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Discrete Dynamics in Nature and Society
Volume 2009 (2009), Article ID 256195, 15 pages
http://dx.doi.org/10.1155/2009/256195
Research Article

A Predator-Prey Gompertz Model with Time Delay and Impulsive Perturbations on the Prey

School of Mathematics and Computer Science, Shanxi Normal University, Shanxi, Linfen 041004, China

Received 28 June 2009; Accepted 5 November 2009

Academic Editor: Leonid Berezansky

Copyright © 2009 Jianwen Jia and Chunhua Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We introduce and study a Gompertz model with time delay and impulsive perturbations on the prey. By using the discrete dynamical system determined by the stroboscopic map, we obtain the sufficient conditions for the existence and global attractivity of the “predator-extinction” periodic solution. With the theory on the delay functional and impulsive differential equation, we obtain the appropriate condition for the permanence of the system.

1. Introduction

It is well known that Gompertz equation [1] describes the growth law for a single species. The model reads as

where is the density of the population, is a positive constant called the intrinsic growth rate, the positive constant is usually referred to as the environment carrying capacity or saturation level, and denotes relative growth rate.

Many systems in physics, chemistry, biology, and information science have impulsive dynamical behavior due to abrupt jumps at certain instants during the evolving processes. This dynamical behavior can be modeled by impulsive differential equations. The theory of impulsive differential systems has been developed by numerous mathematicians [24]. But more realistic models should include some of the past states of these systems, that is, ideally, a real system should be modeled by differential equations with time delays. Recently, the investigation of impulsive delay differential equations is beginning [57].

In this paper, we need to consider a function

which is called Holling III functional response. The coefficient is a positive constant and its biological meaning can be found in [8].

We introduce the model as follows:

where and are densities of the prey and the predator, respectively, is the Gomportz intrinsic growth rate of the prey in the absence of the predator, is the conversion rate, is the death rate of the predator, is mean length of the digest period, and represents impulsive harvest to preys by catching or pesticides at , .

The initial conditions for system (1.3) are

From the biological point of view, we only consider system (1.3) in the biological meaning region .

All outline of this paper is as follows. We give some basic knowledge in Section 2. In Section 3, using discrete dynamical system determined by the stroboscopic map, we obtain the existence and global attractivity of the “predator-extinction” periodic solution. In Section 4, with the theory of delay and impulsive different equations, we obtain the sufficient condition for the permanence of the system. In the last section, we give the numerical simulation and discussion.

2. Basic Knowledge

Let , , , and be the set of all nonnegative integers. Denote that is the map defined by the right-hand side of the first two equations of system (1.3). Let = , continuous on and exist.

Definition 2.1. Let , then for , the upper right derivative of with respect to the impulsive differential system (1.3) is defined as The solution of system (1.3) is a piecewise continuous function. The smoothness properties of guarantee the global existence and uniqueness of the solution of system (1.3) [9].

Lemma 2.2 (see [9]). Considering the following impulsive differential inequalities: where and , are constants, then

Lemma 2.3. There exists a constant such that and for each solution of system (1.3) with large enough.

Proof. Define then . Since , ; in addition, . Thus for large enough.
We calculate the upper right derivative of along a solution of system (1.3) as
So we have that By Lemma 2.2, for , we have that So, is ultimately bounded. Therefore by the definition of there exists a constant such that and for each solution of system (1.3) with large enough. This completes the proof.

Lemma 2.4 (see [10]). Consider a delay equation where are all positive constants and for . Thus one has the following.(1)If , then .(2)If , then .

3. “Predator-Extinction” Periodic Solution

3.1. Existence of a “Predator-Extinction” Periodic Solution

First, we begin analyzing the existence of a “predator-extinction” solution, in which predator is absent from the system, that is,

In this condition, we know the growth of the prey in the time-interval and give some basic properties of the following subsystem of (1.3):

For system (3.2), we let and obtain a linear nonhomogeneous impulsive equation

Solving the first equation of system (3.3) between pulses yields

For the second equation of system (3.3), using discrete dynamical system determined by the stroboscopic map yields

where . From (3.5), we can see that this difference system has an equilibrium , which implies that system (3.3) has a unique T-periodic solution

Since , then is the unique positive T-periodic solution of (3.2).

In the following, we will prove that is globally asymptotically stable. By Lemma 2.3, we find that any solution of (3.2) is ultimately upper bounded, so we need only to prove that

where is the periodic solution of system (3.3) Suppose that with .

since ; thus, .

Theorem 3.1. System (3.2) has unique a positive periodic solution which is globally asymptotically stable. That is, system (1.3) has a “predator-extinction” periodic solution for , .

3.2. Global Attractivity of the “Predator-Extinction” Periodic Solution

Denote that

Theorem 3.2. If , then the “predator-extinction” periodic solution of system (1.3) is globally attractive.

Proof. We denote that be any solution of system (1.3) with initial condition (1.4). From the second equation of system (1.3), we have that
We consider the following comparison equation: If , then ; according to Lemma 2.4, we obtain .
Since for all , by the comparison theorem in differential equation and the positivity of the solution, we have that as .
In the following, we suppose that . Since , we have that
Because the function is monotonically increasing with respect to , we choose a sufficiently small positive constant such that
Noting that for , then we consider the following comparison system:
From Section 3.1, we see that which is unique a globally asymptotically stable positive T-periodic solution of system (3.14).
There exist a positive integer and an arbitrarily small positive constant for all such that
From (3.16) and the second equation of system (1.3), for we have
Considering the following comparison equation: we can see (3.13), that is, ; according to Lemma 2.4, we obtain .
Since for all , by the comparison theorem in differential equation and the positivity of the solution, we have that as .
Next, we will prove that as . Without loss of generality, we may assume that for all . By the first equation of system (1.3), we have that
Then we have that as as , where is unique a positive solution of From Section 3.1, we obtain
By using comparison theorem of impulsive differential equation [11], for any , there exists and such that On the other hand, from the first equation of (1.3), we have that .
Considering the following comparison system: we have that as and . Let , then it follows from (3.22) and (3.25), that we can see for large enough, which implies as . The proof is completed.

