Dynamical Analysis of a Delayed Reaction-Diffusion Predator-Prey System

and Applied Analysis 3 see 1, 4–18 . In this paper, we mainly focus on the effects of both spatial diffusion and time delay factors on system 1.6 with Neumann boundary conditions as follows: ∂u ∂t u ( α − βu) − cuv mv 1 d1Δu, x ∈ Ω, t > 0, ∂v ∂t v ( −γ bu t − τ mv t − τ 1 ) d2Δv, x ∈ Ω, t > 0, ∂u ∂ν ∂v ∂ν 0, x ∈ ∂Ω, t > 0, u x, 0 u0 x ≥ 0, v x, 0 v0 x ≥ 0, x ∈ Ω, 1.7 where Ω is a bounded open domain in R with a smooth boundary ∂Ω, and Δ ∂2/∂x2 denotes the Laplacian operator in R. d1 > 0 and d2 > 0 denote the diffusion coefficients of the prey u and predator v, respectively. ν is the outward unit normal vector on ∂Ω. τ > 0 can be regarded as the gestation of the predator. System 1.7 includes not only the dispersal processes, but also some of the past states of the system. Throughout this paper, we restrict ourselves to the one-dimensional spatial domain Ω 0, π for the sake of convenience. The remaining parts of the paper are structured in the following way. In Section 2, we analyze the distribution of the roots of the characteristic equation and give various conditions on the stability of a unique positive equilibrium and the existence of Hopf bifurcation with time delay. In Section 3, applying the normal form theory 19, 20 and the center manifold reduction of partial functional differential equations 21 , we derive the explicit algorithm in order to determine the direction of the Hopf bifurcation, the stability, and other properties on bifurcating periodic solutions. Finally, a brief discussion is given. 2. Stability of Positive Equilibrium and Existence of Hopf Bifurcation In this section, by analyzing the associated characteristic equation of system 1.7 at the positive equilibrium, we investigate the stability of the positive equilibria of system 1.7 . It is straightforward to see that system 1.7 has the following two boundary equilibria: i E1 0, 0 total extinct which is saddle point, hence it is unstable; ii E2 α/β, 0 extinct of the predator which is saddle point if bα > βγ , or stable if bα < βγ . To find the positive equilibrium, we set u ( α − βu) − cuv mv 1 0, v ( −γ bu mv 1 ) 0, 2.1 which yields mbβu2 c −mα bu − cγ 0. 2.2 4 Abstract and Applied Analysis Obviously, system 1.7 has a unique positive equilibrium E∗ u∗, v∗ with bα > βγ , where u∗ mbα − bc √ 4mbcβγ b2 mα − c 2 2mbβ , v∗ bu∗ − γ mγ . 2.3 Set u u − u∗, v v − v∗ and drop the bars for simplicity of notations, then system 1.7 can be transformed into the following equivalent system: ∂u ∂t u u∗ ( α − β u u∗ ) − c u u∗ v m v v∗ 1 d1Δu, x ∈ Ω, t > 0, ∂v ∂t v v∗ ( −γ b u t − τ u ∗ m v t − τ v∗ 1 ) d2Δv, x ∈ Ω, t > 0, ∂u ∂ν ∂v ∂ν 0, x ∈ ∂Ω, t > 0, u x, 0 u0 x − u∗ ≥ 0, v x, 0 v0 x − v∗ ≥ 0, x ∈ Ω. 2.4 Assume that u0 x , v0 x ∈ C −τ, 0 ;X and X is defined by X { u, v : u, v ∈W2,2 Ω : ∂u ∂ν ∂v ∂ν 0, x ∈ ∂Ω } 2.5 with the inner product 〈·, ·〉. Denote u t , v t u t, x , v t, x andU t u t , v t T . Then system 2.4 can be rewritten as an abstract differential equation in the phase space C −τ, 0 ;X as follows: ∂U t ∂t dΔU t L Ut F Ut , 2.6 where d diag d1, d2 , Ut θ U t θ , −τ ≤ θ ≤ 0, and L : C −τ, 0 ;X → X, F : C −τ, 0 ;X → X are given by L ( φ ) ⎛ ⎜⎜⎝ −βu∗φ1 0 − cu ∗ mv∗ 1 2 φ2 0 bv∗ mv∗ 1 φ1 −τ − mbu ∗v∗ mv∗ 1 2 φ2 −τ ⎞ ⎟⎟⎠, 2.7 F ( φ ) ⎛ ⎜⎜⎜⎜⎝ −βφ2 1 0 − c mv∗ 1 2 φ1 0 φ2 0 cmu∗ cm 1 3 φ2 2 0 ( φ2 0 mv∗ mv∗ 1 φ2 −τ )( b mv∗ 1 φ1 −τ − bmu ∗ mv∗ 1 2 φ2 −τ ) ⎞ ⎟⎟⎟⎟⎠ , 2.8 respectively, where φ θ Ut θ , φ φ1, φ2 T ∈ C −τ, 0 ;X . Abstract and Applied Analysis 5 The linearization of 2.6 is given byand Applied Analysis 5 The linearization of 2.6 is given by ∂U t ∂t dΔU t L Ut , 2.9 and its characteristic equation is λy − dΔy − L ( eλ·y ) 0, 2.10 where y ∈ dom Δ and y / 0, dom Δ ⊂ X. It is well known that the eigenvalue problem −Δψ μψ, x ∈ 0, π , ∂ψ ∂x ∣∣∣ x 0 ∂ψ ∂x ∣∣∣ x π 0, 2.11 has eigenvalues 0 μ0 ≤ μ1 ≤ μ2 ≤ · · · ≤ μn ≤ μn 1 ≤ · · · , with the corresponding eigenfunctions ψn x . Substituting y ∞ ∑ n 0 ψn x ( y1n y2n ) 2.12 into characteristic equation 2.10 , we obtain ⎛ ⎜⎜⎝ −βu∗ − d1μn − cu ∗ mv∗ 1 2 bv∗ mv∗ 1 e−λτ − bmu ∗v∗ mv∗ 1 2 e−λτ − d2μn ⎞ ⎟⎟⎠ ( y1n y2n ) λ ( y1n y2n ) . 2.13 Hence, we can conclude that the characteristic equation 2.10 is equivalent to the sequence of the following characteristic equations: λ2 ( An De−λτ ) λ Bn Cne−λτ 0 n 0, 1, 2, . . . , 2.14 where An βu∗ d1 d2 μn, Bn βu∗d2μn d1d2μn, Cn bd1mμnβu ∗v∗ mv∗ 1 2 bmβ u∗ 2v∗ mv∗ 1 2 bcu∗v∗ mv∗ 1 3 , D bmu∗v∗ mv∗ 1 2 . 2.15 6 Abstract and Applied Analysis The stability of the positive equilibrium E∗ u∗, v∗ can be determined by the distribution of the roots of 2.14 n 0, 1, 2, . . . , that is, the equilibrium E∗ u∗, v∗ is locally asymptotically stable if all the roots of 2.14 n 0, 1, 2, . . . have negative real parts. Note that λ 0 is not a root of 2.14 for any n 0, 1, 2, . . .. Next, we analyze the behaviour of system 1.7 in two situations: with/without delay effect. 2.1. Case τ 0 Equation 2.14 with τ 0 is equivalent to the following quadratic equation: λ2 An D λ Bn Cn 0, 2.16 where An, Bn, Cn, and D are defined as 2.15 . Let λ1 and λ2 be two roots of 2.16 , then for any n 0, 1, 2, . . ., we have λ1 λ2 − An D < 0, λ1λ2 Bn Cn > 0. 2.17 Then we can get the following theorem. Theorem 2.1. If bα > βγ holds, the positive equilibrium E∗ u∗, v∗ of system 1.7 with τ 0 is asymptotically stable. In the following, we prove that E∗ u∗, v∗ of system 1.7 is globally stable with τ 0. Theorem 2.2. If bα > βγ holds, the positive equilibrium E∗ u∗, v∗ of system 1.7 with τ 0 is globally asymptotically stable. Proof. To prove our statement, we need to construct a Lyapunov function. To this end, we define


