Dynamical analysis on a single population model with state-dependent impulsively unilateral diffusion between two patches
© Jiao et al.; licensee Springer 2012
Received: 17 April 2012
Accepted: 20 August 2012
Published: 10 September 2012
Coupling with social progress, traffic development and so on, the natural environment is going into patches. There are differences between population in differential patches. The population living in a patchy environment is affected by the space structures. It is important to depict the dynamical behaviors of the population in the present world. In this work, a state-dependent impulsive differential model, which focuses on impulsively unilateral diffusion between two patches, aims for the simulation of the factual population dynamical behaviors. With the approaches of mathematical analysis, we obtain sufficient conditions of the existence and orbitally asymptotic stability of a periodic solution of the investigated system. Finally, the numerical simulations verify our results.
The population diffusion affecting the dynamical behaviors of populations in differential patches are investigated by many researchers [1–5]. Two unable to be competitive populations can be stabilized by the population diffusion . The persistence of competitive systems of two or three populations can be increased under appropriate diffusion conditions . Some papers assume that the individuals’ mobility is mainly induced by intrinsic factors of population such as genetic characteristics. Nevertheless, more and more researchers have found that the population diffusion could be affected by many factors such as qualities and quantities of food, the pressures from competitors, and the predation risk from enemies. Abrams et al.  found that prey diffusion may be accelerated by poor reproduction conditions and high predation risks in local habitat. Because of attraction from better reproductions or less predation pressure at other patches, the predator may change behaviors on the basis of prey abundance and demographic advantages. Kuang  analyzed a delayed two-stage population model with space limited recruitment.
The theory of impulsive differential equations is much richer than the corresponding theory of differential equations. They generally describe phenomena which are subjected to steep or instantaneous changes. Impulsive equations are found in almost every domain of the applied science and have been studied in many works [8, 9]. Jiao et al.  and Tang et al.  have studied the population models with impulsive perturbation at fixed moments. Jiao et al.  investigated the dynamical behaviors of a stage-structured predator-prey system with birth pulse and impulsive harvesting at different moments. Jiao et al.  provided a fishing policy by investigating a stage-structured model with state-dependent impulsive harvesting.
Coupling with social progress, traffic development and so on, the natural environment is going into patches. There are differences between population in differential patches. For example, when the density of population living in one patch reaches one threshold, that is to say, when the quantity of population reaches the superior limit, intraspecific competition for limited resources increases, the population in this patch will move to another patch. Although Jiao et al.  devoted their work to investigation of the dynamics of a stage-structured predator-prey model with prey impulsively diffusing between two patches, they did not propose a single population model with state-dependent impulsively unilateral diffusion between two patches. Motivated by these biological facts and the previous studies, we propose and investigate a state-dependent impulsive differential model, which focuses on impulsively unilateral diffusion between two patches.
The organization of this paper is as follows. In the next section, we introduce the model and background concepts. In Section 3, some important lemmas are presented. In Section 4, we give the sufficient conditions of the existence and orbitally asymptotic stability of a periodic solution of the investigated system. In Section 5, numerical simulation and a brief discussion are given to conclude this work.
2 The model
where system (2.1) is assumed to be composed of two patches. () denotes the density of population in i th patch. denotes the intrinsic rate of natural increase in the population in Patch 1. denotes the interspecific competition coefficient of the population in Patch 1. denotes the death rate of the population in Patch 2. denotes the diffusive rate coefficient from Patch 1 to Patch 2. is called an environment pressure threshold of the population in Patch 1, that is to say, when the density of the population in Patch 1 reaches the threshold , the population will unilaterally diffuse from Patch 1 to Patch 2. It is also assumed that the population in Patch 2 does not diffuse from Patch 2 to Patch 1.
3 The definitions and lemmas
where is a small parameter. For each , the is a hypersurface in . Suppose consists of q nonintersecting smooth hypersurfaces which are given by the equations ().
Let , be a solution of Equation (3.1) with moments of impulsive effect , , and . Let denote the solution of Equation (3.1) for which , and let denote the right maximal interval of the existence of this solution.
- (i)orbitally stable, if
- (ii)orbitally attractive, if
orbitally asymptotically stable, if it is orbitally stable and orbitally attractive.
Lemma 3.3 
For Equation (3.1) with , the following conditions hold:
() For , Equation (3.1) has a -periodic solution with moments of impulsive effect () and ().
then the -periodic solution of Equation (3.1) with is orbitally asymptotically stable and enjoys the property of asymptotic phase.
and P, Q, , , , , , are calculated at point and , .
Lemma 3.4 
The T-periodic solution of system (3.3) is orbitally asymptotically stable and enjoys the property of asymptotic phase if the multiplier calculated by (3.4) satisfies .
4 The dynamical analysis
system (2.1) has a uniquely τ-periodic solution.
Considering and , and substituting (4.3) into (4.4) and (4.6), we get , and . From the impulsive effect , we cast out . This completes the proof. □
hold, then the periodic solution of system (2.1) is orbitally asymptotically stable.
In view of (4.7), it follows from Lemma 3.4 that the τ-periodic solution of system (2.1) is orbitally asymptotically stable. This completes the proof. □
Research supported by National Natural Science Foundation of China (10961008), the Development Project of Nature Science Research of Guizhou Province Department (No. 2010027) and the Science Technology Foundation of Guizhou (No. 2010J2130).
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