Abstract

A SI-type ecoepidemiological model that incorporates reproduction delay of predator is studied. Considering delay as parameter, we investigate the effect of delay on the stability of the coexisting equilibrium. It is observed that there is stability switches, and Hopf bifurcation occurs when the delay crosses some critical value. By applying the normal form theory and the center manifold theorem, the explicit formulae which determine the stability and direction of the bifurcating periodic solutions are determined. Computer simulations have been carried out to illustrate different analytical findings. Results indicate that the Hopf bifurcation is supercritical and the bifurcating periodic solution is stable for the considered parameter values. It is also observed that the quantitative level of abundance of system populations depends crucially on the delay parameter if the reproduction period of predator exceeds the critical value.

1. Introduction

Ecoepidemiology is a branch in mathematical biology which considers both the ecological and epidemiological issues simultaneously. After the pioneering work of Anderson and May [1], literature in the field of ecoepidemiology has grown enormously [29]. Chattopadhyay and Bairagi [3] studied the following ecoepidemiological model with : In this model, , , and represent the densities of susceptible prey, infected prey, and the predator populations, respectively. Both susceptible and infected preys contribute to the carrying capacity (), but only susceptible prey can reproduce at the intrinsic growth rate . Disease spreads horizontally from infected to susceptible prey at a rate following the law of mass action. Predator preys on infected prey only and predation process follows Holling Type II [10] response function with search rate and half-saturation constant . Here, is the conversion efficiency of the predator defining the increase in predator's number per unit prey consumption. represents the total death rate of infected prey where is the natural death rate and is the virulence of the disease. Predators consume both the susceptible and infected preys; however, the predation rate on infected prey may be very high (31 times) compare to that on susceptible prey [11]. Based on the experimental observation [11], it is assumed that predator consumes infected prey only. Predators may have to pay a cost in terms of extra mortality in the tradeoff between the easier predation and the parasitized prey acquisition, but the benefit is assumed to be greater than the cost [12, 13]. So it is assumed that consumption of infected prey contributes positive growth to the predator population. is the total death rate of predator where is the natural death rate and is the cost due to parasitized prey acquisition. All parameters are assumed to be positive.

Reproduction of predator after consuming the prey is not instantaneous, but mediated by some time lag. Chattopadhyay and Bairagi [3] did not consider this reproduction delay, defined by the time required for the reproduction of predator after consuming the prey, in their model system. It is well recognized that introduction of reproduction delay makes the model biologically more realistic. If (>0) is the time required for the reproduction, the model (1.1) can be written as We study the delay-induced system (1.2) with the following initial conditions:

Hopf bifurcation and its stability in a delay-induced predator-prey system have been studied by many researchers [1419]. In this paper, we study the effect of reproduction delay on an ecoepidemiological system where predator-prey interaction follows Holling Type II response function, and find the direction and stability of the bifurcating periodic solutions, if any.

The organization of the paper is as follows. Section 2 deals with the linear stability analysis of the model system. In Section 3, direction and stability of Hopf bifurcation are presented. Numerical results to illustrate the analytical findings are presented in Section 4 and, finally, a summary is presented in Section 5.

2. Stability Analysis and Hopf Bifurcation

In epidemiology, the basic reproductive ratio , the number of new cases acquired directly from a single infected prey when introduced into a population of susceptible, plays a significant role in the spread of the disease. In particular, if , the disease dies out, but if , it remains endemic in the host population [20]. For the system (1.2), the basic reproductive ratio is given by . In ecology, on the other hand, stress is given on the stability of coexisting equilibrium point. We, therefore, concentrate on the study of the stability of the coexisting or endemic equilibrium point of the system (1.2). The ecoepidemiological system (1.2) has a unique interior equilibrium point , where , and . Note that exists if , exists if and exists if with . Thus, the conditions for coexisting equilibrium point are(i), that is, , (ii).

Let ,  ,  and be the perturbed variables. Then, the system (1.2) can be expressed in the matrix form after linearization as follows: where The characteristic equation of the system (2.1) is given by that is, where Equation (2.4) can be written as where For , (2.5) becomes Here Thus, if . After some algebraic manipulation, can be written as So the sufficient condition for to be positive is Note that is always positive. One can write, Since all the terms in the third bracket are positive, so the sufficient condition for the positivity of is Hence, by Routh-Hurwitz criterion and using existence conditions, we state the following theorem for the stability of the interior equilibrium of the system (1.2) for .

Theorem 2.1. If (i),(ii),
where and , then the system (1.2) is locally asymptotically stable without delay around the positive interior equilibrium .

We now reproduce some definitions given by [21, 22].

Definition 2.2. The equilibrium is called asymptotically stable if there exists a such that implies that where is the solution of the system (1.2) which satisfies the condition (1.3).

Definition 2.3. The equilibrium is called absolutely stable if it is asymptotically stable for all delays and conditionally stable if it is stable for in some finite interval.

