Abstract
We investigate a predator-prey model with impulsive diffusion on predator and stage structure on prey. The globally attractive condition of prey-extinction periodic solution of the system is obtained by the stroboscopic map of the discrete dynamical system. The permanent condition of the system is also obtained by the theory of impulsive delay differential equation. The results indicate that the discrete time delay has influence on the dynamical behaviors of the system. Finally, some numerical simulations are carried out to support the analytic results.
1. Introduction
The dispersal is a ubiquitous phenomenon in the natural world. It is well recognized that the spatial distribution of populations and population dynamics are much affected by spatial heterogeneity and population mobility [1]. The fragmented landscapes are common because the populations of most species occupy mosaic habitats and because of rapid destruction of natural habitats. Briggs and Hoopes [2] identify three mechanisms whereby limited dispersal of hosts and parasitoids combined with other features, such as spatial and temporal heterogeneity, can promote persistence and stability of populations. It is important for us to understand the ecological and evolutionary dynamics of populations mirrored by the large number of mathematical models devoting to it in the scientific literatures [3–6]. In recent years, the analysis of these models focuses on the coexistence of population and local (or global) stability of equilibria [7–13]. Spatial factors play a fundamental role on the persistence and stability of the population, although the complete results have not yet been obtained even in the simplest one-species case. Most previous papers focused on the population dynamical system modeled by the ordinary differential equations; if the population dynamics with the effects of spatial heterogeneity are modeled by a diffusion process, it will be very interesting. While in practice, it is often the case that diffusion occurs in regular pulse. 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 at fixed period of time every year. Thus, impulsive diffusion provides a more natural description. Lately theories of impulsive differential equations [14] have been introduced into population dynamics. Impulsive differential equations are found in almost a domain of applied science [15–24]. Newly, persistence and stability of population dynamical system involving time delay have been discussed by some authors; see, for example, [8, 25, 26] and references cited therein. They obtained some sufficient conditions that guarantee permanence of population or stability of positive equilibria or positive periodic solutions.
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 conditions of global attractivity and permanence for system (2.3). In Section 5, A brief discussion is given in the last section to conclude this work.
2. The Model
Stage-structured models were analyzed in many literatures [7, 27–33]. The following stage-structured Holling mass defence predator-prey model with impulsive perturbations on predators was introduced by Jiao et al. [31]: where , represent the immature and mature pest densities, respectively, denotes the density of nature enemy, represents a constant time to maturity, and and are positive constants. This model is derived as follows. We assume that at any time , birth into the immature population is proportional to the existing mature population with proportionality constant . We then assume that the death rate of immature population is proportional to the existing immature population with proportionality constant . , , and are called the death coefficient of , , and , respectively. We assume that the death rate of mature populations are of a logistic nature, that is, proportional to the square of the population with proportionality constant . is the rate of conversing prey into predator. , is the releasing amount of natural enemies at , and , and is the period of the impulsive immigration of the predator.
Diffusive predator-prey models were analyzed in literatures [16, 34, 35]. In [16], Hui and Chen considered the single species model with impulsive diffusion as follows: where we suppose that the system is composed of two patches connected by diffusion and is the density of species in the th patch. Intrinsic rate of natural increase and density dependence rate of prey population in the first habitat are denoted by , and is the dispersal rate in the th patch. It is assumed here that the net exchange from the th patch to th patch is proportional to the difference of population densities. The pulse diffusion occurs every period (). The system evolves from its initial state without being further affected by diffusion until the next pulse appears; and 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 , , and are positive constants.
Motivated by these ideals from all of the above, we assume that the predator population diffuses between the two patches, and the prey population exists only in one patch. Then, we consider a delayed predator-prey model with impulsive diffusion on predator and stage structure on prey as follows: with initial condition where , represent the immature and mature prey population densities, respectively. represents a constant time to maturity, that is, immature individuals and mature individuals are divided by age . It is assumed that the system is composed of two patches connected by diffusion and occupied by a single species and is the density of predator species in the th patch. Death rate of the predator population in the first patch is denoted by . Intrinsic rate of natural increase and density dependence rate of predator population in the second patch are denoted by and ; denotes the carrying capacity in the second patch. is dispersal rate of the predator population between two patches. It is assumed here that the net exchange from the th patch to th patch is proportional to the difference of predator population densities. is the prey capture rate by mature predator. is the rate of conversion of nutrients into the reproduction rate of the mature predator. is the death rate of immature prey population in the first patch. is the death rate of mature prey population in first patch. The pulse diffusion occurs every period. The system evolves from its initial state without being further affected by diffusion until the next pulse appears. , 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 .
Obviously, (2.3) can be simplified as follows: with initial condition
3. The Lemmas
The solution of (2.3), denoted by , is a piecewise continuous function . is continuous on , , and exists. Obviously, the global existence and uniqueness of solutions of (2.3) is guaranteed by the smoothness of properties of , which denotes the mapping defined by right-side of system (2.3) (see Lakshmikantham et al. [36]). Before having the main results, we need some lemmas which will be used next.
According to the biological meanings, it is assumed that , , , and . Now, we will show that all solutions of (2.3) are uniformly ultimately bounded.
Lemma 3.1. There exists a constant such that , , , and for each solution of (2.3) with all large enough .
Proof. Define . Denote . When , we have
where . When we also have . By Lemma in [14], for , we obtain
So, is uniformly ultimately bounded. Hence, by the definition of , there exists a constant such that , , , for all large enough. The proof is complete.
If , the subsystem of (2.3) is obtained as follows:
It is easy to solve the first two equations of system (3.3) between pulses
By considering the last two equations of System (3.3), we obtain the following stroboscopic map of system (3.3):
Then, two lemmas are obtained as follows.
