A Numerical Bifurcation Analysis of a Dryland Vegetation Model

TL;DR

This paper conducts a detailed bifurcation analysis of a dryland vegetation model, examining how varying parameters like precipitation and infiltration rates affect the stability and existence of different biomass states.

Contribution

It extends previous bifurcation studies by exploring a wider parameter range and introduces a modified model that removes unphysical solutions.

Findings

Infiltration parameters control biomass state stability.

Existence of multiple periodic orbits depending on parameters.

Modified model eliminates unphysical steady states.

Abstract

The dryland vegetation model proposed by Rietkerk and collaborators has been explored from a bifurcation perspective in several previous studies. Our aim here is to explore in some detail the bifurcation phenomena present when the coefficients of the model are allowed to vary in a wide range of parameters. In addition to the primary bifurcation parameter, the precipitation, we allow the two infiltration rate parameters to vary as well. We find that these two parameters control the size and stability of nonhomogeneous biomass states in a way that can be predicted. Further, they control when certain homogeneous and inhomogeneous (in space) periodic (in time) orbits exist. Finally, we show that the model possesses infinitely many unphysical steady state branches. We then present a modification of the model which eliminates these unphysical solutions, and briefly explore this new model for a fixed set of parameters.