Now, we give the following theorem.

Theorem 3.3. If , then the “predator-extinction” periodic solution is globally attractive.
If and , , , then the “predator-extinction” periodic solution is globally attractive, where the critical values , , are listed in Table 1.

tab1
Table 1: Critical values of some parameters of system (1.3) ( must be satisfied).

4. Permanence

In the above section, we have proved that, when , , or , the “predator-extinction” periodic solution is globally attractive. But in natural world, the predator cannot be eradicated totally. In order to save resources, we need to keep the prey and predator coexisting when the prey does not bring about immense economic losses. Next, we will discuss the permanence of system (1.3).

Definition 4.1. System (1.3) is said to be uniformly persistent if there are positive constants and a finite time such that for all solutions with initial values , , , and hold for all .

Definition 4.2. System (1.3) is said to be permanent if there exists a compact region , such that every solution of system (1.3) with initial condition (1.4) will eventually enter and remain in region .
Denote that

Theorem 4.3. If , then there exist two positive constants and such that , for large enough, that is, system (1.3) is uniformly persistent.

Proof. Suppose that is any positive solution of system (1.3) with initial condition (1.4). From Lemma 2.3, we know that , .
Firstly, from the first equation of system (1.3), we know that . Now, we consider the following equation:
therefore we have that There exists a small enough such that, for sufficiently large ,
Secondly, we will find such that .
The second equation of system (1.3) may be rewritten as follows:
Define Calculating the derivative of along the solution of (1.3), it follows from (4.5) that
Since , then we can deduce that Hence, we choose two positive constants and small enough such that where In the following, we will find such that There are two cases.
(1) We claim that the inequality cannot hold for all ; otherwise, there is a positive constant such that for all .
From the first equation of system (1.3), we have that We have that , where is unique a positive solution of From Section 3.1, we have that
By comparison theory [11], for any , there exists a , for , such that From (4.7) and (4.14), we have that
Denote that
We show that for all . Otherwise, there exists a nonnegative constant such that , for , and .
Thus, from the second equation of (1.3) and (4.12), (4.14), we can see that which is a contradiction to . Hence we get that for all . Meanwhile, we can see that , which implies that , .
This is a contradiction to . Thus, for any positive constant , the inequality cannot hold for all .
(2) If holds for all large enough, then our aim is obtained. Otherwise, is oscillatory about . Let .
We will prove that . There exist two positive constants , such that and for .
When is large enough, the inequality holds true for .
Since is continuous and bounded and does not have impulsive effort, we conclude that is uniformly continuous. There exist a constant ( and is independent to ) such that for all .
If , then our aim is obtained.
If , from the second equation of (1.3), we have that for , and then we have that for . Therefore, for .
If , by the second equation of system (1.3), we have that for , the same as above claim; we can obtain for .
Since the interval is arbitrarily chosen, we get that for large enough.
This proof is complete.

Theorem 4.4. If , then system (1.3) is permanent.

Proof. Suppose that is any solution of system (1.3) with initial condition (1.4). By Theorem 4.3, there exist positive constants , , and such that , for . Set
Then is a bounded compact region and int . By Theorem 4.3, every solution of system (1.3) with initial condition (1.4) eventually enters and remains in region .
The proof is complete.

Theorem 4.5. If , , or , then system (1.3) is permanent, where the critical values , , are listed in Table 2.

tab2
Table 2: Critical values of some parameters of system (1.3) ( must be satisfied).

5. Numerical Simulation and Discussion

In this paper, we introduce and discuss a predator-prey system model with Holling III response functional under time delay on the predator and impulsive perturbations on the prey. From Section 3, there exists a predator-extinction periodic solution of system (1.3); when , the predator eradication periodic solution is globally attractive. From Section 4, when , system (1.3) is permanent.

In the following, we will analyze the influence of them on the dynamics of system (1.3). We consider the hypothetical set of parameter values as , , , , and .

Figure 1 is the dynamical behavior of system (1.3) with , , , , , , , and . (a) is the timeseries of prey population for periodic oscillation; (b) is the timeseries of population for extinction; (c) is the phase portrait of the prey and the predator population for global attractivity of the “predator-eradication” periodic solution.

fig1
Figure 1

Figure 2 is the dynamical behavior of system (1.3) with , , , , , , , and . (a) is the timeseries of prey population for permanence; (b) is the timeseries of population for permanence; (c) is the phase portrait (T-periodic solution) of the prey and the predator population of system (1.3).

fig2
Figure 2

By Theorems 3.2 and 4.3, we know that, when , the “predator-eradication” periodic solution is globally attractive (Figure 1); when , system (1.3) is permanence (Figures 2(a) and 2(b)).

In Figure 1(b), we know that the predator population dies ultimately; we know that, although impulsive catching is larger or the period of pulsing is shorter, we kill the prey largely, and the predator population will decrease largely. It is very difficult for it to prey on prey; the predator can die out earlier than the prey. From Figure 2, we suppose that a smaller impulsive catching rate or a longer period can cause the prey and predator populations to coexist.

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