Stability Analysis of a Ratio-Dependent Predator-Prey Model Incorporating a Prey Refuge

A ratio-dependent predator-preymodel 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 τ = τ 0 . 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.


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][2][3][4][5].In [5], Xiao and Chen have considered a ratio-dependent predatorprey 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 (1 − ) 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.Stagestructured models have received great attention in recent years (see, e.g., [6][7][8][9]).
Time delays of one type or another have been incorporated into biological models by many researchers (see, e.g., [8][9][10][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  1 () and  2 () represent the densities of the immature and the mature predator population at time , respectively, the parameters  1 ,  2 , and  1 are positive constants in which  1 and  2 are the death rates of the immature and the mature predator, respectively,  1 denotes the rate of immature predator becoming mature predator, the constant proportion infected prey refuge is (1−), where  ∈ [0, 1) is a constant, and  ≥ 0 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 ((), (),  1 (),  2 ()) 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.

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  ≥ 0.

Predator-Extinction Equilibrium and Its Stability
In this section, we discuss the stability of the predatorextinction equilibrium.
Theorem 3. Let  >  hold; the predator-extinction equilibrium  1 is globally stable provided that Proof.Based on above discussions, we only prove the global attractivity of the equilibrium  1 .Let ((), (),  1 (),  2 ()) 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 lim By comparison, we obtain that lim sup Hence, for  > 0 sufficiently small, there is a  1 > 0 such that if  >  1 , then () ≤  1 + .
It follows from the third and the fourth equations of system (2) that, for  >  1 + , Consider the following auxiliary equations: If  2 ( 1 +  1 ) >  1 , then by Lemma 2.4 in [9], it follows from (23) that lim By comparison, we obtain that lim Hence, for  > 0 sufficiently small, there is a  2 > 0 such that if  >  2 , then  2 () ≤ .

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: then system (2) has a unique coexistence equilibrium  * ( * ,  * ,  * 1 ,  * 2 ), where The characteristic equation of system (2) at the equilibrium  * takes the form where When  = 0, (32) becomes If the following holds: then it is easy to show that , then, by the Routh-Hurwitz theorem, when  = 0, the coexistence equilibrium  * of system (2) is locally asymptotically stable and  * is unstable if If ( > 0) 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 ℎ 3 > 0, ℎ 2 > 0, ℎ 1 > 0 and  0 −  0 > 0, by the general theory on characteristic equation of delay differential equation from [13] (Theorem 4.1),  * remains stable for all  > 0.
In the following, we claim that This will show that there exists at least one eigenvalue with a positive real part for  >  0 .Moreover, the conditions for the existence of a Hopf bifurcation (Theorem 2.9.1 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 sgn { (Re )  } We derive from (36) that Hence, it follows that sgn {  (Re )  } Therefore, the transversal condition holds and a Hopf bifurcation occurs at  =  0 ,  =  0 .
In conclusion, we have the following results.

Conclusion
In this paper, we have incorporated a prey refuge, stage structure for the predator and time delay due to the gestation of the predator into a predator-prey system.Incorporating a refuge into system (1) provides a more realistic model.A refuge can be important for the biological control of a pest; however, increasing the amount of refuge can increase prey densities and lead to population outbreaks.By using the iteration technique and comparison arguments, respectively, we have established sufficient conditions for the global stability of the predator-extinction equilibrium and the globally attractivity for the coexistence equilibrium.As a result, we have shown the threshold for the permanence and extinction of the system.By Theorem 3, we see that the predator population go to extinction if 0 < (1 − ) < (/)( − ) and  > ,  2 ( 1 +  1 ) >  1 .By Theorem 7, we see that if 0 < (1 − ) < (/)(−) and  2 ( 1 + 1 ) <  1  < 2 2 ( 1 + 1 ), then both the prey and predator species of system (2) are permanent.