Patterns in a freshwater tussock sedge model

Abstract

We study pattern formation mechanisms in a reaction diffusion model arising from the interactions between the tussock sedges and the wracks. We first show that the corresponding kinetic system has either a limit cycle or a heteroclinic orbit that connects the extinction state to the coexistent state. This implies that under different conditions, the tussock sedge grasses and the wracks coexist in a cyclic way, or settle down directly to the coexistent equilibrium state regardless of the spatial location. We then investigate the non-constant positive solutions of the corresponding elliptic system in a bounded region in ${\mathbb{R}}^n$ with zero flux boundary condition. We identify the above solutions as patterns. We find that the domain size, by various a priori estimates, the growth rate of the tussock sedge grasses, and the carrying capacity of land all contribute to the formation of patterns. By computing the Leray Schauder degree, we derive the conditions for the existence of the patterns.

Keywords:

• Positive solution
• steady state
• pattern
• Turing pattern
• Leray Schauder degree

AMS Subject Classifications:

• Primary: 35B40
• 35K57
• Secondary: 92D25
• 92D40

Acknowledgments

The authors are grateful to the anonymous reviewers for the helpful suggestions and comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by National Natural Science Foundation of China [grant numbers 11771033, 10971009, 10771196], by the China Scholarship Council [grant numbers 201303070222, 201706020203, 201706020094].