Wave phenomena and edge of chaos in a diffusive predator-prey system under Allee effect

Abstract

We investigate a spatial predator-prey system; predator under Allee effect. The predator-prey community dynamics is assumed to be of Holling-Tanner type. When community’s local dynamics displays sustained periodic oscillations, the system’s spatio-temporal dynamics supports periodic traveling waves. Dispersal of species is assumed to be a random process in one-dimension only. Two dimensional (2D) parameter scan studies reveal that the system dynamics is self-organized at Edges of Chaos (EOC). This is a vital result as existence of EOC has not been demonstrated in a reaction-diffusion system yet. As population dynamics of all animals are governed by two basic processes: a) growth and b) dispersal, existence of EOC puts a recent theory of ecological chaos (Upadhyay, 2009; Rai and Upadhyay, 2006) on robust footing. Role of Allee effect in enhancing chaos presents an opportunity for its application in control of biological invasions and conservation of small populations. Extinction-persistence transitions were also explored.