Abstract
This paper is concerned with a diffusive predator-prey system with Beddington-DeAngelis functional response and delay effect. By analyzing the distribution of the eigenvalues, the stability of the positive equilibrium and the existence of spatially homogeneous and spatially inhomogeneous periodic solutions are investigated. Also, it is shown that the small diffusion can affect the Hopf bifurcations. Finally, the direction and stability of Hopf bifurcations are determined by normal form theory and center manifold reduction for partial functional differential equations.
Highlights
By choosing the delay τ as the bifurcation parameter and analyzing the associated characteristic equation of 1.1 at the positive constant steady state, we investigate the stability of the positive constant steady state of 1.1 and obtain the conditions under which 1.1 undergoes Hopf bifurcation
Equation 2.13 with τ 0 is equivalent to the following quadratic equations: λ2 d1k2 d2k2 p s λ d1d2k4 b3d1k2 a3d2k2 r d2sk[2] q 0
By Lemma 2.1 and the theorem proved by Ruan and Wei 18, all roots of 2.13 have negative real parts
Summary
We will study the stability and Hopf bifurcations of a diffusive predator-prey system with Beddington-DeAngelis functional response and delay effect as follows: ut d1Δu t, x u t, x 1 − u t − τ, x − sP u, v , t > 0, x ∈ Ω, vt d2Δv t, x rP u, v − dv t, x , t > 0, x ∈ Ω, 1.1 ∂νu ∂νv 0, t > 0, x ∈ ∂Ω, u 0, x u0 x ≥ 0, v 0, x v0 x ≥ 0, x ∈ Ω, where u and v denote the population densities of prey and predator species at time t and space x, respectively; the positive constants d1 and d2 represent the diffusion coefficients of prey and predator species, respectively; s > 0 s is called the capturing rate and r > 0 r is Abstract and Applied Analysis called the conversion rate represent the strength of the relative effect of the interaction on the two species; d denotes the death rate of predator species; P u, v uv/ m u nv is the Beddington-DeAngelis functional response function with m and n are positive numbers; τ ≥ 0 denotes the generation time of the prey species; Ω is a bounded domain in RN N is any positive integer with a smooth boundary ∂Ω; Δ is the Laplacian operator on Ω; ν is the outward normal to ∂Ω; homogeneous Neumann boundary conditions reflect the situation where the population cannot move across the boundary of the domain. There has been an increasing interest in the study of diffusive predator-prey system see 1, 2, 4, 6–14 and references therein with functional response. As is known to all, the Beddington-DeAngelis functional response, proposed by Beddington 6 and DeAngelis et al 8 , is more general than those the above authors considered, and it has been studied extensively in the literature 1–3, 7, 14–16. To the authors’ best knowledge, few researches have been done on the diffusive predator-prey system with BeddingtonDeAngelis functional response and time delay. The aim of this paper is to extend and develop the work in 1, 2 ; that is, we will study the stability and Hopf bifurcation of a diffusive predator-prey system with BeddingtonDeAngelis functional response and delay.
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