Stability and Hopf Bifurcation in a Modified Holling-Tanner Predator-Prey System with Multiple Delays

and Applied Analysis 3 transformed into the following nondimensional form: dx dt x 1 − x t − τ1 − xy a1 bx c1y , dy dt y [ δ − β t − τ2 x t − τ2 ] , 1.4 where a1 a/K, c1 cr/α, δ s/r, β sh/α are the non-dimensional parameters and they are positive. The main purpose of this paper is to consider the effect of multiple delays on system 1.4 . The local stability of the positive equilibrium and the existence of Hopf bifurcation are investigated. By employing normal form and center manifold theory, the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are determined. Finally, some numerical simulations are also included to illustrate the theoretical analysis. 2. Local Stability and the Existence of Hopf Bifurcation In this section, we study the local stability of each of feasible equilibria and the existence of Hopf bifurcation at the positive equilibrium. Obviously, system 1.4 has a unique boundary equilibrium E1 1, 0 and a unique positive equilibrium E∗ x∗, y∗ , where x∗ −[ a1 − b β 1 − c1 δ] √[ a1 − b β 1 − c1 δ]2 4a1β(bβ c1δ) 2 ( bβ c1δ ) , y∗ δ β x∗. 2.1 The Jacobian matrix of system 1.4 at E1 takes the form J E1 ⎛ ⎜⎝−e − 1 a1 b 0 δ ⎞ ⎟⎠. 2.2 The characteristic equation of system 1.4 at E1 is of the form λ − δ ( λ e−λτ1 ) 0. 2.3 Clearly, the boundary equilibrium E1 1, 0 is unable. 4 Abstract and Applied Analysis Next, we discuss the existence of Hopf bifurcation at the positive equilibrium E x∗, y∗ . Let x t z1 t x∗, y t z2 t y∗, and still denote z1 t and z2 t by x t and y t , respectively, then system 1.4 becomes dx dt a11x t a12y t b11x t − τ1 ∑ i j k≥2 f ijk 1 x yx t − τ1 , dy dt c21x t − τ2 c22y t − τ2 ∑ i j k≥2 f ijk 2 y x t − τ2 y t − τ2 , 2.4


Introduction
Predator-prey dynamics has long been and will continue to be of interest to both applied mathematicians and ecologists due to its universal existence and importance 1, 2 . Many population models investigating the dynamic relationship between predators and their preys have been proposed and studied. For example, Lotka-Volterra model 3-5 , Leslie-Gower model 6-10 , and Holling-Tanner model 11-16 . Among these widely used models, Holling-Tanner model plays a special role in view of the interesting dynamics it possesses. Holling-Tanner model for predator-prey interaction is governed by the following nonlinear coupled ordinary differential equations: Abstract and Applied Analysis where X and Y denote the population densities of prey species and predator species at time T , respectively. The first equation in system 1.1 shows that the prey grows logistically with the carrying capacity K and the intrinsic growth rate r in the absence of the predator. And the growth of the prey is hampered by the predator at a rate proportional to the functional response mX/ a X in the presence of the predator. The second equation shows that the predator consumes the prey according to the functional response mX/ a X and grows logistically with the intrinsic growth rate s and carrying capacity X/h proportional to the number of the prey. The parameter m denotes the maximal predator per capita consumption rate. a is a saturation value; it corresponds to the number of prey necessary to achieve one half the maximum rate m. The parameter h denotes the number of prey required to support one predator at equilibrium when y equals X/h. Recently, there has been considerable interest in predator-prey systems with the Beddington-DeAngelis functional response. And it has been shown that the predator-prey systems with the Beddington-DeAngelis functional response have rich but biologically reasonable dynamics. For more details about this functional response one can refer to 17-21 . Zhang 16 , Lu and Liu 22 considered the following modified Holling-Tanner delayed predator-prey system: where τ is incorporated in the negative feedback of the predator density. αXY/ a bX cY is the Beddington-DeAngelis functional response. The parameters α, a, b, and c are assumed to be positive. α is the maximum value at which per capita reduction rate of the prey can attain. a measures the extent to which environment provides protection to the prey. b describes the effect of handling time on the feeding rate. c describes the magnitude of interference among predators. Zhang 16 investigated the local Hopf bifurcation of system 1.2 . Lu and Liu 22 proved that system 1.2 is permanent under some conditions and obtained the sufficient conditions of local and global stability of system 1.2 . Since both the species are growing logistically, it is reasonable to assume delay in prey species as well. Based on this consideration, we incorporate the negative feedback of the prey density into system 1.2 and obtain the following system: where T 1 and T 2 are the feedback time delays of the prey density and the predator density respectively. Let X Kx, Y rK/α y, t rT, τ 1 rT 1 , τ 2 rT 2 , system 1.3 can be Abstract and Applied Analysis 3 transformed into the following nondimensional form: where a 1 a/K, c 1 cr/α, δ s/r, β sh/α are the non-dimensional parameters and they are positive.
The main purpose of this paper is to consider the effect of multiple delays on system 1.4 . The local stability of the positive equilibrium and the existence of Hopf bifurcation are investigated. By employing normal form and center manifold theory, the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are determined. Finally, some numerical simulations are also included to illustrate the theoretical analysis.

