HOPF BIFURCATION ANALYSIS IN A DIFFUSIVE PREDATOR-PREY SYSTEM WITH DELAY AND SURPLUS KILLING EFFECT

. A diﬀusive predator-prey system with a delay and surplus killing eﬀect subject to Neumann boundary conditions is considered. When the de- lay is zero, the prior estimate of positive solutions and global stability of the constant positive steady state are obtained in details. When the delay is not zero, the stability of the positive equilibrium and existence of Hopf bifurcation are established by analyzing the distribution of eigenvalues. Furthermore, an algorithm for determining the direction of Hopf bifurcation and stability of bifurcating periodic solutions is derived by using the theory of normal form and center manifold. Finally, some numerical simulations are presented to illustrate the analytical results obtained.


1.
Introduction. In ecological systems, the interactions between different species can generate rich phenomena. Many models are derived to illustrate the predatorprey system from the view of both mathematics and biology [2,22,28,32]. Meanwhile, it is well known that the spatial structure may further affect the population dynamics of the species [7,8,15]. The spatially homogeneous reaction-diffusion predator-prey model with classical Lotka-Volterra interaction and no flux boundary conditions has been studied by many scholars, and the unique positive steady state solution is globally asymptotically stable in that case [21]. Our work is based on the important contribution of Yi, Wei and Shi [35] in the bifurcation analysis from the constant coexistence equilibrium solution of the following Rosenzweig-MacArthur model with Holling type-II functional response [14,27]: x ∈ Ω, t > 0, which can be demonstrated by different functional responses in the equations.
Here r 1 , r 2 is the intrinsic growth rate of u, v, respectively; K 1 , K 2 is the corresponding carrying capacity of u, v; m 1 is the predation rate, and m 2 is the consumption rate with m 2 = αm 1 where α is a positive constant; the Holling I type functional response m 1 u and the Holling II type functional response m2u γ+u show the " surplus killing" effect: the predators keep hunting whenever they find a potential targets, but the conversion is limited since most of the prey they have captured are abandoned.
u(x, 0) = u 0 (x) ≥ 0( ≡ 0), v(x, 0) = v 0 (x) ≥ 0( ≡ 0), x ∈ Ω. (2) Considering that the biomass that predator consumed cannot convert into the new production for an instant, we add time delay into the functional response of the second equation of (2), and make it conform with natural situation: (3) Here, d 1 , d 2 , θ, γ, m 1 and m 2 are all positive constants, and given that the predators in nature don't usually just eat one species, mostly they have other choices. Based on this, we turn the growth of v into a logistic form which means u is not the only hunting target for the predators, they can still live even though the prey u became extinct.

ZUOLIN SHEN AND JUNJIE WEI
Define the real-valued Sobolev space For sake of discussion, we make the following assumption: (H1) m 1 < 1.
The system (3) always has three non-negative constant equilibrium solutions: E 0 (0, 0), E 1 (0, 1), E 2 (1/γ, 0), meanwhile, under the condition (H1), the system also has a positive equilibrium E * (u * , v * ), where Our main contribution for this article is a detailed and rigorous analysis about the global dynamics of the positive equilibrium of the diffusive predator-prey system (2). Keeping other parameters constant, we use the predation rate m 1 as a variable, and find that different values of m 1 result in distinct tendencies of the two species. Because of the effect of "surplus killing", the prey must raise their fertility rate or reduce the chances of encountering a predator to avoid extinction. For system (3), we shall be employing τ as our bifurcation parameter in our stability analyses to follow. When the spatial domain Ω is one-dimensional, and the parameters satisfy γθ − m 2 m 1 (1 + u * ) 2 < 0, we show the existence of Hopf bifurcation. We derive an explicit formula of the bifurcation point τ n,j where n, j are integers with n has an upper bound. From the Proposition 2.3 of [4], we get that the smallest value of τ n,j is τ 0,0 , which indicates that the periodic solutions bifurcated from the constant steady state solutions at τ = τ 0,0 are homogeneous. We have also studied the direction of Hopf bifurcation and stability of bifurcating periodic solution, one can see the details in Section 4.
The rest of the paper is organized as follows. In Section 2, the existence and priori bound of a positive solution for the reaction diffusion system are given, and the global asymptotically stability of positive equilibrium is proved. In Section 3, the stability of the positive constant steady state is considered, and the existence of the related Hopf bifurcation at the critical points is investigated with delay as the bifurcation parameter. In Section 4, by applying the normal form theory and center manifold reduction of partial functional differential equations, some detailed results of Hopf bifurcation are derived. Some numerical simulations are presented in Section 5. Throughout the paper, we denote N as the set of positive integers.

