Hopf Bifurcations in a Predator-Prey Diffusion System with Beddington-DeAngelis Response
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Hopf Bifurcations in a Predator-Prey Diffusion System with Beddington-DeAngelis Response Jia-Fang Zhang · Wan-Tong Li · Xiang-Ping Yan
Received: 6 November 2008 / Accepted: 3 November 2010 / Published online: 17 November 2010 © Springer Science+Business Media B.V. 2010
Abstract This paper is concerned with a two-species predator-prey reaction-diffusion system with Beddington-DeAngelis functional response and subject to homogeneous Neumann boundary conditions. By linearizing the system at the positive constant steady-state solution and analyzing the associated characteristic equation in detail, the asymptotic stability of the positive constant steady-state solution and the existence of local Hopf bifurcations are investigated. Also, it is shown that the appearance of the diffusion and homogeneous Neumann boundary conditions can lead to the appearance of codimension two Bagdanov-Takens bifurcation. Moreover, by applying the normal form theory and the center manifold reduction for partial differential equations (PDEs), the explicit algorithm determining the direction of Hopf bifurcations and the stability of bifurcating periodic solutions is given. Finally, numerical simulations supporting the theoretical analysis are also included. Keywords Predator-prey system · Diffusion · Stability · Hopf bifurcation · Bogdanov-Takens bifurcation Mathematics Subject Classification (2000) 35K57 · 35B32 · 92D25 1 Introduction In recent years, a great deal of predator-prey models have been proposed and investigated extensively since the pioneering theoretical works by Lotka [13] and Volterra [15]. For inW.-T. Li was supported by NNSF of China (10871085) and the FRFCU (lzujbky-2010-k10). X.-P. Yan was supported by NNSF of China (10961017) and “Qinglan” Talent Program of Lanzhou Jiaotong University (QL-05-20A). J.-F. Zhang · W.-T. Li () School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, People’s Republic of China e-mail: [email protected] X.-P. Yan Department of Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, People’s Republic of China
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stance, when predator and prey two-species living in a bounded domain ⊂ RN (N ≥ 1) with a smooth boundary ∂ and the zero boundary flux are considered, the general twospecies predator-prey system can be modeled by the following system of partial differential equations (PDEs) ⎧ t > 0, x ∈ , ut = d1 u(t, x) + u(t, x)(a − bu(t, x)) ⎪ ⎪ ⎪ ⎪ − sQ(u(t, x), v(t, x))v(t, x), ⎨ vt = d2 v(t, x) + rQ(u(t, x), v(t, x))v(t, x) − δv(t, x), t > 0, x ∈ , ⎪ ⎪ u = ∂ν v = 0, t > 0, x ∈ ∂, ∂ ⎪ ⎪ ⎩ ν x ∈ , u(0, x) = u0 (x) ≥ 0, v(0, x) = v0 (x) ≥ 0,
(1.1)
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 rates of prey and predator species, respectively; a > 0 denotes the intrinsic growth rate of prey species; a > 0 denotes the carrying capacity of prey species; s > 0 (s is called the capturing rate) b and r > 0 (r is called the conver
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