Pattern formation and global analysis of a systematically reduced plant model in dryland environment

TL;DR

This paper analyzes the stability and pattern formation in a reduced plant model for drylands, exploring bifurcations and pattern types, and how parameter control can prevent desertification.

Contribution

It provides a detailed stability and bifurcation analysis of a reduced plant model, including spatial pattern formation and the effects of seed dispersal rate.

Findings

Identification of conditions for stability and limit cycles.

Observation of various spatial patterns like spots, stripes, and gaps.

Demonstration that parameter control can prevent desertification.

Abstract

This paper delves into a systematically reduced plant system proposed by Ja\"ibi et al. [Phys. D, 2020] in arid area. They used the method of geometric singular perturbation to study the existence of abundant orbits. Instead, we deliberate the stability and distributed patterns of this system. For a non-diffusive scenario for the model, we scrutinize the local and global stability of equilibria and derive conditions for the existence or non-existence of the limit cycle. The bifurcation behaviors are also explored. For the spatial model, we investigate Hopf, Turing, Hopf-Turing, Turing-Turing bifurcations. Specially, the evolution process from periodic solutions to spatially nonconstant steady states is observed near the Hopf-Turing bifurcation point. And mixed nonconstant steady states near the Turing-Turing bifurcation point are observed. Furthermore, it's found that there exist gap, spot, stripe and mixed patterns. The seed-dispersal rate enables the transformation of pattern structures. Reasonable control of system parameters may prevent desertification from occurring.