Effect of Slow–Fast Time Scale on Transient Dynamics in a Realistic Prey-Predator System

Abstract

Systems with multiple time scales, often referred to as `slow–fast systems’, have been a focus of research for about three decades. Such systems show a variety of interesting, sometimes counter-intuitive dynamical behaviors and are believed to, in many cases, provide a more realistic description of ecological dynamics. In particular, the presence of slow–fast time scales is known to be one of the main mechanisms resulting in long transients—dynamical behavior that mimics a system’s asymptotic regime but only lasts for a finite (albeit very long) time. A prey–predator system where the prey growth rate is much larger than that of the predator is a paradigmatic example of slow–fast systems. In this paper, we provide detailed investigation of a more advanced variant of prey–predator system that has been overlooked in previous studies, that is, where the predator response is ratio-dependent and the predator mortality is nonlinear. We perform a comprehensive analytical study of this system to reveal a sequence of bifurcations that are responsible for the change in the system dynamics from a simple steady state and/or a limit cycle to canards and relaxation oscillations. We then consider how those changes in the system dynamics affect the properties of long transient dynamics. We conclude with a discussion of the ecological implications of our findings, in particular to argue that the changes in the system dynamics in response to an increase of the time scale ratio are counter-intuitive or even paradoxical.

A Complex Dynamic of an Eco-Epidemiological Mathematical Model with Migration

“…After the pioneer work of Kermack and McKendricK [3], epidemiological models have drawn consideration of many biologists and ecologists. The infectious diseases in the community of prey-predator models has been studied extensively in literature [1,[4][5][6][7][8][9][10][11][12][13][14]. In [14], Tewa et al proposed and investigated a prey-predator model with an SIS infectious disease affecting preys or predators or both.…”

“…For animal species, migration is usually due to the climate change, for habitat, looking for food, predation, cannibalism, reproduction, urbanization, deforestation, etc. Many researches are mainly focused on the effect of infectious diseases in the community of prey-predator models under the influence of migration [4,11,13,[19][20][21]. Indeed, Kant and Kumar [11] proposed and studied a prey-predator model with disease in both species by taking into account the process of migration in only prey population.…”

“…Their mathematical analysis has permitted to establish existence of equilibria and the local and global stability. Chowdhury [13] investigated an ecoepidemiological model describing the effect of fast migration on prey-predator model between two different patches. Mathematically, the author proved the asymptotic stability of the unique fast equilibrium point and the aggregated model.…”