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Two components is too simple: an example of oscillatory Fisher–KPP system with three components

Part of: Local and nonlocal bifurcation theory Parabolic equations and systems

Published online by Cambridge University Press:  24 September 2019

Léo Girardin
Léo Girardin*
Affiliation:
Laboratoire de Mathématiques d'Orsay, Université Paris Sud, CNRS, Université Paris-Saclay, 91405 Orsay Cedex, France ([email protected])
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Abstract

In a recent paper by Cantrell et al. [9], two-component KPP systems with competition of Lotka–Volterra type were analyzed and their long-time behaviour largely settled. In particular, the authors established that any constant positive steady state, if unique, is necessarily globally attractive. In the present paper, we give an explicit and biologically very natural example of oscillatory three-component system. Using elementary techniques or pre-established theorems, we show that it has a unique constant positive steady state with two-dimensional unstable manifold, a stable limit cycle, a predator–prey structure near the steady state, periodic wave trains and point-to-periodic rapid travelling waves. Numerically, we also show the existence of pulsating fronts and propagating terraces.

Keywords

KPP nonlinearityreaction–diffusion systemHopf bifurcationspreading phenomena

MSC classification

Primary: 35K40: Second-order parabolic systems
Secondary: 35K57: Reaction-diffusion equations 37G10: Bifurcations of singular points 92D25: Population dynamics (general)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 6 , December 2020 , pp. 3097 - 3120
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Copyright
Copyright © 2019 The Royal Society of Edinburgh

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