Introduction
The study on the dynamics of predator-prey systems is one of the dominant subjects in ecology and mathematical ecology due to its universal existence and importance 1 .A prototypical predator-prey interaction model is of the following form: where u t and v t are the densities of the prey and predator at time t > 0, respectively.
Abstract and Applied Analysis Furthermore, the function a u is growth rate of the prey in the absence of predation, which is given by a u αu min 1, If ε 0, this reduces to the traditional logistic form a u αu 1 − u/K , see 2 and the references therein.Here, the parameter α stands for the specific growth rate of the prey u, and K for carrying capacity of the prey in the absence of predators.
The product f u g v gives the rate at which prey is consumed, and f u g v /v is termed as the functional response 3 .In particular, these functions can be defined by where c denotes the capture rate, and m the half capturing saturation constant.
The proportionality constant σ is the rate of prey consumption.And the function z v is given by z v γv lv 2 , γ > 0, l ≥ 0, 1.4 where γ denotes the natural death rate of the predators, and l > 0 can be used to model predator in traspecific competition that is not direct competition for food, such as some type of territoriality, see 2 .In this paper, we discuss the case of l 0, which is used in a much more traditional case.Based on the above discussions, we can obtain the following model:
In recent years, the models involving time delay and spatial diffusion have been extensively studied by many authors and many interesting results have been obtained, including the stability, the existence of Hopf bifurcation, and direction of bifurcating periodic solutions, see 1, 4-18 .In this paper, we mainly focus on the effects of both spatial diffusion and time delay factors on system 1.6 with Neumann boundary conditions as follows: where Ω is a bounded open domain in R with a smooth boundary ∂Ω, and Δ ∂ 2 /∂x 2 denotes the Laplacian operator in R. d 1 > 0 and d 2 > 0 denote the diffusion coefficients of the prey u and predator v, respectively.ν is the outward unit normal vector on ∂Ω.τ > 0 can be regarded as the gestation of the predator.System 1.7 includes not only the dispersal processes, but also some of the past states of the system.
Throughout this paper, we restrict ourselves to the one-dimensional spatial domain Ω 0, π for the sake of convenience.
The remaining parts of the paper are structured in the following way.In Section 2, we analyze the distribution of the roots of the characteristic equation and give various conditions on the stability of a unique positive equilibrium and the existence of Hopf bifurcation with time delay.In Section 3, applying the normal form theory 19, 20 and the center manifold reduction of partial functional differential equations 21 , we derive the explicit algorithm in order to determine the direction of the Hopf bifurcation, the stability, and other properties on bifurcating periodic solutions.Finally, a brief discussion is given.