Local Stability and the Existence of Hopf Bifurcation
In this section, we study the local stability of each of feasible equilibria and the existence of Hopf bifurcation at the positive equilibrium. Obviously, system 1.4 has a unique boundary equilibrium E 1 1, 0 and a unique positive equilibrium E * x * , y * , where

2.1
The Jacobian matrix of system 1.4 at E 1 takes the form The characteristic equation of system 1.4 at E 1 is of the form λ − δ λ e −λτ 1 0.

Abstract and Applied Analysis
Next, we discuss the existence of Hopf bifurcation at the positive equilibrium E x * , y * . Let x t z 1 t x * , y t z 2 t y * , and still denote z 1 t and z 2 t by x t and y t , respectively, then system 1.4 becomes

2.5
Then we can obtain the linearized system of system 1.4

2.6
The characteristic equation of system 2.6 is  It is easy to verify that Therefore, if H 1 : A B D > 0, the roots of 2.8 must have negative real parts. Then, we know that the positive equilibrium E * x * , y * of system 1.4 is locally stable in the absence of delay, if H 1 holds.
The associated characteristic equation of the system is Multiplying e λτ on both sides of 2.11 , we can obtain Now, for τ > 0, if λ iω ω > 0 be a root of 2.13 . Then, we have

2.14
It follows from 2.14 that

2.15
Then we have where

6
Abstract and Applied Analysis Let v ω 2 , then 2.16 becomes v 4 e 3 v 3 e 2 v 2 e 1 v e 0 0.

2.18
Next, we give the following assumption. H 2 : 2.18 has at least one positive real root.
Suppose that H 2 holds. Without loss of generality, we assume that 2.18 has four real positive roots, which are defined by v 1 , v 2 , v 3 , and v 4 , respectively. Then 2.16 has four positive roots ω k √ v k , k 1, 2, 3, 4. Therefore, Then we can know that ±iω k are a pair of purely imaginary roots of 2.11 with τ τ j k . Define Let λ τ α τ iω τ be the root of 2.11 near τ τ 0 which satisfies α τ 0 0, ω τ 0 ω 0 . Taking the derivative of λ with respect to τ in 2.13 , we obtain

2.25
Abstract and Applied Analysis 7 Noting that

2.26
Therefore, we make the following assumption in order to give the main results: The associated characteristic equation of the system is We consider 2.27 with τ 2 in its stable interval, regarding τ 1 as a parameter. Without loss of generality, we consider system 1.4 under the case considered in 16 , and τ 2 ∈ 0, τ 20 . τ 20 is defined as in 16 and can be obtained by

2.30
Let λ iω ω > 0 be a root of 2.27 . Then we obtain

2.31
It follows from 2.31 that With

2.33
Then we have
To verify the transversality condition of Hopf bifurcation, we take the derivative of λ with respect to τ 1 in 2.27 , we can obtain

2.39
Abstract and Applied Analysis 9 where

2.40
Next, we make the following assumption: Thus, by the discussion above and by the general Hopf bifurcation theorem for FDEs in Hale 25 , we have the following results.

