous; that is, the dispersal is always happening at any time.
Actually, real dispersal behavior is very complicated and is always influenced by environmental change and human activities. In many practical situations, it is often the case that maybe one of the species suffers a significant loss or increase in density for some reason at some transitory time slots. These short-term perturbations are often assumed to be in the form of impulses in the modeling process. For example, when winter comes, birds will migrate between patches in search for a better environment, whereas they do not diffuse in other seasons, and the excursion of foliage seeds occurs during a fixed period of time every year. Therefore, impulsive differential equations [13] provide a natural description of such system. With the developments and applications of impulsive differential equations, theories of impulsive differential equations have been introduced into population dynamics, and many important studies have been performed [14–20].
In [14], the authors studied the following autonomous single-species model with impulsively bidirectional diffusion: where , are the intrinsic growth rate and density-dependent parameters of the population and is the dispersal rate in the th patch. Consider , where represents the density of population in the th patch immediately after the th diffusion pulse at time , while represents the density of population in the th patch before the th diffusion pulse at time ( the period of dispersal between any two pulse events is a positive constant, ). It is assumed here that the net exchange from the th patch to th patch is proportional to the difference of population densities. The dispersal behavior of populations between two patches occurs only at the impulsive instants . Obviously, in this model, species inhabits, respectively, two patches before the pulse appears, and when the time at the pulse comes, species in two patches disperses from one patch to the other. The boundedness and global stability of positive periodic solution were obtained.
Time delay often appears in many control systems (such as aircraft, chemical, or process control systems) either in the state, the control input, or the measurements. In order to reflect the dynamical behaviors of models that depend on the past history of system, it is often necessary to incorporate time delays into systems [21]. There have been extensive theoretical works on delay differential equations in the past three decades. The research topics include global asymptotic stability of equilibria, existence of periodic solutions, complicated behavior, and chaos (e.g., [8, 22–24]).
Takeuchi et al. in [25] studied the following population model with time delays that introduced the dispersal time for individuals to move from one patch to other patches: where , , are positive constants, , , , are nonnegative constants, satisfy , , , are constants, and some of them may be negative. In this paper, the authors took account of dispersal delay; however, they assumed that the dispersal is continuous.
It is well known that the application of impulsive delay differential equations to population dynamics has been an interesting topic since it is reasonable and correct in modelling the evolution of population, such as pest management [26].
However, in all of the impulsive dispersal models studied up till now, there are few papers considering the dispersal delay, which is really a pity. Actually, in the real world, the migration between patches is usually not immediate; that is, dispersal processes often involve time delay. For example, elks move from higher to lower elevations to escape cold in winter, and ungulates migrate annually among grazing areas to follow spatiotemporal changes in rainfall. Obviously, this kind of dispersal delay between patches extensively exists in the real world. Therefore, it is a very basilic problem to research this kind of population dynamic systems.
Moreover, in the above impulsive dispersal models, it is assumed that the dispersal occurs between homogeneous habitat patches; that is, the dispersal rate between any two patches is equal or symmetrical [1, 11], which is really too idealized for a real ecosystem. Actually, in the real world, due to the heterogeneity of the spatiotemporal distributions in nature, movement between fragments of patches is usually not the same rate in both directions. In addition, once the individuals leave their present habitat, they may not successfully reach a new one, due to predation, harvesting, or other reasons, so that there are traveling losses. Thus, the dispersal rates among these patches are not always the same. Rather, in real ecological situations they are different (or dissymmetrical) [27, 28].
Therefore, it is our basilic goal to investigate a single species model with dissymmetric impulse dispersal and dispersal delay. Motivated by the calculation hereinbefore, in this paper, we extend system (2) with dispersal delay and dispersal loss and consider it where is the rate of population emigrating from the th patch and is the rate of population immigrating from the th patch. Here we assume , which means that there possibly exists mortality during migration between two patches. stands for the time delay; that is, a period of time of species dispersing between patches.
The organization of this paper is as follows. In Section 2, as preliminaries, the definition of permanence and some useful lemmas are introduced. From discrete dynamic system theory, we establish the stroboscopic map of system (4), by which we can obtain the dynamical behaviors of it. In Section 3, the results of permanence and extinction for the system are presented. The existence and the uniqueness of the positive periodic solution for system (4) are established in Section 4. In Section 5, using the discrete dynamic system theory in [29], we can get the global stability of the positive periodic solution for the system. Finally, we give a brief discussion and our theoretical results are conformed by numerical simulations.
