An Organizing Center of Codimension Four in a Predator-Prey Model with Generalist Predator: From Tristability and Quadristability to Transients in a Nonlinear Environmental Change

Abstract

In this paper, we take the Rosenzweig–MacArthur (RM) model with generalist predator as an example in a constant or changing environment. When the environment is fixed, we provide a more easily verifiable classification, in terms of the coefficients of the system with nilpotent linear part and general higher terms, to determine the types and codimension of nilpotent singularities in a general planar system. Second, by using the existing classification and some algebraic methods, we show that the highest codimension of a nilpotent focus is 4 and the sample RM model with generalist predator can exhibit nilpotent focus bifurcation of codimension 4. Our results indicate that generalist predation can cause not only richer bifurcations and dynamics (such as multitype tristability and quadristability, a figure-eight loop) but also the possible extirpation of prey. When the environment is changing, we study the impact of the rate and intensity of a nonlinear environmental change on dynamics. The key observations on the asymptotic and transient dynamics include (i) transient tracking on unstable steady states or oscillations, and transient-related regime shifts; (ii) slow and fast regime shifts; (iii) regulation of transient dynamics by the environmental change parameters and ; (iv) slow negative or fast positive environmental change can delay or even avoid population extirpation.