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In a recent paper by Cantrell, Cosner and Yu, two-component KPP systems withcompetition of Lotka--Volterra type were analyzed and their long-time behaviorlargely settled. In particular, the authors established that any constantpositive steady state, if unique, is necessarily globally attractive. In thepresent paper, we give an explicit and biologically very natural example ofoscillatory three-component system. Using elementary techniques orpre-established theorems, we show that it has a unique constant positive steadystate with two-dimensional unstable manifold, a stable limit cycle, apredator--prey structure near the steady state, periodic wave trains andpoint-to-periodic rapid traveling waves. Numerically, we also show theexistence of pulsating fronts and propagating terraces.

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