Stability of Positive Equilibrium and Existence of Hopf Bifurcation
In this section, by analyzing the associated characteristic equation of system 1.7 at the positive equilibrium, we investigate the stability of the positive equilibria of system 1.7 .
It is straightforward to see that system 1.7 has the following two boundary equilibria: i E 1 0, 0 total extinct which is saddle point, hence it is unstable; ii E 2 α/β, 0 extinct of the predator which is saddle point if bα > βγ, or stable if bα < βγ.
To find the positive equilibrium, we set Obviously, system 1.7 has a unique positive equilibrium E * u * , v * with bα > βγ, where and drop the bars for simplicity of notations, then system 1.7 can be transformed into the following equivalent system:

2.4
Assume that u 0 x , v 0 x ∈ C −τ, 0 ; X and X is defined by   Hence, we can conclude that the characteristic equation 2.10 is equivalent to the sequence of the following characteristic equations: where

2.15
The stability of the positive equilibrium E * u * , v * can be determined by the distribution of the roots of 2.14 n 0, 1, 2, . . ., that is, the equilibrium E * u * , v * is locally asymptotically stable if all the roots of 2.14 n 0, 1, 2, . . .have negative real parts.Note that λ 0 is not a root of 2.14 for any n 0, 1, 2, . ... Next, we analyze the behaviour of system 1.7 in two situations: with/without delay effect.

Case τ 0
Equation 2.14 with τ 0 is equivalent to the following quadratic equation: where A n , B n , C n , and D are defined as 2.15 .
Let λ 1 and λ 2 be two roots of 2.16 , then for any n 0, 1, 2, . .., we have 2.17 Then we can get the following theorem.
In the following, we prove that E * u * , v * of system 1.7 is globally stable with τ 0.

