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

Pranali Roy Chowdhury,Malay Banerjee,Sergei Petrovskii

doi:10.3390/math10050699

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.

Highlights

The study of population dynamics of the interacting species in an ecosystem plays an important role in understanding the survival and long term existence of various species
Small stable limit cycles exist for β < βGH and parameter values taken below the Hopf bifurcation curve
The green curve represents the saddle-node bifurcation of limit cycles which exists for βGH < β < βSN, and it intersects with a global bifurcation curve and a homoclinic bifurcation curve at β = βSN

Summary

Introduction

The study of population dynamics of the interacting species in an ecosystem plays an important role in understanding the survival and long term existence of various species. Comprehending the intricate dynamics of the system through mathematically tractable yet realistic models are necessary In this regard, presence of multiple time scales due to the difference in growth rates when measured with respect to a fixed time scale plays a crucial role. The prey–predator models with a small time scale parameter ε (0 < ε 1) can be characterized as a singularly perturbed differential equation, with ε being the singular parameter These types of systems were initially observed in many chemical systems where the reaction rates of the reactants differs widely. Rinaldi and Muratori first studied the slow–fast prey predator models where the cyclic existence of the slow–fast limit cycle was discussed [1,2] They analyzed the cyclical fluctuation in population densities of three species model in a slow–fast setting with one and multiple time scale parameter

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