Analytical and Numerical Bifurcation Analysis of a Forest-Grassland Ecosystem Model with Human Interaction

TL;DR

This paper combines analytical and numerical bifurcation analysis to explore complex dynamics, including multiple steady states and oscillations, in a forest-grassland ecosystem model influenced by human perception, revealing mechanisms for abrupt ecological shifts.

Contribution

It introduces a comprehensive bifurcation analysis of a coupled ecosystem-human interaction model, identifying critical parameter conditions for various bifurcations and oscillatory behaviors.

Findings

Multiple steady states and regimes with stable and unstable limit cycles.

Identification of critical bifurcations such as transcritical, saddle-node, and Hopf.

Detection of turning points in limit cycles indicating potential for catastrophic shifts.

Abstract

We perform both analytical and numerical bifurcation analysis of a forest-grassland ecosystem model coupled with human interaction. The model consists of two nonlinear ordinary differential equations incorporating the human perception of forest/grassland value. The system displays multiple steady states corresponding to different forest densities as well as regimes characterized by both stable and unstable limit cycles. We derive analytically the conditions with respect to the model parameters that give rise to various types of codimension-one criticalities such as transcritical, saddle-node, and Andronov-Hopf bifurcations and codimension-two criticalities such as cusp and Bogdanov-Takens bifurcations. We also perform a numerical continuation of the branches of limit cycles. By doing so, we reveal turning points of limit cycles marking the appearance/disappearance of sustained oscillations. These far-from-equilibrium criticalities that cannot be detected analytically give rise to the abrupt loss of the sustained oscillations, thus leading to another mechanism of catastrophic shifts