Stability and Restoration phenomena in Competitive Systems

TL;DR

This paper investigates how conservation laws influence stability and rhythmic population cycles in nonlinear dynamical systems, applying the model to ecological data like lynx and hare populations.

Contribution

It introduces a 2D nonlinear dynamical model with conservation laws to explain population cycles and recovery phenomena in ecological systems.

Findings

The model reproduces the 10-year lynx-hare cycle.

Conservation laws underpin the stability and rhythmic behavior.

A characteristic 'standard rhythm' emerges from the system's dynamics.

Abstract

A conservation law and stability, recovering phenomena and characteristic patterns of a nonlinear dynamical system have been studied and applied to biological and ecological systems. In our previous study, we proposed a system of symmetric 2n-dimensional conserved nonlinear differential equations with external perturbations. In this paper, competitive systems described by 2-dimensional nonlinear dynamical (ND) model with external perturbations are applied to population cycles and recovering phenomena of systems from microbes to mammals. The famous 10-year cycle of population density of Canadian lynx and snowshoe hare is numerically analyzed. We find that a nonlinear dynamical system with a conservation law is stable and generates a characteristic rhythm (cycle) of population density, which we call the {\it standard rhythm} of a nonlinear dynamical system. The stability and restoration phenomena are strongly related to a conservation law and balance of a system. The {\it standard rhythm} of population density is a manifestation of the survival of the fittest to the balance of a nonlinear dynamical system.