Complex oscillatory patterns near singular Hopf bifurcation in a two-timescale ecosystem

Abstract

<p style='text-indent:20px;'>We consider an ecological model consisting of two species of predators competing for their common prey with explicit interference competition. With a proper rescaling, the model is portrayed as a singularly perturbed system with one fast (prey dynamics) and two slow variables (dynamics of the predators). The model exhibits a variety of rich and interesting dynamics, including, but not limited to mixed-mode oscillations (MMOs), featuring concatenation of small and large amplitude oscillations, relaxation oscillations and bistability between a semi-trivial equilibrium state and a coexistent oscillatory state. More interestingly, in a neighborhood of <i>singular Hopf</i> bifurcation, long lasting transient dynamics in the form of chaotic MMOs or relaxation oscillations are observed as the system approaches the periodic attractor born out of supercritical Hopf bifurcation or a semi-trivial equilibrium state respectively. The transient dynamics could persist for hundreds or thousands of generations before the ecosystem experiences a regime shift. The time series of population cycles with different types of irregular oscillations arising in this model stem from a biological realistic feature, namely, by the variation in the intraspecific competition amongst the predators. To explain these oscillations, we use bifurcation analysis and methods from <i>geometric singular perturbation theory</i>. The numerical continuation study reveals the rich bifurcation structure in the system, including the existence of codimension-two bifurcations such as fold-Hopf and generalized Hopf bifurcations.