Direction and Stability of Bifurcated Periodic Solutions
In this section, we shall investigate the direction of the Hopf bifurcation and the stability of bifurcating periodic solution of system 1.4 w.r. to τ 1 for τ 2 ∈ 0, τ 20 , and τ 20 is defined by 2.29 . The idea employed here is the normal form and center manifold theory described in Hassard et al. 26 . Throughout this section, it is considered that system 1.4 undergoes the Hopf bifurcation at τ 1 τ 1 * , τ 2 ∈ 0, τ 20 at E * x * , y * . Let τ 1 τ 1 * μ, μ ∈ R so that the Hopf bifurcation occurs at μ 0. Without loss of generality, we assume that τ 2 * < τ 1 * , where τ 2 * ∈ 0, τ 20 .
Let u 1 t x t − x * , u 2 t y t − y * , and rescaling the time delay t → t/τ 1 , Then system 1.4 can be transformed into an FDE in C C −1, 0 , R 2 as: where u t u 1 t , u 2 t T ∈ R 2 and L μ : C → R 2 , F : R × C → R 2 are given, respectively, by

10
Abstract and Applied Analysis with A a 11 a 12 0 0 ,

3.7
Then system 3.1 can be transformed into the following operator equatioṅ u t A μ u t R μ u t , 3.8 where u t u t θ u 1 t θ , u 2 t θ . For ϕ ∈ C 1 0, 1 , R 2 * , where R 2 * is the 2-dimensional space of row vectors, we further define the adjoint operator A * of A 0 : and a bilinear inner product: where η θ η θ, 0 .

3.11
Then q * , q 1, q * , q 0. In the remainder of this section, Following the algorithms given in 26 and using similar computation process to that in 16 , we can get that the coefficients which will be used to determine the important qualities of the bifurcating periodic solutions, with W 20 θ ig 20 q 0 τ 1 * ω * e iτ 1 * ω * θ ig 02 q 0 3τ 1 * ω * e −iτ 1 * ω * θ E 20 e 2iτ 1 * ω * θ ,

3.13
The period of the bifurcating periodic solutions is determined by the sign of T 2 : if T 2 > 0 T 2 < 0 , the period of the bifurcating periodic solutions increases (decreases).

Numerical Example
In order to support the analytic results obtained above, we give some numerical simulations in this section. We only study the most important steady state, namely, the positive steady state. We consider the following system by taking the same coefficients as in 16 : where we get the positive equilibrium E * 0.6328, 1.1074 . For system 4.1 , we can get that A B D 3.9017 > 0, namely, the condition H 1 holds. For τ 1 τ 2 τ / 0. By a simple computation, we obtain that 2.18 has two positive roots: v 1 12.0256, v 2 0.3915. Thus, we know that the condition H 2 holds. Further, we get ω 0 3.4678, τ 0 0.4292. In addition, we have Λ 1 × Λ 3 Λ 2 × Λ 4 94.4826 > 0. Therefore, the condition H 3 is satisfied. Hence, from Theorem 2.1, we conclude that the positive equilibrium E * 0.6328, 1.1074 is asymptotically stable when τ ∈ 0, τ 0 . The corresponding waveform and the phase plot are illustrated by Figure 1. When the delay τ passes through the critical value τ 0 the positive equilibrium E * 0.6328, 1.1074 will lose its stability and a Hopf bifurcation occurs, and a family of periodic solution bifurcates from the positive equilibrium E * 0.6328, 1.1074 . This property is illustrated by the numerical simulation in Figure 2.

Conclusion
In this paper, a delayed predator-prey system with Beddington-DeAngelis functional response has been investigated. The bifurcation of a predator-prey system with single delay has been studied by many researchers 16, 27-30 . However, there are few papers considering the bifurcation of a predator-prey system with multiple delays see 31-33 . Compared with the system considered in 16 , the system in this paper accounts for not only the feedback delay of the prey density but also the feedback delay of the predator. The sufficient conditions for the stability of the positive equilibrium and the existence of periodic solutions via Hopf bifurcation at the positive equilibrium are obtained when τ 1 τ 2 and τ 1 / τ 2 with τ 2 ∈ 0, τ 20 . Special attention is paid to the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions. By computation, we find that the feedback delay of the predator is marked because the critical value of τ 2 is much smaller when we only consider it. The feedback delay of the prey is unremarkable because the critical value of τ 1 is much bigger when we consider it with τ 2 in its stable interval. Furthermore, Zhang 16 has obtained that the two species in system 1.4 with only the feedback delay of the predator could coexist. However, we get that the two species could also coexist with some available feedback delays of the prey and the predator. This is valuable from the view of ecology. Unfortunately, the existence of the periodic solutions remain valid only in a small neighborhood of the critical value. It is definitely an interesting work to investigate whether these nonconstant periodic solutions which are obtained through local Hopf bifurcation can still exist for large values of the corresponding parameter time delay. The global continuation of the local Hopf bifurcation is left as the future work.