The Effect of Delay on a Diffusive Predator-Prey System with The Effect of Delay on a Diffusive Predator-Prey System with Holling Type-Ii Predator Functional Response Holling Type-Ii Predator Functional Response

. A delayed diﬀusive predator-prey system with Holling type-II predator functional response subject to Neumann boundary conditions is considered here. The stability/instability of nonnegative equilibria and associated Hopf bifurcation are investigated by analyzing the characteristic equations. By the theory of normal form and center manifold, an explicit formula for determining the stability and direction of periodic solution bifurcating from Hopf bifurcation is derived.


Introduction.
A diffusive predator-prey system with Holling type-II functional response [9] is a prototypical reaction-diffusion model describing a pair of species with consumer-resource interaction [15,26].The equation is in the form Here Ω is a bounded domain in R N , N ≥ 1, with a smooth boundary ∂Ω; ν is the outward unit normal vector on ∂Ω; u(x, t) and v(x, t) are the densities of the prey and predator at time t > 0 and a spatial position x ∈ Ω respectively; d 1 , d 2 > 0 are the diffusion coefficients of the species; k > 0 is the carrying capacity of prey; r > 0 is the mortality rate of predator; m > 0 is the measure of the interaction strength between two species.This reaction-diffusion model (1.1) has been widely used in ecological and biological applications (see e.g.[14,15,17]).More biological explanation of this predator-prey system can be found in [15,26].
System (1.1) has been analyzed or simulated by some researchers.Medvinsky et.al.[15] showed that (1.1) possesses a rich structure of spatiotemporal dynamics through extensive numerical simulations.In [26], Yi, Wei and Shi investigated the bifurcations of non-constant equilibria and periodic orbits of (1.1) with parameter β, and they obtained that in some situations spatially nonhomogeneous periodic orbits and nonhomogeneous steady state solutions exist in (1.1).Peng and Shi [16] gave some further results on the steady state solutions, that is when m is sufficient large, system (1.1) had no nonconstant positive steady state solutions.System (1.1) was also considered in [12].
Time-delay in some interactions of an evolution system may have significant impact on the underlying dynamics.Hence reaction-diffusion systems with time delays have been proposed as models for the population ecology and biology in recent years (see [3,6,10,11,19,20,21,24,25]).There are many results of various delayed diffusive predator-prey systems, regarding bifurcations at equilibria, and local/global stability of the constant equilibrium (see e.g.[3,10,19,21,24,25] and references therein).As for predator-prey systems, the delay effect on the growth rate per capita of predator or prey is often considered (see [10,23,27]).In this paper we consider the delay effect on the growth rate per capita of predator, then system (1.1) becomes x ∈ ∂Ω, t > 0, u(x, t) = u 0 (x, t) ≥ 0, v(x, t) = v 0 (x, t) ≥ 0, x ∈ Ω, t ∈ [−τ, 0], (1.2) where τ ≥ 0 represents the delay effect on growth rate per capita of predator.
Our result shows that a stable oscillatory pattern in (1.2) can be induced by either a larger delay τ or a smaller β, hence a combined impact of the delay τ , the interaction strength m and the predator mortality rate r can destabilize the positive equilibrium state so the system (1.2) exhibits oscillatory behavior.Such delay-induced Hopf bifurcations occur in (1.2) for all the β values for which (β, v β ) is locally asymptotically stable for system (1.1).This is different from a similar diffusive Leslie-Gower predator-prey system [1] studied by the authors recently, in which the global stability of the constant equilibrium persists for all delay values τ > 0. For system (1.2), it is not known whether or not (β, v β ) can be globally asymptotically stable when τ > 0, although the global stability of (β, v β ) when τ = 0 can be established by a Lyapunov functional for k − 1 < β < k (see [26]).
The rest of this paper is organized as follows.In Section 2, we analyze the stability/instability of nonnegative equilibria of system (1.2) through the study of associated characteristic equations and show the occurrence of Hopf bifurcation at the positive equilibrium (β, v β ).We also give a detailed description of the distribution of the characteristic values of the associated characteristic equations of (β, v β ).In Section 3, we investigate the stability and direction of bifurcating periodic orbits by using normal form [8,22] and the center manifold theorem due to Lin, So and Wu [13].Some numerical simulations are also presented in Section 3. Throughout the paper, we denote by N 0 the set of all the nonnegative integers and R + the set of all the positive real numbers.