2. Preliminaries
In this section we introduce a definition and some notations and state some results which will be useful in subsequent sections.
For any fixed , let be a piecewise continuous function such that is continuous in , and exist. For any fixed , let denote the Banach space of all such piecewise continuous functions with the norm . Further, let = for all and for .
Motivated by the biological background of system (4), in this paper we always assume that all solutions of system (4) satisfy the following initial conditions: where . For any , by the fundamental theory of impulsive functional differential equations [30, 31], system (4) has a unique solution satisfying the initial conditions (5).
Definition 1. System (4) is said to be permanent, if there are positive constants and such that for any positive solution of system (4).
We consider the following scalar impulsive differential equation: where , , is an impulsive time sequence, , is continuous, and is a nondecreasing function. We have the following comparison theorem [30, 31] for (7).
Lemma 2. Let be a solution of system (7) defined on and satisfy
If , then for all .
Lemma 3 (see [32]). Consider the following nonlinear impulsive system: where and are bounded and continuous -periodic functions defined on , for all and impulsive coefficients , is a fixed positive integer, and . If , then there exists a unique positive periodic solution of system (9), which is globally asymptotically stable.
Remark 4. If system (9) degenerates into the following autonomous impulsive differential equation: where , , are positive constants, . As a consequence of Lemma 3, we have the following result: if , then system (10) has a unique positive periodic solution , which is globally asymptotically stable. In fact, here .
Lemma 5 (see [29]). Let be continuous, in int , and suppose exists with . In addition, assume(a), if ;(b), if .
If , let . If , then, for every as ; if , then either as for every or there exists a unique nonzero fixed point of . In the latter case, and, for every , as .
If , then either as for every or there exists a unique fixed point of . In the latter case, and, for every , as .
Next, we analyze system (4). Integrating and solving the first two equations of system (4) between pulses, we have
Similarly, considering the last two equations of system (4), we obtain the following stroboscopic map: here , , , .
Remark 6. System (12) is a difference system, which means that densities of population in two patches have values at the previous pulse. We are, in other words, stroboscopically sampling at its pulsing period. The dynamical behavior of system (12), coupled with (11), determines the dynamical behavior of system (4). In the following sections, we will focus our attention on system (12) and investigate various aspects of its dynamical behavior.
To write system (12) as a map, we define the map :
The set of all iterations of the map is equivalent to the set of all density sequences generated by system (12); is the map evaluated at the point . Consequently, in system (12), describes the population densities in the time .
On the positivity of solutions of system (4) we have the following result.
Lemma 7. The solution of system (4) with initial condition (5) is positive, that is, on the interval of the existence.
The proof of Lemma 7 is simple; we hence omit it here.
3. Permanence and Extinction
In this section, we present conditions to ensure that system (12) is permanent and extinct which will imply the permanence and extinction of system (4). The permanence plays an important role in mathematical ecology since the criterion of permanence for ecological systems is a condition ensuring the long-term survival of all species. So, we firstly prove system (12) is permanent.
Theorem 8. Suppose hold; then system (12) is permanent.
Proof. Let be the solution of system (4) satisfying the initial conditions (5). From the first equation of system (12), we have
Similarly, we have
Hence, by (14) and (15) we know that system (12) has an ultimately upper bound.
Next, we prove that all the solutions of system (4) are ultimately below bounded. Since , from the third equation of system (4), we have
Similarly,
Thus, system (4) becomes
From (18), we find that there is no relation between and . Therefore, we will discuss them, respectively;
If holds, from Remark 4, we can obtain that the auxiliary system
has a unique positive periodic solution which is globally asymptotically stable.
Let be the solution of system (20) with initial value . By Lemma 2, we have
Hence, for any sufficiently small, there exists a such that
Similarly, if holds, for above , there exists a such that
Denote and ; then we have and , . Finally, we can determine that there exist constants , , such that , and . The proof of Theorem 8 is completed.
Next, we present condition to ensure that system (12) is extinct.
Theorem 9. System (12) is extinct if
Proof. Let us consider the system (13). Obviously, is continuous in , and . We obtain
Obviously, ; if ; if , . We have the characteristic equation of :
Let ; then we have
Assume ; then by (27) we can obtain
that is,
which contradicts with (24). Therefore we have . By Lemma 5, we can get as , which means that system (12) is extinct. This completes the proof.