Note that the system (1.2) will be stable around the equilibrium if all the roots of the corresponding characteristic equation (2.5) have negative real parts. But (2.5) is a transcendental equation and has infinite number of roots. It is difficult to determine the sign of these infinite number of roots. Therefore, we first study the distribution of roots of the cubic exponential polynomial equation (2.5).

We know that is a root of (2.5) if and only if satisfies Separating real and imaginary parts, we get This two equations give the positive values of and for which (2.5) can have purely imaginary roots.

Squaring and adding, we obtain where If we assume , then (2.16) reduces to Denote Note that and . Thus, if , then (2.18) has at least one positive root.

From (2.18), we have Clearly, if , then the function is monotonically increasing in . Thus, for and , (2.18) has no positive roots for . On the other hand, when and , the equation has two real roots Obviously, and . It follows that and are the local minimum and the local maximum, respectively. Hence we have the following lemma.

Lemma 2.4. Suppose that and . Then (2.17) has positive roots if and only if .

Proof. Noticing that ,   is the local minimum of and , we immediately know that the sufficiency is true. So we have to prove now the necessity. In contrary, we suppose that either or and . Since is increasing for and , it follows that has no positive real roots for and . If and , since is the local maximum value, it follows that . Thus, cannot have any positive real roots when and . This completes the proof.

Summarizing the above discussions, we obtain the following.

Lemma 2.5. One has the following results on the distribution of roots of (2.17). (i)If , then (2.17) has at least one positive root;(ii)if , and , then (2.17) has no positive root;(iii)if , and , then (2.17) has positive roots if and only if and , where .

Suppose that (2.17) has positive roots. Without loss of generality, we assume that it has three positive roots, defined by , , and , respectively. Then, (2.16) has three positive roots , and .

From (2.15), we have Thus, if we denote where ;  , then is a pair of purely imaginary roots of (2.5). Define We reproduce the following result due to Ruan and Wei [23] to analyze (2.5).

Lemma 2.6. Consider the exponential polynomial where and , are constants. As vary, the sum of the order of zeros of on the open right half hand can change only if a zero appears on or crosses the imaginary axis.

Using Lemmas 2.5 and 2.6, we can easily obtain the following results on the distribution of roots of the transcendental (2.5).

Lemma 2.7. For the third degree exponential polynomial equation (2.5), one has(i)if , and , then all roots with positive real parts of (2.5) have the same sum as those of the polynomial equation (2.6) for all ,(ii)if either or , and , then all roots with positive real parts of (2.5) have the same sum as those of the polynomial equation (2.6) for all .

Let where and are real, be the roots of (2.5) near satisfying Then the following transversality condition holds.

Lemma 2.8. Suppose that and , where is defined by (2.18). Then, and the sign of is consistent with that of .

Proof. Differentiating (2.5) with respect to , we obtain This gives It follows from (2.15) that Using (2.31) in (2.30), we get where . Thus, we have Since are positive, we conclude that the sign of is determined by that of . This proves the lemma.

From and , we have Thus, from Lemmas 2.7 and 2.8, we have the following theorem.

Theorem 2.9. Let and are defined by , (2.34), and (2.23), respectively. Suppose that conditions of Theorem (2.1) hold. Then the following results hold.(i)When , and , then all roots of (2.5) have negative real parts for all and the equilibrium of the system (1.2) is absolutely stable for all .(ii)If either or and hold, then has at least one positive root and all roots of (2.5) have negative real parts for all , then the equilibrium of the system (1.2) is conditionally stable for .(iii)If all the conditions as stated in (ii) and hold, then the system (1.2) undergoes a Hopf bifurcation at when , .

3. Direction and Stability of the Hopf Bifurcation

In the previous section, we obtained some conditions under which system (1.2) undergoes Hopf bifurcation at . In this section, we assume that the system (1.2) undergoes Hopf bifurcation at when , that is, a family of periodic solutions bifurcate from the positive equilibrium point at the critical value . We will use the normal form theory and center manifold presented by Hassard et al. [24] to determine the direction of Hopf bifurcation, that is, to ensure whether the bifurcating branch of periodic solution exists locally for or , and determine the properties of bifurcating periodic solutions, for example, stability on the center manifold and period. Throughout this section, we always assume that system (1.2) undergoes Hopf bifurcation at the positive equilibrium for and then is corresponding purely imaginary roots of the characteristic equation.

Let , ,  , , , where is defined by (2.23) and . Dropping the bars for simplification of notations, system (1.2) can be written as functional differential equation (FDE) in as where , and ,   are given, respectively, by By the Riesz representation theorem, there exists a matrix, whose elements are bounded variation functions such that In fact, we can choose where is the Dirac delta function defined by

For , define the operator as Then system (3.1) is equivalent to where .

For , define and a bilinear inner product where = . Then and are adjoint operators. By Theorem 2.9, we know that are eigenvalues of . Thus, they are also eigenvalues of . We first need to compute the eigenvalues of and corresponding to and , respectively.

