A Single Species Model with Symmetric Bidirectional Impulsive Diffusion and Dispersal Delay

In the natural ecosystem, impulsive diffusion provides a more natural description for population dynamics. In addition, dispersal processes often involve with time delay. In view of these facts, a single species model with impulsive diffusion and dispersal delay is formulated. By the stroboscopic map of the discrete dynamical system and other analysis methods, the permanence of the system is investigated. Moreover, sufficient conditions on the existence and uniqueness of a positive periodic solution for the system are derived from the intermediate value theorem. We also demonstrate the global stability of the positive periodic solution by the theory of discrete dynamical system. Finally, numerical simulations and discussion are presented to validate our theoretical results.


Introduction
Species dispersal in patchy environment is one of the most prevalent phenomena of nature, and many empirical works and monographs on population dynamics in a spatial heterogeneous environment have been done.The persistence and extinction for ordinary differential equation and delayed differential equation models were investigated in [1][2][3][4][5][6].Global stability of equilibrium and periodic solution for diffusing model were studied in [7][8][9][10][11][12].Particularly, the predator-prey system with the prey dispersal also were studied in [13][14][15].Regretfully, in all of above population dispersing systems, they always assumed that the dispersal occurs at every time.For example, in [16], Mahbuba and Chen proposed the following two patches single species diffusion system: x t x t b t a t x t t D t x t x t i j where presents the dispersal rate from patch j to patch i at time t.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.It usually occurs stochastically or discontinuously [17], and it is often the case that species dispersal occurs at some transitory time slots when indi-viduals move among patches to search for mates, food, refuge, etc.
Animal movements between regions or patches of habitat are rarely continuous in time.They may occur during short time slots within seasons or within the lifetimes of animals.This short-time scale dispersal is more appropriately assumed to be in the form of impulses in the modeling process, in order to be in much better agreement with the real ecological situation.For example, when winter comes, birds will migrate between patches in search for a better environment, whereas they do not diffuse in other season, and the excursion of foliage seeds occurs during a fixed period of time every year.Thus impulsive dispersal provides a more natural description.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 [18][19][20][21][22][23][24].
In [19], Wang and Chen studied the following autonomous single-species model with impulsively bidirectional diffusion: where presents the population density or size at time t, is the intrinsic growth rate of the population d is the dispersal rate in the ith patch.   the period of dispersal between any two pulse events is a positive constant, ).It is assumed here that the net exchange from the jth patch to ith patch is proportional to the difference N N  of population densities.The dispersal behavior of populations between two patches occurs only at the impulsive instants n .Obviously, in this model, species N inhabits respectively two patches before the pulse appears, when the time at the pulse comes, species N both in two patches disperse from one patch to the other.Sufficient criteria were obtained for the permanence, existence, uniqueness and global stability of positive periodic solutions by using discrete dynamical system theory.
It is well known that the time delay is quite common for a natural population.In order to reflect the dynamical behaviors of models that depend on the past history of system, it is often necessary to take the effect of time delay into account in forming a biologically meaningful mathematica model.Delay differential equations have attracted a significant interest in recent years due to their frequent appearance in a wide range of applications.They serve as mathematical models describing various phenomena in physics, biology, physiology, and engineering, see, e.g., [25,26] and references therein.There has been an 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, see, e.g., [8,14,27,28].
Zhang and Teng in [14] introduced the following two species time-delayed predator-prey Lotka-Volterra type dispersal system with periodic coefficients (1.3): where , conditions on the boundedness, permanence and existence of positive periodic solution for system (1.3) are established.
Recently, the application of impulsive delay differential equations to population dynamics is also an interesting topic since it is reasonable and correct in modelling the evolution of population, such as pest management [29].
However, in all of the impulsive dispersal models studied up to 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 with time delay.For example, elks move from higher to lower elevations to escape cold in winter, and ungulates migrate annually among grazing areas to following spatio-temporal changes in rainfall.In these cases, movement is unidirectional during each migration period and takes place over fairly short time periods.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.
Motivated by the calculation hereinbefore, in this paper, we consider the following impulsive dispersal system with dispersal delay: where we suppose that the system is composed of two patches connected by diffusion.The pulse diffusion occurs every  period, the system evolves from its initial state without being further affected by diffusion until the next pulse appears;  stands for the time delay, that is, a period of species N disperse between patch i and j.
 denote the population density of prey species in ith patch and y is the population density of predator species.In this paper, the authors took dispersal delay into account, however, they assumed that the dispersal is continuous.Sufficient 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 (1.4), by which we can obtain the dynamical

t a t b t x t c t k s y t s s t d t x t t d t x t t x t x t a t b t x t t d t x t t d t x t t y t y t e t f t k s x t s s i n t
behaviors of it.The theorem on the permanence for sys-, let tem (1.4) is stated and proved in Section 3. In section 4, the existence and uniqueness of positive periodic solution for the system are obtained by the intermediate value theorem.In Section 5, the global stability of the positive periodic solution for system is established by the discrete dynamic system theory in [30].Finally, we give a brief discussion and our theoretical results are conformed by numerical simulations.

