Dynamical Analysis of a Delayed Predator-Prey System with Birth Pulse and Impulsive Harvesting at Different Moments
© Jianjun Jiao and Lansun Chen. 2010
Received: 21 August 2010
Accepted: 22 September 2010
Published: 27 September 2010
We consider a delayed Holling type II predator-prey system with birth pulse and impulsive harvesting on predator population at different moments. Firstly, we prove that all solutions of the investigated system are uniformly ultimately bounded. Secondly, the conditions of the globally attractive prey-extinction boundary periodic solution of the investigated system are obtained. Finally, the permanence of the investigated system is also obtained. Our results provide reliable tactic basis for the practical biological economics management.
The biological meaning of the parameters in System (1.1) can refer to Literature .
where denotes the harvesting effort.
where is the density of the population. is the death rate. The population is birth pulse as intrinsic rate of natural increase and density dependence rate of predator population are denoted by , respectively. The pulse birth and impulsive harvesting occurs every period ( is a positive constant). . represents the birth effort of predator population at , , . represents the harvesting effort of predator population at , .
where represent the immature and mature populations densities, respectively, 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 . We also assume that the death rate of mature population is of a logistic nature, that is, proportional to the square of the population with proportionality constant . In this paper, we consider a delayed Holling type II predator-prey system with birth pulse and impulsive harvesting on predator population at different moments.
The organization of this paper is as follows. In the next section, we introduce the model. In Section 3, some important lemmas are presented. In Section 4, we give the globally asymptotically stable conditions of prey-extinction periodic solution of System (2.1), and the permanent condition of System (2.1). In Section 5, a brief discussion is given in the last section to conclude this paper.
2. The Model
where represent the densities of the immature and mature prey populations, respectively. represents the density of predator population. is the intrinsic growth rate of prey population. represents a constant time to maturity. is the natural death rate of the immature prey population. is the natural death rate of the mature prey population. is the natural death rate of the predator population. The predator population consumes prey population following a Holling type-II functional response with predation coefficients , and half-saturation constant . is the rate of conversion of nutrients into the reproduction rate of the predators. The predator population is birth pulse as intrinsic rate of natural increase and density dependence rate of predator population are denoted by , respectively. The pulse birth and impulsive harvesting occurs every period ( is a positive constant). . represents the birth effort of predator population at , , . represents the harvesting effort of predator population at , . In this paper, we always assume that .
3. The Lemma
is continuous in and , for each , , and exist.
is locally in .
The solution of (2.1), denote by , is a piecewise continuous function : , is continuous on and , . Obviously, the global existence and uniqueness of solutions of (2.1) is guaranteed by the smoothness properties of , which denotes the mapping defined by right-side of system (2.1) Lakshmikantham et al. . Before we have the the main results. we need give some lemmas which will be used as follows.
Now, we show that all solutions of (2.1) are uniformly ultimately bounded.
There exists a constant such that , , for each solution of (2.1) with all large enough.
So is uniformly ultimately bounded. Hence, by the definition of , there exists a constant such that , for large enough. The proof is complete.
If , the fixed point is globally asymptotically stable;
if , the fixed point is globally asymptotically stable.
- (i)If , is the unique fixed point, we have(3.12)
then is globally asymptotically stable.
- (ii)If , is unstable. For(3.13)
then is globally asymptotically stable. This complete the proof.
It is well known that the following lemma can easily be proved.
If , the triviality periodic solution of System (3.7) is globally asymptotically stable;
- (ii)if , the periodic solution of System (3.7)(3.14)
Lemma 3.5 (see [ 22]).
4. The Dynamics
In this section, we will firstly obtain the sufficient condition of the global attractivity of prey-extinction periodic solution of System (2.1) with (2.2).
from (4.5), we have . According to Lemma 3.5, we have .
Therefore, for any (sufficiently small), there exists an integer such that for all .
for large enough, which implies as . This completes the proof.
The next work is to investigate the permanence of the system (2.4). Before starting our theorem, we give the definition of permanence of system (2.4).
System (2.1) 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 .
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.
is uniformly continuous. The positive solutions of (2.3) are ultimately bounded and is not affected by impulses. Hence, there is a and is dependent of 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 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 (2.3) satisfies for all sufficiently large . This completes the proof of the theorem.
From Theorems 4.1 and 4.3, we can easily obtain the following theorem.
In this paper, considering the fact of the biological source management, we consider a delayed Holling type II predator-prey system with birth pulse and impulsive harvesting on predator population at different moments. We prove that all solutions of System (2.1) with (2.2) are uniformly ultimately bounded. The conditions of the globally attractive prey-extinction boundary periodic solution of System (2.1) with (2.2) are obtained. The permanence of the System (2.1) with (2.2) is also obtained. The results show that the successful management of a renewable resource is based on the concept of a sustain yield, that is, an exploitation does not the threaten the long-term persistence of the species. Our results provide reliable tactic basis for the practical biological resource management.
This work was supported by National Natural Science Foundation of China (no. 10961008), the Nomarch Foundation of Guizhou Province (no. 2008035), the Science Technology Foundation of Guizhou Education Department (no. 2008038), and the Science Technology Foundation of Guizhou (no. 2010J2130).
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