  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.
