#### Abstract

A ratio-dependent predator-prey model incorporating a prey refuge with disease in the prey population is formulated and analyzed. The effects of time delay due to the gestation of the predator and stage structure for the predator on the dynamics of the system are concerned. By analyzing the corresponding characteristic equations, the local stability of a predator-extinction equilibrium and a coexistence equilibrium of the system is discussed, respectively. Further, it is proved that the system undergoes a Hopf bifurcation at the coexistence equilibrium, when . By comparison arguments, sufficient conditions are obtained for the global stability of the predator-extinction equilibrium. By using an iteration technique, sufficient conditions are derived for the global attractivity of the coexistence equilibrium of the proposed system.

#### 1. Introduction

Since the pioneering work of Kermack-Mckendrick on SIRS [1], epidemiological models have received much attention from scientists. Mathematical models have become important tools in analyzing the spread and control of infectious disease. It is of more biological significance to consider the effect of interacting species when we study the dynamical behaviors of epidemiological models. Ecoepidemiology which is a relatively new branch of study in theoretical biology, tackles such situations by dealing with both ecological and epidemiological issues. It can be viewed as the coupling of an ecological predator-prey model and an epidemiological SI, SIS, or SIRS model. Following Anderso and May [2] who were the first to propose an ecoepidemiological model by merging the ecological predator-prey model introduced by Lotka and Volterra, the effect of disease in ecological system is an important issue from mathematical and ecological point of view. Many works have been devoted to the study of the effects of a disease on a predator-prey system [1–5]. In [5], Xiao and Chen have considered a ratio-dependent predator-prey system with disease in the prey. Consider where and represent the densities of susceptible and infected prey population at time , respectively, and represents the density of the predator population at time . The parameters , , , , , , , and are positive constants representing the prey intrinsic growth rate, carrying capacity, transmission rate, the infected prey death rate, capturing rate, half capturing saturation constant, conversion rate, and the predator death rate, respectively. A periodic solution can occur whether the system (1) is permanent or not; that is, there are solutions which tend to disease-free equilibrium while bifurcating periodic solution exists.

Recently, the qualitative analysis of predator-prey models incorporating a prey refuge has been done by many authors, see [3, 4]. In [3], Pal and Samanta incorporated a prey refuge into system (1). Sufficient conditions were derived for the stability of the equilibria of the system.

We note that it is assumed in system (1) that each individual predator admits the same ability to feed on prey. This assumption seems not to be realistic for many animals. In the natural world, there are many species whose individuals pass through an immature stage during which they are raised by their parents, and the rate at which they attack prey can be ignored. Moreover, it can be assumed that their reproductive rate during this stage is zero. Stage-structure is a natural phenomenon and represents, for example, the division of a population into immature and mature individuals. Stage-structured models have received great attention in recent years (see, e.g., [6–9]).

Time delays of one type or another have been incorporated into biological models by many researchers (see, e.g., [8–11]). In general, delay differential equations exhibit much more complicated dynamics than ordinary differential equations since a time delay could cause the population to fluctuate. Time delay due to gestation is a common example, because, generally, the consumption of prey by the predator throughout its past history governs the present birth rate of the predator. Therefore, more realistic models of population interactions should take into account the effect of time delays.

Based on the above discussions, in this paper, we incorporate a prey refuge, stage structure for the predator, and time delay due to the gestation of predator into the system (1). To this end, we study the following differential equations: where and represent the densities of the immature and the mature predator population at time , respectively, the parameters , , and are positive constants in which and are the death rates of the immature and the mature predator, respectively, denotes the rate of immature predator becoming mature predator, the constant proportion infected prey refuge is , where is a constant, and is a constant delay due to the gestation of the predator.

The initial conditions for system (2) take the form where .

It is well known by the fundamental theory of functional differential equations [12] that system (2) has a unique solution satisfying initial conditions (3).