2.
Analysis of solution for model without delay.

2.1.
Existence and priori bound of solution. In this section, we shall investigate the existence of a positive solution for system (3) with delay τ = 0, and give a priori bound of the solution. Clearly, the system (3) with τ = 0 is Theorem 2.1. Suppose that d 1 , d 2 , θ, γ, m 1 , m 2 are all positive, Ω ⊂ R n is a bounded domain with smooth boundary. Then (a) the system (4) has a unique solution (u(x, t), v(x, t)) satisfying and (b) for any solution (u(x, t), v(x, t)) of system (4), Proof. Define Obviously, for any (u, v) ∈ R 2 + = {(u, v)|u ≥ 0, v ≥ 0}, one can see that thus system (4) is a mixed quasi-monotone system(see [23]). Let

2.2.
Global stability of positive equilibrium. In this section, we shall give the conditions to ensure that the positive constant equilibrium E * (u * , v * ) is globally asymptotically stable by utilizing the upper-lower solution method introduced by Pao [24].
are satisfied. Then the positive equilibrium E * (u * , v * ) of (4) is globally asymptotically stable, that is E * (u * , v * ) is stable, and for any initial values Proof. In Section 2.1, we get that From (H2), we can choose an ε 0 > 0 satisfying Letc 1 = 1/γ + ε 0 , without loss of generality, there exists a T 1 > 0, such that u(x, t) ≤ 1/γ + ε 0 for any t > T 1 , and this in turn implies Again we have for t > T 2 . Since m 1 < 1 and ε 0 satisfies (6), then Finally, using the similar method shown above, we have for t > T 3 , and we can ensure the following inequalities hold since ε 0 chosen as in (6), Therefor for t > T 4 , we obtain that and c 1 ,c 1 , c 2 ,c 2 satisfy Then (c 1 ,c 2 ) and (c 1 , c 2 ) are a pair of coupled upper and lower solutions of system (4), and when c 1 ≤ u ≤c 1 , c 2 ≤ v ≤c 2 , from the boundedness of the partial derivative of f i (i = 1, 2) with respect to u, v, we know that f i satisfies the Lipschitz condition. Here we denote the Lipschitz constants by L i , i = 1, 2.
To investigate the asymptotic behavior of the positive equilibrium, we define two where Then for m ≥ 1, we know that Simplify the equations, we get Then we obtain If we assume thatū = u, then we can get the following relation from Eq.(9) From Eq. (8), we can also have Substituting the first equation of Eq.(11) and Eq.(10) into the second equation of Eq.(11), it follows that This is a contraction to the condition (H2). Henceū = u, and consequentlyv = v. Then from the Theorem 3.3 in [25] and Corollary 2.1 in [24], we can get the asymptotic behavior of the positive solution. Now we investigate the local stability of positive equilibrium E * (u * , v * ). Recall that by writing the vector (u(x, t), v(x, t)) T as Then the system (4) can be rewritten aṡ where is defined by We consider the linearization at E * (u * , v * ) foṙ where and its characteristic equation satisfies It is well known that the eigenvalue problem with corresponding eigenfunctions ϕ n (x) = cos(nx/l), n ∈ N 0 . Let be an eigenfunction for (15). Then from a straightforward computation, we obtain that the eigenvalues of (15) can be given by the following equations For all n ∈ N 0 we have