Lemma 3.2 (see [37, Theorem ]). If , the trivial equilibrium of (3.5) is globally asymptotically stable. If , there exists a unique positive equilibrium , which is globally asymptotically stable, where and
Lemma 3.3 (see [37, Theorem ]). If , then (3.3) has a -periodic positive solution , which is globally asymptotically stable. Here can be expressed as where and are defined as in Lemma 3.2.
Lemma 3.4 (see [38]). Consider the following delay equation: one assumes that , for . Assuming that , then
4. The Dynamics
From the above discussion, we can easily know that there exists a prey-extinction boundary periodic solution of system (2.3). In this section, we will prove that the prey-extinction boundary periodic solution of system (2.3) is globally attractive.
Theorem 4.1. If hold, the prey-extinction boundary periodic solution of (2.3) is globally attractive. Here,
Proof. It is clear that the global attractivity of the predator-extinction boundary periodic solution of system (2.3) is equivalent to the global attractivity of predator-extinction boundary periodic solution of system (2.5). So, we devote ourselves to investigate system (2.5). Since we can choose sufficiently small such that
Since we know from the second equation of system (2.5) that , we consider the following comparison an impulsive differential system:
In view of Lemma 3.3 and (3.6), we obtain the -periodic solution of system (4.3)
which is globally asymptotically stable, where and
From Lemma 3.3 and comparison theorem of impulsive equation [36], we have and as . Then there exists an integer , such that
that is,
From (2.5), we get
Consider the following comparison differential system:
So we have . According to Lemma 3.4, we have .
Let be the solution of system (2.5) with its initial conditions and , is the solution of system (4.8) with initial condition . By the comparison theorem, we have
Incorporating into the positivity of , we know that Therefore, for any (sufficiently small), there exists an integer such that for all .
For system (2.5), we have
Then, and , as , where and are the solutions of
respectively. Here
where
Therefore, for any , there exists an integer , such that Let . So we have for large enough, which implies and as . This completes the proof.
The next work is to investigate the permanence of the system (2.3). Before starting the following theorem, we give the following definition.
Definition 4.2. System (2.3) is said to be permanent if there are constants (independent of initial value) and a finite time , such that for all solutions with all initial values , , , , , , , , holds for all . Here may depend on the initial values .
Theorem 4.3. If hold, there is a positive constant such that each positive solution of (2.5) satisfies for large enough . can be confirmed by the equation Here
Proof. The fourth equation of (2.5) can be rewritten as
Let us consider any positive solution of system (2.5). According to (4.18), is defined as
We calculate the derivative of along the solution of (2.5)
Since we can easily know that there exists sufficiently small such that
We claim that for any , it is impossible that for all . Suppose that the claim is not valid. Then there is a such that for all . It follows from the first equation of (2.5) that for all ,
Consider the following comparison an impulsive system for all
By Lemma 3.2 and the condition we obtain
where
is the unique positive periodic solution of (4.23), which is globally asymptotically stable. By the comparison theorem for impulsive differential equation [36], we know that there exists such that the inequalities and hold for . Thus for all . We make notation as for convenience. Also
Then
for all . Setting we will show that for all . Supposing the contrary, then there is a such that for , and . Hence, the first equation of system (2.5) implies that
This is a contradiction. Thus, for all . As a consequence, for all . This implies that as , . It is a contradiction to . Hence, the claim is complete.
By the claim, we are left to consider two case. First, for all large enough. Second, oscillates about for large enough .
Define
where . We hope to show that for all large enough. The conclusion is evident in the first case. For the second case, let and satisfy and for all where is sufficiently large such that for , is uniformly continuous. The positive solutions of (2.5) are ultimately bounded, and is not affected by impulses. Hence, there is a and is dependent on the choice of such that for . If , there is nothing to prove. Let us consider the case . Since and , it is clear that for . Then, proceeding exactly as the proof for the above claim, we see that for . Because the kind of interval, is chosen in an arbitrary way (we only need to be large). We concluded that for all large . In the second case, in view of our above discussion, the choice of is independent of the positive solution, and we proved that any positive solution of (2.5) satisfies for all sufficiently large . This completes the proof of this theorem.
Theorem 4.4. If hold, system (2.3) is permanent. Here
Proof. Let be any solution of system (2.3). From system (2.3) and Theorem 4.3, we have
By the same argument as those in the proof of Theorem 4.1, we have that and and is small enough. Here
In view of Theorem 4.1, the third equation of system (2.3) becomes
It is easy to obtain and is small enough, where . By Theorem 4.3 and the above discussion, system (2.3) is permanent. The proof of Theorem 4.4 is complete.
5. Discussion
In this paper, we investigate a delayed predator-prey model with impulsive diffusion on predator and stage structure on prey. We analyze that the prey-extinction periodic solution of system (2.3) is globally attractive, and we also obtain the permanent condition of system (2.3). It is assumed that , , , , , , , , , , , , and . Then the prey-extinction periodic solution is a global attractivity (see Figure 1). We also assume that , , , , , , , , , , , , , , then, system (2.5) is permanent (see Figure 2). From Theorems 4.1 and 4.4, we can easily guess that there must exist a threshold . If , the prey-extinction periodic solution of (2.3) is globally attractive. If , system (2.3) is permanent. The results show that the diffusion and the mature time of the prey play important roles for the permanence of system (2.3) and provide tactical basis for the biological resource protection.
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Acknowledgments
This paper is supported by National Natural Science Foundation of China (10961008), the Nomarch Foundation of Guizhou Province (2008035), and the Science Technology Foundation of Guizhou (2010J2130).