Dynamics of Performances in a Competitive Model

TL;DR

This paper analyzes a complex resource-consumer dynamical model combining Lotka-Volterra and prey-predator dynamics, revealing bifurcations, coexistence mechanisms, and transitions to chaos through bifurcation analysis and simulations.

Contribution

It introduces an integrated LV-RM model with realistic resource and consumer dynamics, and applies bifurcation theory to understand complex behaviors and coexistence.

Findings

Identification of Hopf and zero-Hopf bifurcations

Coexistence facilitated by differentiated time scales

Transition from periodic to chaotic dynamics

Abstract

A competitive resource-consumer dynamical model is analyzed based on an integrated model of a competitive Lotka-Volterra model and a prey-predator Rosenzweig-MacArthur model that we call that LV-RM model throughout this paper. Resource growth in the absence of consumers is logistic, and competing consumers' type II Holling's functional response made the model structure more realistic. We used the normal form and the center manifold theorems for bifurcation analysis of the presented model, identified Hopf and zero-Hopf bifurcations and their directions, and discussed their biological interpretations. We hypothesized that differentiated time scales of the competing consumers' predatory that lead to asymmetry in competition are the mechanisms that promote coexistence through relaxation-oscillation dynamics. Though, other performance parameters of both competitors are the same. Graphical representation of variations of the first Lyapunov coefficient, after competition coefficients interplay, shows various dynamics with growing complexity from the periodic state towards chaotic motion like R\"ossler attractor. We presented simulations to visualize the theoretical results obtained through bifurcation analysis.