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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 371685, 16 pages
http://dx.doi.org/10.1155/2012/371685
Research Article

Analysis of an Ecoepidemiological Model with Prey Refuges

1School of Mathematics and Computer Science, Northwest University for Nationalities, Lanzhou 730030, China
2School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China

Received 10 August 2012; Accepted 18 October 2012

Academic Editor: Wan-Tong Li

Copyright © 2012 Shufan Wang and Zhihui Ma. 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

An ecoepidemiological system with prey refuges and disease in prey is proposed. Bilinear incidence and Holling III functional response are used to model the contact process and the predation process, respectively. We will study the stability behavior of the basic system from a local to a global perspective. Permanence of the considered system is also investigated.

1. Introduction

Ecoepidemiology is the branch of biomathematics that understands the dynamics of disease spread on the predator-prey system. Modeling researches on such ecoepidemiological issues have received much attention recently [113]. Anderson and May [1] investigated a prey-predator model with prey infection and observed destabilization due to the spread of infectious diseases within animal and plant communities. Chattopadhyay analyzed predator-prey system with disease in the prey [5] and applied the ecoepidemiological study to the Pelicans at risk in the Salton Sea [6]. Bairagi et al. [13] made a comparative study on the role of prey infection in the stability aspects of a predator-prey system with several functional responses. An ecoepidemiological model with prey harvesting and predator switching was investigated by Bhattacharyya and Mukhopadhyay [14]. Kooi et al. [15] studied stabilization and complex dynamics in a predator-prey system with disease in predator. Most of the above-mentioned studies focused on the role of disease in regulating the dynamical consequences of the interacting populations concerned, such as disease-induced stabilization and destabilization of population states [13].

In fact, the dynamical consequences of the predator-prey model can be determined by much ecological effect, such as the Allee effect and prey refuge. Theoretical research and field observations on population dynamics of prey refuges lead to the conclusion that prey refuges have two influences (stabilizing and destabilizing effect) on predator-prey models and prey extinction can be prevented by the addition of prey refuges [1334]. Here, stabilization (destabilization) of stability refers to cases where an equilibrium point changes from an attractor (a repeller) to a repeller (an attractor) due to increase in the value of a control parameter [17]. Ruxton [16] proposed a continuous-time predator-prey model under the assumption that the rate of prey moving to refuges is proportional to predator density and the results showed that the hiding behavior of prey has a stabilizing effect. The stabilizing effect was also observed in a simple predator-prey system by González-Olivares and Ramos-Jiliberto [17]. Ma et al. [23] formulated a predator-prey model with a class of functional response incorporating the effect of prey refuges and observed the stabilizing and destabilizing effect due to the increases in the prey refuges.

In the present research, we formulate a mathematical model of prey-predator interaction with prey refuges and disease in prey. We mainly study the positivity and boundedness, the stability behavior of the disease-free equilibrium point, and the permanence of the basic model.

2. Model Formulation

The basic model comprises two population subclasses—(i) prey population with density and (ii) predators with density . To formulate our model, we make the following assumptions.(1)The prey population increases logistically with intrinsic growth rate and environmental carrying capacity .(2)The prey population is divided into two subclasses—the susceptible prey and the infected prey due to infectious disease. We also assumed that at any instant of time .(3)The susceptible prey is capable of reproducing only and the infected prey is removed by death at a rate .(4)The disease is spread only among the prey population and the disease is not genetically inherited. The infected prey does not become immune.(5)Susceptible prey becomes infected with the simple mass action law , where measures the force of infection.(6)The predators vanish due to natural death at a constant rate . They consume susceptible and infected prey following the Holling III functional response with predation coefficients and , respectively. The consumed prey is converted into predator with efficiency .(7)It is assumed that there is a quantity of (the susceptible and the infected) prey population incorporating refuges. denotes that a constant proportion of (the susceptible and the infected) prey use refuges. All the above-mentioned parameters are assumed to be positive.

With the previously mentioned assumptions, the generalized predator-prey system with prey refuges and disease in prey can be represented by the following equations:

Defining , it is easy to show that the set is the positively invariable set of system (2.1).

3. The Positivity and Boundedness

Theorem 3.1. All solutions of system (2.1) initiating are positive and ultimately bounded.

Proof. Let be one of the solutions of system (2.1).
Integrating (2.1) with initial conditions , we have
Hence all solutions starting in remain in for all .

Next, we will prove the boundedness of the solutions.

Because , then we have

Let , then we obtain that where .

Hence, we have That is

Thus, all curves of system (2.1) will enter the following region:

4. The Equilibrium Point

All equilibrium points of system (2.1) can be obtained by solving the following equations: These points are as follows:(1)the trivial equilibrium point , (2)the equilibrium point , (3)the predator-extinction equilibrium point , (4)the disease-free equilibrium point , where

Let . It is clear to show that the disease-free equilibrium point has its ecological meaning when .

Let , the predator-extinction equilibrium point is nonnegative when .

5. The Stability Property

In this section, we will study the local and global stability of the equilibrium points of system (2.1).

Theorem 5.1. Let , , then one has the following. (1)The trivial equilibrium point is always unstable. (2)If , , then the predator-extinction equilibrium point is globally asymptotically stable. If or , the equilibrium point is unstable. (3)If and , the predator-extinction equilibrium point is locally asymptotically stable. If , the predator-extinction equilibrium point is locally asymptotically stable. If , the predator-extinction equilibrium point is unstable.

Proof. The Jacobian matrix of system (2.1) at the trivial equilibrium point is

Clearly, the trivial equilibrium point is unstable.

