ISRN Applied Mathematics

Volume 2012 (2012), Article ID 598539, 14 pages

doi:10.5402/2012/598539

Research Article

The Study of a Class of Pest Control Pollution Model with Stage-Structure and Time Delay

Jianwen Jia and Bo Wu

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

Received 1 March 2012; Accepted 26 March 2012

Academic Editors: A. Bellouquid and C. Lu

Copyright © 2012 Jianwen Jia and Bo Wu. 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 construct a pest control pollution model with stage-structure for pests and with epidemic in the predator by spraying pesticides and releasing susceptible predators together. We assume that only the pests and infective predators are affected by pesticide. We show that there exists a globally attractive pest-extinction periodic solution and we get the condition of global attractiveness of the pest-extinction periodic solution by applying comparison theorem of impulsive differential equation. Further, the condition for the permanence of the system is also given.

1. Introduction

With the rapid development of modern technology, industry, and agriculture, it is of great interest to consider the effects of toxicant on ecological communities from both an environmental and conservational point of view. Qualitatively estimating the effect of a toxicant on a population by mathematical models is a relatively new field that began only in the early 1980s [1–3]. Population toxicant coupling has been applied in several contexts including Lotka-Volterra and chemostat-like environments, resulting in ordinary, integrodifferential, and impulsive differential equation and stochastic models. So in this paper, we consider the above effects and introduce the pollution model to model the process of pest control problems and study its dynamics, and this is different from the previous pest control model which assumed that pests were reduced proportionally by spraying pesticides [4–6].

In the natural world, many species have a life history that takes their individual members through two stages: immature and mature; the authors of [7] studied an ecological model with stage-structure for predator. A general functional response was considered, and the authors analyzed the stability and the permanence of the system. The authors [8, 9] analyzed prey-predator models with age structure and impulsive control. The authors [10] investigated the dynamics of a pest control model with age structure for pests by introducing a constant periodic pesticide input and releasing natural enemies at different fixed moments. they analyzed the conditions for the global attractivity of the pest-extinction periodic solution and the permanence of the system.

Modeling studies on disease-dominated ecological populations have addressed issues like disease-related mortality, reduction in reproduction, change in population sizes, and disease-induced oscillation of population states. Chattopadhyay and Arino [11] formulated a prey-predator model with prey infection and observed destabilization due to infection. Venturino analyzed prey-predator models with disease in the prey [12] and the predator [13].

Motivated by the above, in this paper, we construct a pest control model with epidemic in the predator by spraying pesticides and releasing susceptible predators at the same time. The pest is stage-structured, and the effects of spraying pesticides into the environment and into the organism are considered. So the pollution model provides a natural description of such a system and should be introduced to our model.

The organization of this paper is as follows: in Section 2, we introduce a pest control pollution model with stage-structure for pest and with epidemic in the predator by introducing a constant periodic pesticide input and releasing susceptible predators together. In Section 3, we will introduce some definitions and lemmas which will be used in the paper. In Section 4, sufficient conditions are obtained for the global attractiveness of pest-extinction periodic solution. In Section 5, sufficient conditions are obtained for the permanence of the system. We give a brief conclusion of our results in the last section.

2. Model Formulation

In this paper, we suppose that pesticides hardly have influence on the susceptible predators, and the susceptible predators only feed on mature pests. Now we consider the following impulsive differential equation:

Here and represent the density of the immature and mature pest (the prey) at time , respectively; and represent the density of susceptible predator and infective predator at time , respectively; represents the concentration of pesticide in the environment at time ; represents the concentration of pesticide in the organism at time ; we use a special functional response, that is, when the number of the prey captured is less, the digestive capacity of the predator will increase with the density of the prey. Here is the birth rate of the immature pest; is the death rate of the mature pest; and are the death rates of the susceptible predator and the infective predator, respectively; is the mean length of the juvenile period; represents the capturing rate of the susceptible predator; represents digestive time of the susceptible predator; is the transformation rate of the susceptible predator; represents the intraspecific competition coefficient of mature pest; represents saturation incidence rate; is the period of the impulsive effect; is the releasing amount of the susceptible predator at ; is the amount of the pesticides spraying at every impulsive period ; represent the decreasing rate of the intrinsic growth rate associated with the uptake of the pesticide in the organism for the immature pest, mature pest, susceptible predator, and infective predator, respectively; represents an organism’s net uptake of toxin from the environment; and represent the digestion and depuration rates of pesticide in an organism, respectively; represents the loss of pesticide in the environment due to natural degradation. All the coefficients are positive constants.

The initial conditions of system (2.1) are

where . To assure the continuity of the initial values, we assume that . This suggests that if we know the properties of , then the properties of can be obtained.

Note that the variable does not appear in the second, third, forth, fifth, and sixth equations of system (2.1), hence we only need to consider the subsystem of (2.1) as follows:

The initial conditions for system (2.3) are

where .

3. Definitions and Lemmas

Let , and be the set of all nonnegative integers, let the map defined by the right of system (2.3). The solution of system (2.3), denoted by , is continuously differentiable on . Let , then V is said to belong to class if;

(1) is continuous on , and for each and

exist;

(2) is locally Lipschitzian in.

Definition 3.1. Let , then for , the upper right derivative of with respect to impulsive differential system (2.3) is defined as

Definition 3.2. System (2.3) is said to be permanent if there are constants (independent of initial value) and a finite time such that for every positive solution with initial conditions of system (2.3) satisfies , , , , for all , here may depend on the initial condition of system (2.3).

