Canards, relaxation oscillations, and pattern formation in a slow-fast ratio-dependent predator-prey system

Abstract

The prey-predator system is an elementary building block in many complicated models of ecological dynamics that exhibit complex dynamical behaviour and as such it remains to be a focus of intense research. Here we consider a new model that incorporates multiple timescales (in the form of a ‘slow-fast’ system) with the ratio-dependent predator response. In the nonspatial case, we study this model exhaustively using an array of analytical tools to demonstrate the existence of canard cycles and relaxation oscillation passing through the close vicinity of the complicated singular point of the system. In the spatial case, the model exhibits a wide variety of spatio-temporal patterns that have been studied using extensive numerical simulations. In particular, we reveal the explicit dependence of the slow-fast timescale parameter on the Turing instability threshold and show how this affects the properties of the emerging stationary population patches. We also show that the self-organized spatial heterogeneity of the species can reduce the risk of extinction and can stabilize the chaotic oscillations. We argue that incorporating multiple timescales can enhance our understanding for studying realistic scenarios of local extinction and periodic outbreaks of the species. Our results suggest that the ubiquitous complexity of ecosystem dynamics observed in nature stems from the elementary level of ecological interactions such as the prey-predator system.