2. Stability analysis of equilibria and bifurcation.In this section, we consider system (1.2) on the spatial domain Ω = (0, lπ), with l ∈ R + , In this section, we assume that m > (1+k)r k , then system (2.1) has three nonnegative equilibria (0, 0), (k, 0) and (β, v β ) (defined in (1.3)).It is easy to show that the equilibria (0, 0) and (k, 0) are unstable with respect to the ODE dynamics, hence they are also unstable with respect to (2.1).In the remaining part of this section, we shall analyze the stability/instability of positive constant equilibrium (β, v β ).
Since we use β as the bifurcation parameter, we substitute m = r β + r into system (2.1).Note that if m > (1+k)r k , then 0 < β < k.Transforming the positive equilibrium (β, v β ) to the origin via the translation û = u − β, v = v − v β and dropping the hats for simplicity of notation, then we have In the abstract space C([−τ, 0], X), system (2.2) can be regarded as the following abstract functional differential equation where where Then the linearization of system (2.2) near (β, v β ) is (2.4) From Wu [22], we obtain that the characteristic equation for the linearized system (2.4) is λy − d∆y − L(e λ• y) = 0, y ∈ dom(d∆), y = 0. (2.5) It is well known that the eigenvalue problem into characteristic equation (2.5), we obtain Therefore the characteristic equation (2.5) is equivalent to where The stability/instability of positive equilibrium (β, v β ) can be determined by the distribution of the roots of Eqs.(2.5 n ), n = 0, 1, 2, • • • , that is, the equilibrium (β, v β ) is locally asymptotically stable if all the roots of Eqs.(2.5 n ), n = 0, 1, 2, • • • have negative real parts.From the result of Ruan and Wei [18,Corollary 2.4], the sum of the multiplicities of the roots of Eq. (2.5 n ) in the open right half plane changes only if a root appears on or crosses the imaginary axis.
In the following, fixing parameters d 1 , d 2 , k, r, l in (2.2), we use τ as the main bifurcation parameter while the value of β may vary in different places.It can be verified that if 0 < k ≤ 1, then 0 is not a root of Eqs.(2.5 If ±iσ(σ > 0) is a pair of roots of Eq. (2.5 n ), then we have which leads to where (2.8) Since lim n→∞ (B 2 n − C 2 ) = +∞ for any 0 < β < k, then there exists a minimal N 0 (β) ≥ 0 such that Eq. (2.7 n ) has no positive root for n > N 0 (β) and Eq.(2.7 n ) has one positive root at most for 0 ≤ n ≤ N 0 (β).
For 0 ≤ n ≤ N 0 (β), if Eq. (2.7 n ) has a positive root σ n satisfying then Eq. (2.5 n ) has a pair of imaginary roots ±iσ n when where τ 0 n satisfies (2.11) From dependence of A n , B n and C on β, τ j n = τ j n (β) can be regarded as a function of β.Let N = max 0<β<k N 0 (β).Here we give some properties of curves τ = τ j n (β), for 0 Proof.From Eq. (2.11), we know that τ 0 n (β) is the minimal positive τ -value for Eq.(2.5 n ) possessing a couple of purely imaginary roots.Eq. (2.5 n ) has a couple of purely imaginary roots if and only if Eq. (2.7 n ) has a positive root.Since A 2 n − 2B n is always nonnegative, Eq. (2.5 n ) has a couple of purely imaginary roots if and only if B 2 n − C 2 < 0. Then we obtain that the domain of Since the domain of τ j n (β) is same as that of τ 0 n (β) when j ≥ 1, then the domain of Then we have the following transversality condition.
To prove that τ 0 p (β) is strictly increasing in β, we observe that p is strictly decreasing in β, is strictly decreasing in β,

Bp
C is strictly increasing in β, and A 2 p − 2B p is strictly increasing in β, we can obtain that σ p and σ 2 p C are strictly decreasing in β.So from (2.13), τ j p (β) is a strictly increasing function when β ∈ (0, β p ) for 0 ≤ p ≤ n, j ∈ N 0 .
