Dynamical behaviour of an ecological system with Beddington-DeAngelis functional response

TL;DR

This paper systematically analyzes the dynamical behavior of an ecological system with Beddington-DeAngelis functional response, focusing on stability, bifurcations, and the effects of parameters on species interactions.

Contribution

It provides a thorough mathematical analysis of the model's stability, bifurcations, and conditions for equilibrium, including explicit formulas and numerical validation.

Findings

System dynamics are highly sensitive to parameters and initial populations.

Conditions for Hopf bifurcation are identified.

Explicit formulas for stability and bifurcation properties are derived.

Abstract

The objective of this paper is to study the dynamical behaviour systematically of an ecological system with Beddington-DeAngelis functional response which avoids the criticism occurred in the case of ratio-dependent functional response at the low population density of both the species. The essential mathematical features of the present model have been analyzed thoroughly in terms of the local and the global stability and the bifurcations arising in some selected situations as well. The threshold values for some parameters indicating the feasibility and the stability conditions of some equilibria are also determined. We show that the dynamics outcome of the interaction among the species are much sensitive to the system parameters and initial population volume. The ranges of the significant parameters under which the system admits a Hopf bifurcation are investigated. The explicit formulae for determining the stability, direction and other properties of bifurcating periodic solutions are also derived with the use of both the normal form and the central manifold theory (cf. Carr \cite{Carr}). Numerical illustrations are performed finally in order to validate the applicability of the model under consideration.