Stability and Local Hopf Bifurcation for a Predator-Prey Model with Delay

A predator-prey system with disease in the predator is investigated, where the discrete delay τ is regarded as a parameter. Its dynamics are studied in terms of local analysis and Hopf bifurcation analysis. By analyzing the associated characteristic equation, it is found that Hopf bifurcation occurs when τ crosses some critical values. Using the normal form theory and center manifold argument, the explicit formulae which determine the stability, direction, and other properties of bifurcating periodic solutions are derived.


Introduction
Many models in ecology can be formulated as system of differential equations with time delays.The effect of the past history on the stability of system is also an important problem in population biology.Recently, the properties of periodic solutions arising from the Hopf bifurcation have been considered by many authors 1-4 .
May 5 first proposed and discussed the delayed predator-prey system dx dt x t r 1 − a 11 x t − τ − a 12 y t , dy dt y t −r 2 a 21 y t − a 22 y t , 1.1 where x t and y t can be interpreted as the population densities of prey and predator at time t, respectively; τ ≥ 0 is the feedback time delay of the prey to the growth of the species itself; r 1 > 0 denotes the intrinsic growth rate of the prey, and r 2 > 0 denotes the death rate of the predator; the parameter a ij i, j 1, 2 are all positive constants.System 1,1 shows that, in the absence of predator species, the prey species are governed by the well-known delayed logistic equation dx/dt x t r 1 − a 11 x t − τ and the predator species will decrease in the absence of the prey species.There has been an extensive literature dealing with system 1,1 or the system similar to 1.1 , regarding boundedness of solutions, persistence, local and global stabilities of equilibria, and existence of nonconstant periodic solutions 6-9 .
Recently, Faria 7 investigated the stability and Hopf bifurcation of the following system with instantaneous feedback control and two different discrete delays: where τ 1 > 0 and τ 2 > 0. But, as pointed out by Kuang 8 , in view of the fact that in real situations, instantaneous responses are rare, and thus, more realistic models should consist of delay differential equations without instantaneous feedbacks.Based on this idea, in the present paper, we combine the model 1.1 and 1.2 and consider the following delayed prey-predator system with a single delay: where X, S, I denote, respectively, the population of prey species, susceptible predator species and infected predator species.In addition, the coefficients r1, r2, p, k, σ, c 1 , c 2 in model 1.3 are all positive constants and their ecological meaning are interpreted as follows: r 1 denotes the intrinsic growth rate of prey and r 1 /r 2 denotes the carrying capacity of prey; p, k, c 1 and c 2 represent the predating coefficient of predator to prey, absorbing rate of predator to prey, and the death rate of susceptible and infected predator, respectively.The main purpose of this paper is to investigate the effects of the delay on the dynamics of model 1.3 with the following initial conditions: where 3 0, { x, y, z | x ≥ 0, y ≥ 0, z ≥ 0}.We will take the delay τ as the bifurcation parameter and show that when τ passes through a certain critical value, the positive equilibrium loses its stability and a Hopf bifurcation will take place.Furthermore, when τ takes a sequence of critical values containing the above critical value, the positive equilibrium of system 1.3 will undergo a Hopf bifurcation.In particular, by using the normal form theory and the center manifold, the formulae determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are also obtained.
The organization of this paper is as follows.In Section 2, we discuss the stability of the positive solutions and the existence of the Hopf bifurcations.In Section 3, the direction of the Hopf bifurcation and the stability of bifurcated periodic solutions are obtained by using the normal form theory and the center manifold theorem.In Section 4, we do some numerical simulations to validate our theoretical results.

Stability of Positive Equilibrium and Hopf Bifurcation
System 1.3 has a unique positive equilibrium E X * , S * , I * provided that the condition

2.1
Linearizing system 1.2 at E gives the following linear system:

2.2
The characteristic matrix of this system 2.2 is Thus, the characteristic equation of system 2.2 is given by

2.5
Then we rewrite 2.4 as:

2.7
Separating the real and imaginary parts, we have

2.8
By calculating, we have obtained

2.14
Since lim z → ∞ G z ∞ and f 5 < 0, then we can get the following conclusion.
H 2 Equation 2.13 has at least one positive real root.
Without loss of generality, we assume that it has five positive roots, defined by z 1 , z 2 , z 3 , z 4 , z 5 , respectively.Then 2.13 has five positive roots where k 1, . . ., 5; j 0, 1, . .., then ±iω k is a pair of purely imaginary roots of 2.6 with τ j k .Define

2.18
Note that when τ 0, 2.6 becomes By Routh-Hurwitz criterion, we know that all the roots of 2.19 have negative real parts, that is, the positive equilibrium E is locally asymptotically stable for τ 0.
In order to give the main results, it is necessary to make the following assumption: Differentiating two sides of 2.6 in respect to τ, we get