Suppose that is the eigenvector of corresponding to . Then = . It follows from the definition of and (3.2), (3.4), and (3.5) that Thus, we can easily obtain where Similarly, let be the eigenvector of corresponding to . By the definition of and (3.2), (3.3), and (3.4), we can compute In order to assure , we need to determine the value of . From (3.10), we have Thus, we can choose as In the remainder of this section, we use the theory of Hassard et al. [24] to compute the conditions describing center manifold at . Let be the solution of (3.8) when . Define On the center manifold , we have where and are local coordinates for center manifold in the direction of and . Note that is real if is real. We only consider real solutions. For solution of (3.8), since , we have We rewrite this equation as where We have and , so from (3.17) and (3.19) it follows that and then we have

It follows together with (3.3) that

Comparing the coefficients with (3.22), we have Since there are and in , we still need to compute them.

From (3.8) and (3.17), we have where Substituting the corresponding series into (3.27) and comparing the coefficients, we obtain From (3.27), we know that for , Comparing the coefficients with (3.28), we get From (3.29) and (3.31) and the definition of , it follows that Notice that , hence where is a constant vector. Similarly, from (3.29) and (3.32), we obtain where is also a constant vector.

In what follows, we will seek appropriate and . From the definition of and (3.29), we obtain where .

By (3.27), we have Substituting (3.34) and (3.38) into (3.36) and noticing that we obtain This leads to Solving this system for , we obtain where Similarly, substituting (3.35) and (3.39) into (3.37), we get and hence where

Thus, we can determine and from (3.34) and (3.35). Furthermore, in (3.26) can be expressed by the parameters and delay. Thus, we can compute the following values: which determine the qualities of bifurcating periodic solution in the center manifold at the critical value .

Theorem 3.1. determines the direction of the Hopf bifurcation. If , then the Hopf bifurcation is supercritical and the bifurcating periodic solutions exist for . If , then the Hopf bifurcation is subcritical and the bifurcating periodic solutions exist for . determines the stability of the bifurcating periodic solutions: the bifurcating periodic solutions are stable if and unstable if . determines the period of the bifurcating periodic solutions: the period increase if and decrease if .

4. Numerical Simulations

In this section, we present some numerical simulations to illustrate the analytical results observed in the previous sections. We consider the following set of parameter values: For the above parameter set, the system (1.2) has a unique coexistence equilibrium point . When , the system (1.2) satisfies all conditions of the Theorem 2.9(i). Consequently, the coexistence equilibrium point becomes absolutely stable. Figure 1 shows the behavior of the system (1.2) when , and Figure 2 depicts the same for and . If we change the value of from 0.45 to 0.72 in the given parameter set, then conditions of the Theorem 2.9(ii) are satisfied and the system (1.2) becomes conditionally stable around the coexistence equilibrium point for (see, Figure 3(a)) and unstable for (see, Figure 3(b)).

For the given parameter set with , one can evaluate that and , so the system (1.2) undergoes a Hopf bifurcation at when following the condition (iii) of Theorem 2.9. We have constructed a bifurcation diagram (see, Figure 4) to observe the dynamics of the system when varies. For this, we have run the system (1.2) for 500 time-steps and have plotted the successive maxima and minima of the prey and predator populations with as a variable parameter. This figure shows that the coexisting equilibrium is stable if is less than its critical value and unstable if and a Hopf bifurcation occurs at .

Using Theorem 3.1, one can determine the values of and . For the given parameter set with , one can evaluate that (>0), (<0), and (>0). Since and , the Hopf bifurcation is supercritical and the bifurcating periodic solutions exist when crosses from left to right. Also, the bifurcating periodic solution is stable (as ) and its period increases with (as ). From the bifurcation diagram (Figure 4), it is clear that when the delay, , exceeds the critical value ( days approximately), the system (2.4) bifurcates from stable focus to stable limit cycle. One can also notice that the amplitude of the oscillations increases with increasing .

5. Summary

In this paper, we have studied the effects of reproduction delay on an ecoepidemiological system where predator-prey interaction follows Holling Type II response function. We have obtained sufficient conditions on the parameters for which the delay-induced system is asymptotically stable around the positive equilibrium for all values of the delay parameter and if the conditions are not satisfied, then there exists a critical value of the delay parameter below which the system is stable and above which the system is unstable. By applying the normal form theory and the center manifold theorem, the explicit formulae which determine the stability and direction of the bifurcating periodic solutions have been determined. Our analytical and simulation results show that when passes through the critical value , the coexisting equilibrium losses its stability and a Hopf bifurcation occurs, that is, a family of periodic solutions bifurcate from . Also, the amplitude of oscillations increases with increasing . For the considered parameter values, it is observed that the Hopf bifurcation is supercritical and the bifurcating periodic solution is stable. The quantitative level of abundance of system populations depends crucially on the delay parameter if the reproduction period of predator exceeds the critical value .

Acknowledgment

Research is supported by DST (PURSE seheme), India; no. SR/54/MS: 408/06. The author wishes to thank the anonymous referee for careful reading of the paper.