Lemma 3.3 (the comparison theorem of impulsive differential equation [14]). Let . Assume that

where is continuous in , and for each exists and is finite; is nondecreasing.

Let be the maximal solution of the scalar impulsive differential equation defined on , then

So implies that

where is any solution of system (2.3).

Lemma 3.4 (see [15]). Consider the following equation:

where , for , one has

(i)if , then ,(ii)if , then .

Remark 3.5 (see [16]). are the concentration of toxicant. To assure , it is necessary that .

Remark 3.6 (see [16]). From the point of the biological meaning, we assume that .

Lemma 3.7. Consider the following subsystem of system (2.3)

Then, system (3.6) has a unique positive T-periodic solution and for each solution of system (3.6), and as . Where

for and .

Lemma 3.8. There exists a constant such that for each solution of system (2.1) with all large enough.

Proof. Define . Choose , we have and

Hence there exists a positive constant such that

by Lemma 3.3, for , we have

Then .

So is uniformly ultimately bounded. By the definition of , there exists a constant such that for large enough. The proof is complete.

4. The Global Attractivity of Periodic Solution

In this section, the sufficient conditions are obtained for the global attractivity of the pest-extinction periodic solution.

We first demonstrate the expression of the pest-extinction solution of system (2.3), in which the pest individual and infective predator individual are entirely absent from the model, that is, for all .

When and , satisfies the following system:

. Clearly, we can obtain the unique positive periodic solution with the form

Therefore, , is the solution of system (4.1) with initial value .

Denote .

Theorem 4.1. Let be any solution of system (2.3) with positive initial values, if

then is globally attractive.

Proof. From system (2.3), we have

Consider the following comparison system

Obviously, system (4.5) has a positive periodic solution

which is globally asymptotically stable. By Lemma 3.3, we conclude that for an arbitrary positive constant small enough, there exists an such that

From which, we get

That is,

By Lemma 3.7, we conclude that for a sufficiently small , there exists an such that

that is,

Let , from the first equation of systems (2.3), (4.9), and (4.11). We have

Now consider the following comparison equation:

Since the first condition of the theorem holds, we can choose the above small enough such that

By (4.14) and Lemma 3.4, we have .

By Lemma 3.3, we get

Incorporating the positivity of , we get .

Then for a sufficiently small and large enough, we have , without loss of generality, we may assume as .

From the second equation of system (2.3), we have

Consider the following system:

Obviously, system (4.17) has a positive periodic solution

which is globally asymptotically stable. Thus, for a sufficiently small , when is large enough, we have

By the second condition of the theorem, when is large enough, we have

By (4.19), when is large enough,

Combining the third equation of system (2.3) with (4.21), we obtain

By (4.20) and , we have as .

Further, since , for an arbitrary positive constant small enough, we have as is large enough. Then

Consider the following system:

So system (4.24) has a positive periodic solution

which is globally asymptotically stable. Therefore, for an arbitrary positive constant small enough, when is large enough, we have

Combining (4.26) with (4.19), we obtain , since are all sufficient small constants, we know

By Lemma 3.7, we get

The proof is complete.

Remark 4.2. Obviously, we know that the global attractiveness of pest-eradication periodic solution of system (2.2) is equivalent to the global attractiveness of mature pest-eradication periodic solution of system (2.3).

5. Permanence

Theorem 5.1. System (2.3) is permanent provided that

Proof. By Lemma 3.8, we know that there exists an and such that , , , , for all , we will prove the theorem through the following five steps.

Step 1. From (4.9), we know that there exists an such that for large enough.

Step 2. From (4.11), we know that there exists an such that for large enough.

Step 3. From Lemma 3.7, for an arbitrary positive constant small enough, when is large enough, .

Step 4. We will prove that there exists an such that for large enough, we will do it in the following two steps.

(i) By the condition of the theorem, we can select positive constants and small enough such that

Now we will prove that cannot hold for all , otherwise,

Let

By Lemma 3.3, and , where . So there exists a such that

for . From Lemma 3.7, we know that when is large enough,

for a sufficiently small , then

by Lemma 3.4, , as . This is a contradiction to the boundedness of . So there exists a such that .

(ii) If for all , our aim is obtained. Otherwise, we consider that is oscillating about .

Let , then for , since is continuous, ; and because is oscillating about , we know that there exists a , then for , from the continuity of , we get ; and continue we can obtain the time sequence , which satisfies

(a)when ,(b)when ,

(c)when .

We claim that there exists a , , otherwise, there exists a subsequence , such that ; obviously, (5.7) holds when .

By Lemma 3.4, we get ; this is a contradiction to .

By the boundedness of the system, we have

for . It is clear that

for , Let , then we have for all .

Step 5. We will prove that there exists an such that for large enough, we will do it in the following two steps.

(i) From the condition of the theorem, let and be small enough such that

We will prove cannot hold for all . Otherwise, we have

Consider the following system:

By Lemma 3.3, we have and , as , where .

So there exists a , when .

Hence,

from (5.10), we get , as , this is a contradiction to the boundedness of .So there exists a such that .

(ii) Similar to the method of step (ii) of Step 4, we can find an such that for all .

Therefore, for large enough, where . The proof is complete.

6. Conclusion

In this paper, we propose and analyze a pest control model with age structure for pest and pulse spraying pesticides and pulse releasing infective predators. By Lemma 3.8, we know that any solution of system (2.1) is bounded for large enough and get the specific form of the upper boundedness. From Theorem 4.1, we get the sufficient condition of global attractiveness of the pest-extinction periodic solution:

By Theorem 5.1, we get the sufficient condition for the permanence of the system:

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