Stability and Direction of Hopf Bifurcation
In this section, we will derive the explicit formulae determining the properties of the Hopf bifurcation at the critical value using the normal form theory and center manifold theorem introduced by Hassard et al. 11 .Without loss of generality, let τ τ k μ, where τ k is defined by 2.17 , μ ∈ R, then system 1.3 can be rewritten as where u t u 1 t , u 2 t , u 3 t T U τt , V τt , W τt T ∈ R 3 , u t θ u t θ and L μ : C → R 3 , f : R × C → R 3 are given, respectively, by By the Riesz representation theorem, there exists a function η θ, μ of bounded variation for θ ∈ −1, 0 , such that In fact, we can choose where δ denote the Dirac delta function.For φ ∈ C −1, 0 , R 3 , define

3.6
Then system 3.1 is equivalent to u t A μ u t R μ u t .

3.7
For ψ ∈ C 0, 1 , R 3 * , define and a bilinear inner product where η θ η θ, 0 .Then A 0 and A * are adjoint operators.By the discussion in Section 2, we know that ±iω 0 τ k are eigenvalues of A 0 .Hence, they are also eigenvalues of A * .We first need to compute the eigenvectors of A 0 and A * corresponding to iω 0 τ k and −iω 0 τ k , respectively.Suppose q θ 1, a 1 , a 2 T e iω 0 τ k θ is the eigenvector of A 0 corresponding to iω 0 τ k , then A 0 q θ iω 0 τ k q θ .It follows from the definition of A 0 and 3.2 , 3.4 and 3.5 , we have

3.11
On the other hand, suppose that q * s J 1, a * 1 , a * 2 e iω 0 τ k S is the eigenvector of A * corresponding to −iω 0 τ k , by the similar method, we have

3.12
In order to assure q * s , q θ 1, we need to determine the value of J. From 3.9 , we have

3.13
Therefore, we can choose J as

3.14
Next we will compute the coordinate to describe the center manifold C 0 at μ 0. Let u t be the solution of 3.1 with μ 0. Define z t q * , u t , W t, θ u t θ − z t q θ − z t q * θ u t θ − 2 Re z t q θ .

3.15
On the center manifold C 0 , we have where z and z are local coordinates of center manifold C 0 in the direction of q * and q * .Note that W is real if u t is real.We consider only real solutions.For solution u t ∈ C 0 of 3.7 , since μ 0, we have z t q * , u t q * , Au t Ru t q * , Au t q * , Ru t A * q * , u t q * , Ru t iω 0 τ k z q * 0 f 0 , W z, z, 0 2 Re zq 0 iω 0 τ k z q * 0 f 0 z, z .

3.18
We rewrite this equation as Noticing u t θ W t, θ zq θ z q θ and q θ 1, a 1 , a 2 T e iω 0 τ k θ , we have

3.21
It follows together with 3.3 , that g z, z q * 0 f 0 z, z q * 0 f 0 0, u t .

3.22
Comparing the coefficients with 3.20 , we have

3.24
Since there are W 20 θ and W 11 θ in g 21 , we need to determine them.From 3.7 and 3.15 , we have where Substituting the corresponding series into 3.25 and comparing the coefficients, we have

3.28
Comparing the coefficients with 3.26 , we have

3.36
So we obtain

3.37
By 3.37 , we have

3.43
It follows that

3.44
Similarly, we can get Therefore, we can determine W 20 0 and W 11 0 , hence we can obtain g 21 .Thus, we can compute these values which determine the qualities of bifurcating periodic solution in the center manifold at the critical values τ k , so we have the following results.
(iii) T 2 determines the period of the bifurcating periodic solutions: the period increases (decreases) if T 2 > 0 T 2 < 0 .

Discussion and Numerical Example
In this section,we present some numerical results of system 1.3 at different values of τ.Form Section 3, we may determine the direction of a Hopf bifurcation and the stability of the bifurcating periodic solutions.We consider the following system: When τ passes through the critical value τ 0 .E loses its stability and a Hopf bifurcation occurs, that is, a family of periodic solutions bifurcate from E .Since μ 2 > 0 and β 2 < 0, the Hopf bifurcation is supercritical and the direction of the bifurcation is τ > τ 0 and these bifurcating periodic solutions from E at τ 0 are stable,which are depicted in Figures 2 a and

Figure 1 :
Figure 1: When τ 1.11 < τ 0 , the positive equilibrium E 2.8333, 0.8333, 1.8056 is asymptotically stable.a shows the trajectories graphs of the system 4.1 with initial data X t 2, S 1 t 2, I 2 t 2. b shows the phase portrait of system 4.1 .

Figure 2 :
Figure 2: When τ 1.13 > τ 0 , bifurcation periodic solutions form E .a shows the trajectory graphs of system 4.1 with initial data x t 2, y 1 t 2, y 2 t 2. b shows the phase portrait of system 4.1 .
. . ., m are constants.As τ 1 , τ 2 , . . ., τ m vary, the sum of the order of the zeros of P λ, e −λτ 1 , . . ., e −λτ m on the open right half plane can change only if a zero appears on or crosses the imaginary axis.