Preliminaries
For ), in this paper we always assume that all solutions of system (1.4) satisfy the following initial conditions: where [31,32], system (1.4) has a unique  in its maximal interval of the existe tion 2.1.System (1.4) is said to be perman for any positive solution of anence, existence and Next, to study the perm uniqueness of positively periodic solutions for system (1.4), we take By integrating and solving the first two equations of system (2.2) between pulse, we have Similarly, considering the last two equations of system (2.2), we obtain the following stroboscopic map of system (2.2): ) is a difference system.It describes the densities of population in two patches at a pulse in terms of values at the previous puls other words, stroboscopically sampling period.The dynamical behavior of system (2.4), coupled ), determine the dynamical behavior of system the following sections, we will focus our attention on system (2.4), and investigate various aspects of its dynamical behavior.
Next, In order to establish the permanence of system (2 e.We are, in at its pulsing with (2.3 (1.4).In .4),we introduce the following two lemmas.Lemma 2.1 Without loss of generality we assume that Proof.We f ov t the fi ) irst pr e tha rst part of Case (1 holds.If 0 0 x y  , 0 1 y  , then from (2.4), we have Hence, the first part of Case (1) is true for 1 n  .Next, we assume that the conclusion holds for , that is, Then, for , we get From the above discussion, we can see that the first part of Case (1) is true.Similarly, we can prove that the second part of Case (1) and the Case (2) are also true, so we omit the proof.This completes the proof.
By the same method, we have the following Lemma 2.2.
Lemma 2.2 Without loss of generality we assume that .Let  .Lastly, in order to establish the global stability of wing [30]) positively periodic solution, we introduce the follo lemma:   0, if 0; as n  nonzero fixed p ev oi ery nt q of 0  F ; o there e the la que .In tter case, 0 q  and for every 0

Permanence
anence plays an im a y since the cr n rm ecol erma nce of sys ( 1The perm portant role in m thematical ecolog iterio of pe anence for ogical systems is a condition ensuring the long-term survival of all species.In this section, we prove system (2.4) is permanent which will imply the p ne tem .4).Theorem 3.1 For By Lemmas 2.1 and Lemmas 2.2, we can get that for n N  , large enough, there exists constants i m , Proof.Corresponding to (2.4), let us consider th lowing system (4.1):  by the intermediate value theorem, we can see that there It follows from ( 4.3) that we obtain By (4.2) and assumption, we have   . So, we obtain using the intermediate value theorem, there exists a unique point which, together with , leads to x     there exists a By , we uniqu

Global Stability
Now, we will t the positive fix int   , x y   ma 2.3, of system (4.1) is globally stable by using Lem which means that the positive periodic s tion of system (1.4) is globally stable.
fies all the conditions of Lemma 2.3, then for every 0 , we have , 0 x y  , the trajectory of system .4)will trend to (2   , x y   .This completes the proof.

Numerical Simulation and Discussion
In this paper, we have investigated a single species model with symmetric bidirectional impulsive diffusion an criteria for the So, all conditions of Theorem 3.1 and Theorem 4.1 hold, which means system (2.2) is permanent and has an unique globally stable periodic solution.From Figure 1, we can see that in this four different cases the species x


The period of dispersal between two pulse events 10 nd y are both permanent.Moreover, the longer the dura-a tion of the time delay ( 0      ), the larger the limit inferior of x and the lower the limit superior of y (see   2(a) and (b)), we find that all of the solutions of system (2.2) which through these initial points will converge to the positive periodic solution.Therefore, we can conclude that under the assumptions of Theorem (4.1) system (2.2) has a unique po tive pe-In addition, the periodic solution is lager in Case II than in Case I whic tim y is beneficial to species m g Figu s 2( time delay is more complicated than without.ditions si riodic solution which is globally asymptotically stable.h indicates that the duration of the e dela x living again.Co c) and (d), we realize that system (1.4) with parin re Figure 2. Dynamical behavior of system (2.2). we take a series initial points, such as (0.9, 0.88), (0.92, 0.9), (1, 1.03).
F a) and (b)).thecase with rsal d eficial to sp ing and harm species y.Next, in .

Table 2, if
  gure 1(c , species x and y are both permanent, too (see Fi )).If we take 1 0.6