For k > 1 and 0 < β ≤ (k−1)/2, the constant equilibrium (β, v β ) is unstable even when τ = 0.In [26], Yi, Wei and Shi studied the Hopf bifurcations and equilibrium bifurcations of system (2.2) when τ = 0 and using β, (0 < β ≤ (k − 1)/2) as bifurcation parameter.In this parameter range, the curves τ j n (β) are not always defined, and they may have vertical asymptotes where the curves blow up to infinity.But we point out that the values β so that τ 0 n (β) = 0 are coincident to the Hopf bifurcation points found in [26], since when τ = 0, the characteristic equation λ 2 + A n λ + B n + Ce −λτ = 0 becomes λ 2 + A n λ + B n + C = 0, which is the same as the characteristic equation at (β, v β ) without delay effect (see Eq. (2.39) of [26]).Hence the Hopf bifurcations described in [26] also exist for the system with delay effect (2.2).Similar to Theorem 2.6, this shows that the Hopf bifurcations are jointly driven by two parameters τ (delay) and β (internal system parameter).We will not describe the Hopf bifurcations for this parameter range in details, but only state the following stability results from our discussions: Proposition 2.8.Suppose d 1 , d 2 , k, l, r are all positive constants, and k, β satisfy k > 1 and 0 < β < (k − 1)/2.Then for any τ ≥ 0, the positive equilibrium (β, v β ) is unstable with at least two roots of Eqs.(2.5 n ), (n ≥ 0) with positive real parts.Moreover whenever τ increases through one of curves τ j n (β), 0 ≤ n ≤ N, j ∈ N 0 , the sum of the multiplicities of the roots of Eqs.(2.5 n ) with positive real parts will increase by two.
Setting τ = τ 0 + µ, then µ = 0 is the Hopf bifurcation value of system (2.3).Re-scaling the time by t → t τ to normalize the delay, system (2.3) can be written in the form where From Section 2, we know that ±iσ 0 τ 0 is a pair of simple purely imaginary eigenvalues of the linear system and the linear functional differential equation By Riesz representation theorem, there exists a 2 × 2 matrix η(θ, µ), (θ ∈ [−1, 0]), whose elements are of bounded variation functions such that In fact, we have where Then A(0) and A * are adjoint operators under the bilinear form Hassard et.al.[8]).
Then the center subspace of system (3.3) is P = span{q(θ), q(θ)}, and the adjoint subspace is where By using the notation from Wu [22], we also define c Then the center subspace of linear system (3.2) is given by P CN C, where and C = P CN C ⊕ P S C, where P s C is the stable subspace.
From Wu [22], we know that the infinitesimal generator A U of linear system (3.2) 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 (3.1) and obtain a center manifold with the range in P S C. The flow of system (3.1) in the center manifold can be written as follows: where We rewrite (3.6) as ż(t) = iσ 0 τ 0 z(t) + g(z, z) and then from Taylor formula we have where O(4) = O(||(u, v)|| 4 ), and G(φ, 0) = τ 0 (G 1 , G 2 ) T , where (3.10) From (3.8) and (3.10), we have So in order to compute g 21 , we need to compute W 20 (θ) and W 11 (θ).Since W (z(t), (z(t)) satisfies then by using the chain rule therefore from (3.12) From (3.12) with θ = 0, the definition of A U and where Then g 21 can be determined.
A general conclusion regarding the direction and stability of bifurcating periodic orbits cannot be stated due to the complicated nature of computation.But for given parameter values, a calculation can be carried out by using the formulas above.In the following, we present some numerical simulations to illustrate the analytic results.We use a set of parameters as in Section 2: and we also choose m = 2, r = 1.In this case β = 1, and we examine the effect of delay on the dynamics of the system (2.1).We can compute that τ 0 0 = 3.6276, σ 0 = 0.2887, and Re(C 1 (0)) < 0. From Theorem 2.6 and Theorem 3.1 we obtain that for τ ∈ (0, 3.6276), the positive equilibrium (1, 1/3) is stable.When τ is in a small right-side neighborhood of 3.6276, system (2.1) has stable periodic solutions which bifurcate from the constant equilibrium (1, 1/3).The results are illustrated in Fig. 4 and Fig. 5 in which the left panel shows the graph of u(x, t) and the right panel shows the one of v(x, t).
Finally we indicate that how the procedure described in this section can be modified for the Hopf bifurcation in more general situation when τ = τ j n ∈ P(β).
Since 2σ n is not the eigenvalue of characteristic equation (2.5), then E 1 can be uniquely determined by where d∆ and L is defined in (2.3).Similarly E 2 is uniquely determined by Then we can easily obtain E 1 has the expression as e 1 +e 2 cos 2nx l where e 1 , e 2 ∈ C 2 , and E 2 has the expression as f 1 + f 2 cos 2nx l where f 1 , f 2 ∈ C 2 , and we omit the detailed computation.With these modifications, we are able to compute the direction of Hopf bifurcation in the more general situation where the diffusion term also plays a role.