Theorem 2.2. If bα > βγ holds, the positive equilibrium E *
u * , v * of system 1.7 with τ 0 is globally asymptotically stable.Proof.To prove our statement, we need to construct a Lyapunov function.To this end, we define where

2.19
We claim that V 1 u, v is positive definite.In fact, set we can obtain u, v u * , v * .And the Hessian Matrix at u * , v * is given by Hence The time derivative of V along the solutions of system 1.7 with τ 0, we have
It is enough to see that dV/dt satisfies Lyapunov's asymptotic stability theorem, hence the positive equilibrium E * u * , v * of system 1.7 with τ 0 is globally asymptotically stable.

Case τ / 0
In the following, we prove the stability of the positive equilibrium E * u * , v * of system 1.7 and the existence of Hopf bifurcation at the positive equilibrium Proof.Let λ μ τ iω τ be a root of the characteristic equation 2.14 , then we have where λ, μ, ω are functions of τ.A necessary condition for the stability of E * u * , v * is that the characteristic equation has a purely imaginary solution λ iω.Let μ τ 0 and ω τ / 0, then we can reduce 2.24 to where τ 0 0 is defined as 2.32 , such that, for system 1.7 , the following statements are true.
iii τ τ j 0 n 0, 1, 2, . . .are Hopf bifurcation values of system 1.7 and these Hopf bifurcations are all spatially homogeneous.Proof.Let λ iω τ be a root of the characteristic equation 2.14 .By the same way in Theorem 2.3, then ω satisfies the following equation: 28 has a unique positive root ω 0 satisfying and from 2.25 we obtain
Hence the transversality condition holds and accordingly Hopf bifurcation occurs at τ τ 0 0 , and τ τ 0 j j 0, 1, 2, . . .are Hopf bifurcation values of system 1.7 and these Hopf bifurcations are all spatially homogeneous.This completes the proof.

Direction and Stability of Hopf Bifurcation
In the previous section, we have already obtained that system 1.7 undergoes Hopf bifurcation at the positive equilibrium E * u * , v * when τ crosses through the critical value τ 0 j j 0, 1, 2, . . . .In this section, we will study the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions by employing the normal form method 19, 20 as well as center manifold theorem 21 for partial differential equations with delay.Then we compute the direction and stability of the Hopf bifurcation when τ 0 τ 0 j for fixed j ∈ {0, 1, 2, . ..}.Without loss of generality, we denote the critical value of τ by τ 0 and set τ τ 0 μ, then μ 0 is the Hopf bifurcation value of system 2.6 .Rescaling the time by t → t/τ to normalize the delay, system 2.6 can be written in the following form: where From Section 2, we know that ±iω 0 τ 0 are a pair of simple purely imaginary eigenvalues of the liner system ∂U t ∂t τ 0 dΔU t τ 0 L U t , 3.3 and the following liner functional differential equation: By the Riesz representation theorem, there exists a 2 × 2 matrix function η θ, τ 0 of bounded variation for θ ∈ −1, 0 , such that 3.5 In fact, we can choose where and δ is the Dirac delta function.

3.9
Then A 0 and A * are adjoint operators under the following bilinear form: where η θ η 0, θ .We note that ±iω 0 τ 0 are the eigenvalues of A 0 .Since A 0 and A * are two adjoint operators, ±iω 0 τ 0 are also eigenvalues of A * .We will first try to obtain eigenvector of A 0 and A * corresponding to the eigenvalue iω 0 τ 0 and −iω 0 τ 0 , respectively.
Let q θ 1, ρ T e iω 0 τ 0 θ , θ ∈ −1, 0 be the eigenvector of A 0 corresponding to the eigenvalue iω 0 τ 0 .Then we have A 0 q θ iω 0 τ 0 q θ by the definition of eigenvector.Therefore, from 3.6 , 3.10 , and the definition of A 0 , we can get Abstract and Applied Analysis here, On the other hand, suppose that q * S D 1, r e iω 0 τ 0 S is the eigenvector of A * corresponding to the eigenvalue −iω 0 τ 0 .By the definition of A * , we have and we also assume that q * S , q θ 1.To obtain the value of D, from 3.10 we have