The organization of this paper is as follows. In the next section, we show the positivity and the boundedness of solutions of system (2) with initial conditions (3). In Section 3, we investigate the global stability of the predator-extinction equilibrium. In Section 4, we establish the local stability and the global attractivity of the coexistence equilibrium of system (2). Further, we study the existence of Hopf bifurcation for system (2) at the positive equilibrium. A brief discussion is given in Section 5 to conclude this work.

#### 2. Preliminaries

In this section, we show the positivity and the boundedness of solutions of system (2) with initial conditions (3).

Theorem 1. *Solutions of system (2) with initial conditions (3) are positive, for all .*

*Proof. *Let be a solution of system (2) with initial conditions (3). It follows from the first and the second equations of system (2) that

Let us consider and , for . Since for , we derive from the third equation of system (2) that
Since , a standard comparison argument shows that
that is, for . For , it follows from the fourth equation of (2) that
Since , a standard comparison argument shows that
that is, for . In a similar way, we treat the intervals . Thus, , , , and , for all . This completes the proof.

Theorem 2. *Positive solutions of system (2) with initial conditions (3) are ultimately bounded.*

*Proof. *Let be any positive solution of system (2) with initial conditions (3). Denote . Define
Calculating the derivative of along positive solutions of system (2), it follows that
which yields
If we choose and , then
This completes the proof.

#### 3. Predator-Extinction Equilibrium and Its Stability

In this section, we discuss the stability of the predator-extinction equilibrium.

It is easy to show that if , system (2) admits a predator-extinction equilibrium , where

The characteristic equation of system (2) at the equilibrium is of the form where . When , if , then is locally asymptotically stable and if , then is unstable. It is easily seen that Hence, if , by Lemma B in [11], it follows that the equilibrium is locally asymptotically stable for all . If , then is unstable for all .

Theorem 3. *Let hold; the predator-extinction equilibrium is globally stable provided that
*

*Proof. *Based on above discussions, we only prove the global attractivity of the equilibrium . Let be any positive solution of system (2) with initial conditions (3). It follows from the first and the second equations of system (2) that
Consider the following auxiliary equations:
If , then by Theorem 3.1 in [4], it follows from (19) that
By comparison, we obtain that
Hence, for sufficiently small, there is a such that if , then .

It follows from the third and the fourth equations of system (2) that, for ,
Consider the following auxiliary equations:
If , then by Lemma 2.4 in [9], it follows from (23) that
By comparison, we obtain that
Hence, for sufficiently small, there is a such that if , then .

It follows from the first and the second equations of system (2) that for :
Consider the following auxiliary equations:
If , and , then by Theorem 3.1 in [4], it follows from (27) that
By comparison, for sufficiently small, we obtain that
which, together with (21), yields

Hence, if hold, then the equilibrium is globally stable.

#### 4. Coexistence Equilibrium and Its Stability

In this section, we discuss the stability of the coexistence equilibrium and the existence of a Hopf bifurcation. It is easy to show that if the following holds:(H1), ,then system (2) has a unique coexistence equilibrium , where

The characteristic equation of system (2) at the equilibrium takes the form where

When , (32) becomes If the following holds:(H2), ,then it is easy to show that If , then, by the Routh-Hurwitz theorem, when , the coexistence equilibrium of system (2) is locally asymptotically stable and is unstable if .

If is a solution of (34), separating real and imaginary parts, we have By squaring and adding the two equations of (36), it follows that where If and , by the general theory on characteristic equation of delay differential equation from [13] (Theorem 4.1), remains stable for all .

If and , then (37) has a unique positive root ; that is, (34) admits a pair of purely imaginary roots of the form . From (36), we see that By Theorem in [13], we see that remains stable for .

In the following, we claim that This will show that there exists at least one eigenvalue with a positive real part for . Moreover, the conditions for the existence of a Hopf bifurcation (Theorem in [13]) are then satisfied yielding a periodic solution. To this end, by differentiating equation (34) with respect to , it follows that Hence, a direct calculation shows that We derive from (36) that Hence, it follows that Therefore, the transversal condition holds and a Hopf bifurcation occurs at .

In conclusion, we have the following results.