The Jacobian matrix of system (2.1) at the equilibrium point is

According to the theorem about the local stability, the local stability of the equilibrium point is determined only by the sign of and .

Therefore, if and , the equilibrium point is locally asymptotically stable.

Next, we will prove the global stability of the equilibrium point .

Defining the Lyapunov function , then we obtain that

Hence, if and , .

Having , the maximum invariable set of system (2.1) is .

According to the LaSalle invariable set theorem, , .

Thus, the limit equation of system (2.1) is

Clearly, the equilibrium point is globally asymptotically stable.

According to the limit system theorem, if and , the equilibrium point is globally asymptotically stable.

Setting , the Jacobian matrix of system (2.1) at the predator-extinction equilibrium point is

The characteristic equation of system (2.1) at the predator-extinction equilibrium point is where

According to the Routh-Hurwitz rule, the predator-extinction equilibrium point is locally asymptotically stable when .

Next, we will prove the global stability of the predator-extinction equilibrium point .

Defining the Lyapunov function , then we have

Let in which .

Thus

Clearly, and on the set .

Thus, on the set , if and only if and , then , .

If , then . If , then . If , then ; that is, .

Therefore, the maximum value of the function is obtained at the point or .

Hence, if , ; that is, .

According to the LaSalle invariable set theorem, . Thus, the limit equation of system (2.1) is

According to the results of the appendix section, if , the equilibrium point is globally asymptotically stable.

According to the limit system theorem, if and , the predator-extinction equilibrium point is globally asymptotically stable.

Theorem 5.2. Let , . If , then the disease-free equilibrium point is nonnegative, (1)if and , then the disease-free equilibrium point is locally asymptotically stable; (2)if and , then the disease-free equilibrium point is globally asymptotically stable; (3)if or , then the disease-free equilibrium point is unstable.

Proof. Assuming , the Jacobian matrix of system (2.1) at the disease-free equilibrium point is where

The characteristic equation of system (2.1) at the disease-free equilibrium point is

Clearly, if , .

Again, we have

Hence, if , then .

According to the Routh-Hurwitz rule, if and , then the disease-free equilibrium point is locally asymptotically stable.

Next, we will study the global stability of the disease-free equilibrium point .

Defining the Lyapunov function , then we obtain

Thus, if , and if and only if ; that is, .

According to the LaSalle invariable set theorem, when . The limit system of system (2.1) is

Clearly, the equilibrium points of the system (5.16) are in which and are similar as the equilibria expression of system (2.1).

It is easy to show that the equilibrium point has its ecological meaning when .

According to the Routh-Hurwitz rule, the equilibrium point is unstable. If , then is unstable.

Again, the Jacobian matrix of system (5.16) at the equilibrium point is where

The characteristic equation of system (5.16) at the equilibrium point is

According to the above study, if , then .

Hence, if , then the equilibrium point is locally asymptotically stable in the region by the Routh-Hurwitz rule.

It is easy to note that the globally asymptotically stability of the equilibrium point implies that there is no close orbit in the region for the considered system.

Let and rewrite , , and into , , and , then system (5.16) becomes as follows: where , , , .

Thus, the positive equilibrium point of system (5.16) becomes the positive equilibrium point of system (5.22), where

Considering the Dulac function , then we have

In order to prove the global stability, we will prove only that there exists a real number such that .

Clearly, if or , then since .

Let  (, ), that is, , then we obtain that if , then . Otherwise, .

Therefore, the function has the maximum value at the point and .

Hence, there exists only one real number , such that and

Thus, we will prove only that there exists , such that where , .

The discriminant of the cubic equation is

It is easy to show that if , then .

Again, if and , then and . If , then .

According to Shengjin's distinguishing means, the cubic equation has one negative real root and two positively real roots.

If , the cubic equation has three roots which are not equal.

According to the Descartes rule of signs, the cubic equation has two positively real roots and one negatively real root at most.

Therefore, the cubic equation has at least one positively real root. That is to say, there exists a number , such that .

Furthermore, we obtain that

According to the Bendixson-Dulac theorem, there does not exist the limit cycle for the limit system.

Hence, the equilibrium point is globally asymptotically stable.

Therefore, if and , then the equilibrium point is globally asymptotically stable according to the limit system theorem.

6. Permanence

Theorem 6.1. If , , , and , then system (2.1) is permanent.

Proof. Considering the average Lyapunov function , where is positive, then in the region , we have

In order to prove the permanence of system (2.1), we only indicate the following results: the function for all boundary equilibrium points.

Let , then

Hence, if and , then . If , then . If , then .

Therefore, system (2.1) is permanent when , , , and by the average Lyapunov function theorem [34].

By simple computation, is equivalent to .

Hence, if , then implies .

Appendix

Considering the following model where and are density of susceptible and infected prey population at time . The parameters , , , and are all positive.

The positive equilibrium point of this system is in which

Let .

Clearly, if , then the equilibrium point is positive.

Theorem A.1. Let . If , then the positive equilibrium point is globally asymptotically stable.

Proof. The Jacobian matrix of system (A.1) at the positive equilibrium point is

Clearly, if , then the positive equilibrium point is locally asymptotically stable.

Next, defining the Dulac function , then we have where , .

Therefore, there does not exist closed curve in the region according to the Dulac theorem.

This implies that the positive equilibrium point is globally asymptotically stable.

Acknowledgments

This work was supported by the Fundamental Research Funds for the Central Universities (no. lzujbky-2011-48) and the National Natural Science Foundation of China (no. 11126183, 30970478, 30970491, 31100306).

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