Stability and bifurcation in a ratio-dependent Holling-III system with diffusion and delay ∗

Abstract. A diffusive ratio-dependent predator-prey system with Holling-III functional response and delay effects is considered. Global stability of the boundary equilibrium and the stability of the unique positive steady state and the existence of spatially homogeneous and inhomogeneous periodic solutions are investigated in detail, by the maximum principle and the characteristic equations. Ratio-dependent functional response exhibits rich spatiotemporal patterns. It is found that, the system without delay is dissipative and uniformly permanent under certain conditions, the delay can destabilize the positive constant equilibrium and spatial Hopf bifurcations occur as the delay crosses through some critical values. Then, the direction and the stability of Hopf bifurcations are determined by applying the center manifold reduction and the normal form theory for partial functional differential equations. Some numerical simulations are carried out to illustrate the theoretical results.


Introduction
The well-known Lotka-Volterra type prepator-prey system describing predator-prey interactions has long been and will continue to be one of the dominant themes in both ecology and mathematics [1].Generally, a predator-prey model can take the form where u, v stand for prey and predator density, respectively.a, a/b, d, s/c are positive constants which denote prey intrinsic growth rate, carrying capacity, predator death rate and conversion rate, respectively.g(u) is the prey-dependent functional response, particularly, g(u) = cu 2 /(m 2 + u 2 ) is so-called Holling-type III, first proposed by Holling [2].Some biologists have questioned the functional response solely depending on prey density, for example [3][4][5], especially when predators have to search for food (and therefore, have to share or complete for food).A more suitable predator-prey theory should be the so-called ratio-dependent theory.We replace g(u) by g(u/v).That is, the per capita predator growth rate is a function of the ratio of prey to predator biomass, which is strongly supported by numerous fields and experimental data [3,4].The ratio-dependent predator-prey system (1) with Holling-type III takes the form where u 2 /(m 2 v 2 +u 2 ) denotes the predator response function, s stands for the conversion rate.
The above model can produce the richer and more reasonable dynamics biologically [3,6,7].It is well known that the past history and current state can affect the dynamics of models.The effect of gestation delay has been studied by many authors [8][9][10][11].Assuming the reproduction of the predators after eating the prey will be not transient but need some discrete time lag, we introduce a delay to the system (2) to make it close to reality.
Particularly, by using the continuation theorem of coincidence degree theory, Wang and Li [12] investigated the existence of positive periodic solutions for a delayed ratiodependent predator-prey model with Holling-type III functional response in the following form: where a(t), b(t), c(t) and d(t) are all positive ω-period functions, m > 0 and k(s) : R + → R + is a measurable, ω-periodic, normalized function such that , Xu, Gan and Ma [13] studied the global stability of the equilibria of the above system and the existence of Hopf bifurcation.Our work is an extension of the work in [13].Assuming that predators and preys are in an isolate patch in which the impact of migration, including immigration and emigration, can be neglected, we only consider the diffusion of the spatial domain.It has been shown that the reaction-diffusion system can generate more complex spatiotemporal patterns [14][15][16][17][18][19][20][21][22][23].We consider the ratio-dependent Holling-III predator-prey system with delay and diffusion where ∆ = ∂ 2 /∂x 2 denotes the usual Laplacian operator in the one-dimension space, d 1 , d 2 denote the diffusion rate of prey and predator, respectively.We assume that d 1 d 2 , which implies that the prey has some hope of escaping.ν is the outward unit normal on ∂Ω, and the homogeneous Neumann boundary conditions imply the closed domain where the populations can not move across the boundary of the domain.Let ū(x, t) = (b/a)u(x, t), v(x, t) = (b/a)mv(x, t) and h = c/m, φ(x, t) = (b/a)φ(x, t), ψ(x, t) = (b/a)mψ(x, t), and drop the bars for simplicity of notations, then (4) can be transformed into the following system: x ∈ (0, lπ), t 0, www.mii.lt/NA In addition, what is more important in ecosystem is whether the species would survive in the long run.That is, whether the system is permanent.In the present paper, we aim to study the uniform permanence and dissipativeness and investigate the effects of the delay and diffusion on system (5).The main results are that, the boundary equilibrium is globally asymptotically stable if s < d and 2a > h, and unstable if s > d for all τ 0. The delay and diffusion have no effects on the stability of the boundary equilibrium.In addition, the system without delay is dissipative when s > d and is uniformly permanent when s > d, 2a > h.And, if s > d, there is τ > 0 such that the positive constant equilibrium is stable for τ ∈ [0, τ ) and unstable for τ ∈ (τ , +∞).And a family of homogeneous, as well as inhomogeneous periodic solutions, may bifurcate from the positive constant steady state under some certain conditions, by virtue of the effects of diffusion and delay.
The rest of the paper is organized as follows.In Section 2, the global stability of the boundary equilibrium and the stability of the positive coexistence of system (5) are studied in detail.And the dissipativeness and uniform permanence of the system without delay are investigated by the comparison principle.And we also consider the existence of Hopf bifurcations as the delay τ crosses a sequence of critical values.In Section 3, the direction and stability of the Hopf bifurcations are given.Some numerical simulations are carried out to illustrate the theoretical results in Section 4.
2 Some dynamical behaviors of the system Firstly, we make the following assumptions: It is easy to check that system (5) always has a unique boundary equilibrium point E 1 (1, 0).And the system has a unique positive constant steady state E(u * , v * ) if and only if (H2) holds, where Next, we study the stability of the boundary equilibrium.

