Dynamics of the generalized Rosenzweig–MacArthur model in a changing and patchy environment

In this paper, we propose a generalized Rosenzweig–MacArthur predator–prey model with two patches and Allee effects in predators. In a constant environment, for the single-patch model we show the existence of degenerate Bogdanov–Takens bifurcation with codimension up to 4 and degenerate Hopf bifurcation with codimension up to 3, which indicate that the nilpotent cusp of codimension 4 is the organizing center of the bifurcation set. Our results show that Allee effects in predators can not only induce much more complex bifurcations and dynamics (such as three limit cycles, multitype tristability), but also serve as a destabilizing factor to make predators extinction (deterministically) when the environment is sufficiently enriched, which provides strong support for the so-called "paradox of enrichment". For the two-patch model, we numerically reveal the existence of eight positive steady states including two new coexisting attractors, which means that population dispersal can help species coexist or increase the population abundance. In a changing environment, for the single-patch model we find that the population can track unstable states, fast positive or slow negative environmental change can be beneficial to predators by delaying or even avoiding extinction; while for the two-patch model we observe some interesting phenomena: fast negative (or slow positive) environmental change can make predators in different patches trend to extinction synchronously, and rate-induced different regime shifts.