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

## TL;DR

This paper presents an explicit three-component oscillatory Fisher--KPP system demonstrating complex dynamics such as stable limit cycles, traveling waves, and pulsating fronts, expanding understanding beyond two-component models.

## Contribution

It introduces a natural three-component system exhibiting oscillatory behavior and complex wave phenomena, which was not previously demonstrated.

## Key findings

- Existence of a stable limit cycle
- Presence of pulsating fronts and propagating terraces
- Complex dynamics including predator-prey structures

## Abstract

In a recent paper by Cantrell, Cosner and Yu, two-component KPP systems with competition of Lotka--Volterra type were analyzed and their long-time behavior 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 traveling waves. Numerically, we also show the existence of pulsating fronts and propagating terraces.