3.15
Thus, we can choose such that q * S , q θ 0 and q * S , q θ 1, that is to say that let φ q θ , q θ , ψ q * S , q * S T , then ψ, φ I, where I is the unit matrix.Then the center subspace of system 3.4 is P span{q θ , q θ }, and the adjoint subspace P * span{q * S , q * S }.Let f 0 f 1 0 , f 2 0 , where Abstract and Applied Analysis 13 by using the notation from 20 , we also define And the center subspace of linear system 3.4 is given by P CN C, where

3.19
and C P CN C ⊕ P S C, where P S C is the stable subspace.Following Wu 20 , we know that the infinitesimal generator A U of linear system 3.4 satisfies As the formulas to be developed for the bifurcation direction and stability are all relative to μ 0 only, we set μ 0 in system 3.4 and can obtain the center manifold with the range in P S C. The flow of system 3.4 in the center manifold can be written as follows:

3.24
We rewrite 3.24 as ż t iω 0 τ 0 z t g z t , z t , 3.25 14 Abstract and Applied Analysis with

3.27
Afterwards, from Taylor formula, we have

3.28
From 3.26 and 3.28 , we have

3.29
Since W 11 θ and W 20 θ for θ ∈ −1, 0 appear in g 21 , we need to compute them.It follows from 3.26 that

3.30
In addition, W z, z satisfies where

3.32
Abstract and Applied Analysis Thus, from 3.24 and 2.18 , we can get

3.33
Note that A U has only two eigenvalues ±iω 0 τ 0 , therefore, 3.33 has unique solution W ij in Q given by

3.40
In what follows, we will seek appropriate E 1 and E 2 in 3.39 and 3.40 .From the definition of A U and 3.33 , we have 0 where η θ η 0, θ , then

3.44
then we deduce

3.49
Finally, we arrive at where

3.51
In a similar manner, we can calculate E 1 and E 2 .Then g 21 can be expressed.Based on the above analysis, it is enough to see that each g ij is determined by the parameters.Thus, we can compute the following values which determine the direction and stability of the following bifurcating periodic orbits:

Conclusions and Remarks
In this paper, under homogeneous Neumann boundary conditions, we have analyzed dynamical behaviors of the diffusion predator-prey system 1.7 with and without delay.The value of this study lies in two aspects.First, it presents the stability of positive equilibrium E * u * , v * of system 1.7 with and without delay, and the existence of Hopf bifurcation, which indicates that the dynamical behaviors become rich and complex with delay.Second, it shows the analysis of stability of Hopf bifurcation, from which one can find that small sufficiently delays cannot change the stability of the positive equilibrium and large delays cannot only destabilize the positive equilibrium but also induce an oscillation near the positive equilibrium.
Next, numerical simulations are performed to illustrate results with respect to the theoretical facts under the special example.When τ and d i i 1, 2 are all equal 0, we demonstrate that the positive equilibrium E * is globally asymptotically stable see, Figure 1 , which means that if the intraspecific competitions of the prey and the predator dominate the inter-specific interaction between the prey and the predator, then both the prey and the predator populations are permanent 14 .
In addition, we consider the dynamics of system 1.7 affecting by both spatial diffusion and time-delay factors with fixed parameters d 1 0.01 and d 2 0.1.In this case, τ 0 0 1.0289, Re C 1 0 −0.6689 < 0. If τ < τ 0 0 , the positive equilibrium E * is remain stability see Figure 2 , which indicates that the predators and preys can coexist in stable conditions.
While if τ > τ 0 0 , the positive equilibrium E * losses its stability and Hopf bifurcation occurs, which means that a family of stable periodic solutions bifurcate from E * since μ 2 0.5515 > 0, β 2 −0.1338 < 0 see Figure 3 .The numerical result indicates that the predator coexists with the prey with oscillatory behaviors.
In the present paper, we incorporate time delay into biological system due to the gestation of the predator, which causes stable equilibrium to become unstable and causes the populations to oscillate via Hopf bifurcation.That is to say that the effect of delay for the population dynamics is tremendous.From a biological perspective, the time delay of species may be related to the gestation of the predator, mature stage from juvenile to adult, the interaction time between prey and predator and others.In these cases, the methods and results in the present paper may provide a favorable value on controlling ecological population.It would be more accurate to describe the growth rate of population.
these imply that 2.26 has no positive roots, that is, all roots of 2.14 have negative real parts.
n < 0, then there exists a sequence