Nonlinear Dyn (2010) 60: 265–275 DOI 10.1007/s11071-009-9594-9 ORIGINAL PAPER Self-organized wave pattern in a predator-prey model Gui-Quan Sun · Zhen Jin · Li Li · Bai-Lian Li Received: 2 October 2008 / Accepted: 8 September 2009 / Published online: 25 September 2009 © Springer Science+Business Media B.V. 2009 Abstract In this paper, pattern formation of a predator- prey model with spatial effect is investigated. We ob- tain the conditions for Hopf bifurcation and Turing bifurcation by mathematical analysis. When the val- ues of the parameters can ensure a stable limit cycle of the no-spatial model, our study shows that the spatially extended models have spiral waves dynamics. More- over, the stability of the spiral wave is given by the theory of essential spectrum. Furthermore, although G.-Q. Sun National Key Laboratory For Electronic Measurement Technology, North University of China, Taiyuan, 030051 Shanxi, People’s Republic of China e-mail: gquansun@yahoo.com.cn G.-Q. Sun · Z. Jin ( ) · L. Li Department of Mathematics, North University of China, Taiyuan, Shan’xi 030051, People’s Republic of China e-mail: jinzhn@263.net L. Li e-mail: lili831113@163.com G.-Q. Sun School of Mechatronic Engineering, North University of China, Taiyuan 030051, People’s Republic of China G.-Q. Sun · B.-L. Li Ecological Complexity and Modeling Laboratory, Department of Botany and Plant Sciences, University of California, Riverside, CA 92521-0124, USA B.-L. Li e-mail: bai-lian.li@ucr.edu the environment is heterogeneous, the system still ex- hibit spiral waves. The obtained results confirm that diffusion can form the population in the stable motion, which well enrich the finding of spatiotemporal dy- namics in the predator-prey interactions and may well explain the field observed in some areas. Keywords Predator-prey model · Spiral wave · Spiral spectra · Environmental heterogeneity 1 Introduction Ecological systems are characterized by the interac- tion between species and their natural environment. An important type of interaction which effects pop- ulation dynamics of all species is predation. Thus, predator-prey models have been in the focus of eco- logical science since the early days of this discipline [1, 2]. In the past, investigations have revealed that spatial inhomogeneities like the inhomogeneous dis- tribution of nutrients as well as interactions on spatial scales, which are essentially based on the assumption that the motion of individuals of given population is random and isotropic, i.e., without any preferred direc- tion, can have an important impact on the dynamics of ecological populations [1, 3–6]. And the spatial com- ponent of ecological interactions has been identified as an important factor in how ecological communities are shaped [5, 7].