Global stability analysis of the boundary equilibrium
The linearized equations of system (5) at E 1 (1, 0) are given by From the properties of the Laplacian operator defined on the bounded domain the operator on X has the eigenvalues −n 2 /l 2 with the relative eigenfunctions where and Ω = (0, lπ).
The characteristic equations of the linearized equations ( 6) are given by Then we know that all the characteristic roots of the linearization of system ( 5) at E 1 (1, 0) are negative if s < d.And there is at least one positive characteristic root, if s > d.Combining the stability theory, we can obtain the following result.
Theorem 1. Suppose (H1) is satisfied.Then the boundary equilibrium E 1 (1, 0) of system (5) is locally asymptotically stable if s < d and unstable if s > d for all τ 0.
Proof.It is easy to see that all the solutions (u(x, t), v(x, t)) of system (5) are nonnegative by the maximum principle.And since φ(x, t) > 0, u(x, t) is strictly positive for t > 0. It is well known that if ω(x, t) satisfies By the second equation of system (5), we know that By the comparison principle of parabolic equations, we obtain that By the first equation of system (5), we have From the maximum principle By ( 7) and ( 8), we have that Then, for all ∈ (0, a), there exists T ( ) > 0 such that On the one hand, Again from the comparison principle, we can obtain that Let → 0, we have u(x, t) → 1 as t → +∞ uniformly in Ω.The result is derived.

Dissipativeness and uniform permanence of the system without delay
In this subsection, we analyze the dynamical behavior of the system.We assume that the initial value v(x, 0) > 0 for all x ∈ [0, lπ].The system (5) with τ = 0 is the following form: x ∈ (0, lπ), t 0, In order to obtain the main result, we first give a lemma.
Lemma 1.Consider the ordinary differential equation for a constant a > 0. Then the solution of (10), x(t), satisfies lim t→+∞ x(t) = a.
Note that the system (10) has only two fixed points x ≡ 0 and x ≡ a. Then one can prove the conclusion by the uniqueness of the solution.So we omit the detailed proof here.
Proof.Standard maximum principle of parabolic equations shows the solutions of system (9) always exist and are nonnegative.And since ψ(x, 0) > 0, v(x, t) is strictly positive for t > 0.
By the first equation of ( 9), From the comparison principle we know that lim sup t→+∞ max x∈[0,lπ] u(x, t) 1.
Theorem 4. Suppose that (H1) is satisfied.Then system (9) is uniformly permanent if s > d and 2a > h.
Then, for t > T , .
By virtue of Taylor expansions, (12) can be rewritten as Denote U 1 (t) = u(•, t), U 2 (t) = v(•, t) and U = (U 1 , U 2 ) T .Then (13) can be rewritten as an abstract differential equation in the phase space C = C([−τ, 0], X), where D = d1 d2 and L : C → X, F : C → X are defined by , respectively, for φ = (φ 1 , φ 2 ) T ∈ C .Some computations show that the characteristic equation of the linearization of ( 14) is equivalent to the sequence of the transcendental equations where By (H2), we obtain k > 0 and r > 0. If is reduced to Obviously, λ = 0 is not the root of ( 16) since r > 0, and Bogdanov-Takens singularity does not occur at the positive equilibrium E at the absence of diffusion.
Next we consider the system with diffusion.Equation (15) with τ = 0 is the following sequence of quadratic equations: where This and r > 0 imply that Thus, the roots of Eq. ( 17) have real parts for h ∈ (0, (as This implies that Eq. ( 17) has at least a root with positive real part.
And if h = (as 2 + 2ds(s − d))/(2d (s − d)d), then p − k = 0. Hence we have Thus Eq. ( 17) has a pair of simple purely imaginary roots, and other roots have negative real parts.Meanwhile, by dT 0 (p)/(dp)| p=k = 1 > 0, the transversality condition holds.Summarizing the discussion above, we have the following conclusions.Now we consider the effects of delay τ on the stability of the equilibrium E(u * , v * ) of system (5).We confine h ∈ (0, (as 2 +2ds(s−d))/(2d (s − d)d)), that is, E(u * , v * ) is stable when τ = 0.It is well known that iω (ω > 0) is a root of Eq. ( 15) if and only if ω satisfies the following equation for some n ∈ N 0 : Separating the real and imaginary parts of Eq. ( 18) leads to which implies that where By Q 0 = −r 2 < 0 and D n > 0, there exists n 0 ∈ {1, 2, . . .} such that

Direction and stability of the Hopf bifurcations
In this section, we consider the stability, the direction and the period of bifurcating periodic solutions by using the normal formal theory and the center manifold theorem of partial functional differential equation presented in [25,26] and [27].Without loss of generality, denote any one of the these critical values τ n j , j ∈ {0, 1, 2, . . .}, n ∈ {0, 1, 2, . . ., n 0 − 1}, by τ * at which Eq. ( 15) has a pair of simply purely imaginary roots ±iω n + , denoted by ±iω.

Conclusion
The dynamics of the diffusive Holling-III ratio-dependent predator-prey system with delay effect are investigated under the Neumann boundary conditions.The global stability and instability of the boundary equilibrium are obtained by the maximum principle and comparison principle of the parabolic equations.It shows that, the diffusion and delay have no effects on the stability of the boundary equilibrium and have effect on the positive coexistence.In particular, the system without delay is dissipative and uniformly persistent.We hope that our work could be instructive to study the population.

2. 3
Stability of E(u * , v * ) and existence of Hopf bifurcation In this section, under the assumption s > d, we study the stability of the equilibrium E(u * , v * ) and the existence of Hopf bifurcation, by analyzing the distribution of the eigenvalues.Nonlinear Anal.Model.Control, 2014, Vol. 19, No. 1, 132-153