ZUOLIN SHEN AND JUNJIE WEI
Then all the roots of Eq.(16) have negative real parts. This implies that the positive equilibrium E * (u * , v * ) of (4) is locally asymptotically stable when m 1 < 1.
Combining with the global attractivity proved before, we know that the constant positive equilibrium E * (u * , v * ) is globally asymptotically stable.
The above result indicates that E * (u * , v * ) attracts all feasible solutions under the condition (H1) and (H2). If (H2) doesn't work but (H1) still holds, then E * (u * , v * ) is local asymptotically stable. However, if m 1 > 1, then E * (u * , v * ) disappears while E 1 (0, 1) is global asymptotically stable. The critical curve can be drawn on the (m 1 , γ) plane (see Fig.1). Remark 1. According to the relationship between the original equation (1) and the dimensionless equation (2), we can illustrate the effect of "surplus killing". There are two different functional responses in equation (1), in order to be consistent with the assumptions, let the consumption rate m 2 be fixed. If the predation rate m 1 is sufficiently small(keeping the other parameters constant), then (H1) and (H2) can be satisfied, biologically, the two species can coexist and keep in a certain density. But if m 1 is not small enough such that (H2) holds, this coexistence will be shaken, and only near the equilibrium point, they can maintain this balance. As the parameters continue to change, (H1) is not satisfied, the balance will be completely broken: the population of prey will collapse to zero, and then predator population will grow into its carrying capacity. This's reasonable, because the predator population follows a logistic growth, they will never die out in this case, but the prey doesn't seem to be so fortunate: the predator exhibit a "surplus killing" behaviour, they will keep hunting whenever they can. So, the prey must enhance its fertility rate or reduce the chance of encountering a predator to avoid extinction.
Remark 2. If m 1 > 1, the boundary equilibrium E 1 (0, 1) is global asymptotically stable for system (4). In fact, from the equation of predator in system Eq.(4), we have ∂v ∂t It is well known that the positive solution of latter equation uniformly approach to 1 as t → ∞ under the same initial and boundary conditions. Since m 1 > 1, and the unique solution (u(x, t), v(x, t)) satisfies the conclusions of Theorem 2.1, then we can choose a sufficient small ε > 0 and T 0 > 0 such that 1 − γu − m 1 (1 − ε) < 0 and v(x, t) ≥ 1 − ε for any t > T 0 . Now, consider the equation of prey in system Eq. (4), for t > T 0 . Hence we have u(x, t) → 0 as t → ∞, and this result in turn implies v(x, t) → 0 as t → ∞.
3. Stability of the positive equilibrium and Hopf bifurcation. In this section, we shall study the stability of the positive constant steady state E * (u * , v * ) and the existence of Hopf bifurcation for (3) with τ ≥ 0 by analyzing the distribution of the eigenvalues. Here, we restrict ourselves to the case of one dimensional spatial domain Ω = (0, lπ), for which the structure of the eigenvalues is clear, and let the phase space C := C([−τ, 0], X). The linearization of system (13) at E * (u * , v * ) is given bẏ where L * : C → X is defined by
That is, each characteristic value λ is a root of the equation where Clearly, λ = 0 is not the root of Eq. (19), from the result of Ruan and Wei [29], as parameter τ varies, the sum of the orders of the zeros of Eq. (19) in the open right half plane can change only if a pair of conjugate complex roots appears on or crosses the imaginary axis. Now we would like to seek critical values of τ such that there exists a pair of simple purely imaginary eigenvalues.
Let ±iω(ω > 0) be solutions of the (n + 1)th equation (19). Then we have Separating the real and imaginary parts, it follows that Then we have Denote z = ω 2 . Then (21) can be rewritten in the following form where Hence Eq.(22) has a unique positive root only if D n and B * satisfy D 2 n − B 2 * < 0. From the explicit formula of D n and B * , we know that D n + B * > 0, for all n ∈ N 0 . Since we find a constant n * ∈ N such that for ∀n ∈ N 0 D n − B * < 0, for 0 ≤ n < n * . and D n − B * ≥ 0, for n ≥ n * . Here we denote the set By Eq.(20), we have sin ω n τ > 0, then Following the work of Cooke and Grassman [6], we have Proof. Substituting λ(τ ) into Eq. (19) and taking the derivative with respect to τ , we obtain that By Eq.(20), we have Since the sign of Re dλ dτ is same as that of Re dλ dτ −1 , the lemma follows immediately.
From the Proposition 2.3 of [4], we have that τ n,j ≤ τ n,j+1 , for all j ∈ N 0 , n ∈ S, and τ n,j ≤ τ n+1,j , for all j ∈ N 0 , n ∈ S. Then τ 0,0 is the smallest τ n,j . Denote τ 0,0 as τ * so that the stability will change when τ crosses τ * . Summarizing the above analysis and indicated by Corollary 2.4 of Ruan and Wei [29], we have the following lemma.
for all n ∈ N 0 , then all the roots of (19) have negative real parts for τ ≥ 0.
then for τ = τ n,j , j ∈ N 0 , n ∈ S, the (n + 1)th equation of (19) has a pair of simple pure imaginary roots ±iω n , and all other roots have non-zero real parts. Moreover, all the roots of Eq. (19) have negative real parts for τ ∈ [0, τ * ), and for τ > τ * , Eq. (19) has at least one pair of conjugate complex roots with positive real parts.
From Lemmas 3.1 and 3.2, we have the following theorem.

4.
Direction of Hopf bifurcation and stability of bifurcating periodic solution. In section 3, we obtained some conditions under which the system (3) undergoes a Hopf bifurcation. In this section, we shall study the direction of Hopf bifurcation near the positive equilibrium and stability of the bifurcating periodic solutions. We are able to show more detailed information of Hopf bifurcation by using the normal form theory and center manifold reduction due to [10,13,33].
The initial conditions in all simulations are given by u 0 (x, t) = 1.7 + 0.1 cos 2x,   Remark 3. Fig.2 and Fig.3 come into being on the precondition of (H1), that is to say, when the delay τ is less than the critical value τ * , the population of predator and prey will tend to a relatively stable state (see Fig.2); when the delay τ is a little larger than the critical value τ * , the polulation presents a periodic oscillation phenomenon near the equilibrium point(see Fig.3). If the precondition of (H1) is not satisfied, from Remark 1 and Remark 2, we know that the boundary equilibrium E 1 (0, 1) is global asymptotically stable, the prey will go extinct at last, but the predator will increase and reach its carrying capacity(see Fig.4).