Applicable Analysis Vol. 89, No. 8, August 2010, 1271–1292 Rich dynamical behaviours for predator–prey model with weak Allee effect Xiaohong Lai a , Shengqiang Liu a * and Rongzhen Lin b a The Academy of Fundamental and Interdisciplinary Science, Harbin Institute of Technology, 3041#, 2 Yi-Kuang Street, Harbin 150080, China; b School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China Communicated by R.P. Gilbert (Received 31 December 2009; final version received 22 March 2010) The weak Allee effect on the predator is introduced into the classic predator–prey model of Lotka–Volterra type. Global qualitative and bifurcation analyses are combined to determine the global dynamics of the model. It is shown that the weak Allee effect can bring rich and complicated dynamics to the previous simple model, such as the saddle–node bifurca- tion, subcritical and supercritical Hopf bifurcations, and Bogdanov– Takens bifurcations, implying that weak Allee effect can be one of the simple reasons for many complicated behaviours in the predator–prey communities. Keywords: predator–prey system; weak Allee effect; bifurcation; limit cycle AMS Subject Classifications: 92A15; 34A47 1. Introduction As the fundamental unit of ecological communities [1], the consumer–resource (predator–prey) models have been studied thoroughly since the pioneering work by Lotka [2] and Volterra [3]. Since the complex population dynamical behaviours are very common in the real world, theoreticians and experimentalists have proceeded to investigate the processes that affect the stability of predator–prey systems, it is natural for the ecologists to ask (see Turchin in his monographs [4]): could these complex dynamics have simple causes? Most of the investigations on predator–prey model use the systems of the Lotka– Volterra equation [2,3] with a logistic self-limitation term. Then we have the following the model [1,5]: _ x ¼ rx  1  x K   mxy, _ y ¼ yðd þ amxyÞ, 8 < : ð1Þ *Corresponding author. Email: sqliu@hit.edu.cn ISSN 0003–6811 print/ISSN 1563–504X online ß 2010 Taylor & Francis DOI: 10.1080/00036811.2010.483557 http://www.informaworld.com Downloaded By: [Liu, Shengqiang][Canadian Research Knowledge Network] At: 10:44 8 July 2010