4. Existence and Uniqueness of Positive Periodic Solution
In this part, we will prove the existence and uniqueness of the fixed points of system (12), which means that system (4) has a uniquely positive periodic solution.
Theorem 10. If holds, then there exists a unique positive fixed point of system (12).
Proof. Corresponding to (12), let us consider the following system:
From (30), we have
hence
From (30), we also obtain
Thus
Let ; then . And is an increasing function on the interval . Since , so there exists such that . We can easily find that . By the zero theory of continuous function, there exists such that
Next, we will prove the uniqueness of the fixed point.
It follows from (33) that we obtain
Let
then
By (31), we have ; since , so . Similarly, we have . Therefore, we obtain , which implies that is a decreasing function on the interval .
Since
using the zero theory of continuous function, there exists a unique point such that . Besides,
thus
which, together with , leads to . By , we have that there exists a unique point such that . The proof is completed.
5. Global Stability
Now, we prove that the positive fixed points of system (30) are globally stable by using Lemma 5, which means that the positive periodic solution of system (4) is globally stable.
Theorem 11. If holds, then there exists a unique positive fixed point of the map , and, for every as .
Proof. For any small , , we make the change of variable
By (13), we get the map ; that is,
Now, we show that satisfies the hypotheses of Lemma 5. It is easy to see that is continuous, in , and .
Since
so, . Obviously, if , then ; if , then . It satisfies all the conditions of Lemma 5; then, for every , we have as . Corresponding to coordinate, this means, for , , the system (30) tends to the unique fixed point.
It follows from the permanence of system (12) that we have , for any initial value .
From the above analysis, we can know that, for every , , the trajectory of system (12) will tend to . The proof is completed.
6. Numerical Simulation and Discussion
In order to test the validity of our results, first, for (4) we use the parameters values (Val. 1) in Table 1. We can easily test that the assumptions in Theorems 8 and 10 hold, which means the populations in the two patches are permanent and have a unique periodic solution which is globally stable (see Figure 1(a)). Moreover, if, in Table 1, we consider the influence of time delay, then we can see that the permanence and stability for species unchanged. The details are given in Table 2. However, the longer the duration of the time delay, the lower the limit inferior and the limit superior of (see Figures 1(b), 1(c), and 1(d)). This implies that the case with dispersal delay is harmful to live for species .
(a)
(b)
(c)
(d)
Next, we take the parameters values (Val. 2) in Table 1. We can easily test that assumption in Theorem 9 holds, which implies the populations in the two patches are extinct (see Figure 2). Comparing Figure 2(a) and Figure 2(b), we realize that system (4) with time delay accelerates the extinction comparing with no delay (see Figure 2). This is reasonable from a biological point of view. Without delay means less loss during dispersion, which implies that more members can arrive to other patches. Otherwise, populations go extinct due to much loss. The details are given in Table 3.
(a)
(b)
Furthermore, we take , , , , and keep other parameters unchanged from in Val. 1. Here we can see that the period of individual intrinsic growth rate and density dependence rate is equal to the period of migration . In this case, we have let the period of the environment match the period of migration. If we take , from simulation (see Figure 3(a)), we can see the populations are permanent and have a uniquely periodic solution which is globally stable. However, if we take the migration period , respectively, with other parameters unchanged, from numerical simulations (see Figures 3(b) and 3(c)), we can see that population dynamics change from almost periodic to chaotic. For Case 4 of Table 4, we consider the influence of time delay (see Figure 3(d)). Comparing Figures 3(c) and 3(d), we realize that system (4) with time delay is more complicated than without. The details are given in Table 4.
(a)
(b)
(c)
(d)
Lastly, if we take and keep other parameters unchanged with Figure 3, by numerical simulations (see Figure 4), we find that all of the solutions of system (4) which through the initial points will converge to the positive periodic solution . Therefore, we can guess that under the assumptions of Theorem 10 system (4) has a unique positive periodic solution which is globally asymptotically stable. In addition, the periodic solution with time delay is larger than without delay which indicates that the duration of the time delay is beneficial to live for species compared with species .
(a)
(b)
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (nos. 10901130, 11361059, 10961022, and 11271312), the China Scholarship Council, the Natural Science Foundation of Xinjiang Province of China (2012211B07), and the Scientific Research Programmes of Colleges in Xinjiang (XJEDU2013I03).