Theorem 4. *For system (2), let and hold; we have the following:*(i)*if , , and , then the coexistence equilibrium is locally asymptotically stable, for all ;*(ii)*if , , and , then there exists a positive number , such that the coexistence equilibrium is locally asymptotically stable if and is unstable for ; further, system (2) undergoes a Hopf bifurcation at when ;*(iii)*if , then the coexistence equilibrium is unstable, for all .*

We now give some examples to illustrate the main results above.

*Example 5. *In (2), we let , , , , , and . System (2), with the above coefficients, has a unique coexistence equilibrium . It is easy to show that , , that is the condition holds. We can get ,, , and . By Theorem 4(i), the coexistence equilibrium is locally asymptotically stable, for all . Numerical simulation illustrates our result (see Figure 1).

*Example 6. *In (2), we let , and . System (2), with the above coefficients, has a unique coexistence equilibrium . It is easy to show that , and ; that is, the condition holds. We can get , and . By Theorem 4(ii), there exists a positive number , such that the coexistence equilibrium is locally asymptotically stable if and is unstable for . Numerical simulations illustrate our results (see Figure 2).

**(a)**

**(b)**

Now, we are concerned with the global attractiveness of the coexistence equilibrium .

Theorem 7. *The coexistence equilibrium of system (2) is globally attractive provided that the following conditions hold:*(i)*;*(ii)*.*

That is, the system (2) is persistent, if conditions (i) and (ii) hold.

* Proof. *Let be any positive solution of system (2) with initial conditions (3). Let
We now claim that , . The strategy of the proof is to use an iteration technique.

We derive from the first and the second equations of the system (2) that
Consider the following auxiliary equations:
If , then by Theorem 3.1 in [4], it follows from (47) that
By comparison, we obtain that
Hence, for sufficiently small, there is a such that if , then .

It follows from the third and the fourth equations of system (2) that, for ,
Consider the following auxiliary equations:
If , then by Lemma 2.4 in [9], it follows from (51) that
By comparison, we obtain that
Hence, for sufficiently small, there is a such that if , then .

We derive from the first and the second equations of system (2) that
Since holds, by Theorem 3.1 in [4], it follows from (54) and comparison argument that
Hence, for sufficiently small, there is a such that if , then . We derive from the third and the fourth equations of system (2) that, for ,
Since holds, by Lemma 2.4 of [9], it follows from (56) and comparison argument that
Since these two inequalities hold, for arbitrary sufficiently small, we conclude that , where
Hence, for sufficiently small, there is a , such that if , .

For sufficiently small, we derive from the first and the second equations of system (2) that, for ,
By comparison and Theorem 3.1 in [4], it follows that
Since these two inequalities hold, for arbitrary sufficiently small, we conclude that , where
Therefore, for sufficiently small, there is a such that if , .

For sufficiently small, we derive from the third and the fourth equations of system (2) that, for ,
Since holds, by Lemma 2.4 of [9], it follows from (62) that
Since these two inequalities hold, for arbitrary sufficiently small, we conclude that , where
Therefore, for sufficiently small, there is a such that if , .

For sufficiently small, it follows from the first and the second equations of system (2) that, for ,
By Theorem 3.1 in [4] and comparison argument, we can obtain
Since these two inequalities hold, for arbitrary sufficiently small, we conclude that , where
Hence, for sufficiently small, there is a such that if , . We therefore obtain from the third and the fourth equations of system (2) that, for ,
Since holds, by Lemma 2.4 in [9] and comparison argument, we derive that
Since these inequalities hold for arbitrary sufficiently small, we conclude that , where
Continuing this process, we derive eight sequences , and such that, for ,
It is readily seen that
Noting that the sequences are nonincreasing, and the sequences are nondecreasing. Hence, the limit of each sequence in , and exists. Denote
From (71), we can obtain
It follows from (74) that
(75) minus (76),
Assume that . Then we derive from (77) that
(75) plus (76),
On substituting (78) into (79), it follows that
Note that and . If