Existence and stability of a spike in the central component for a consumer chain model in a two-dimensional domain

In this paper, we study a three-component consumer chain model which is based on Schnakenberg type kinetics in a two-dimensional domain. In the model there is one consumer feeding on the producer and a second consumer feeding on the first consumer. Through a rigorous analysis, we show that there exist two different single spike solutions if the feed rates are small. Further, we also establish the stability results: If the time-relaxation constants for both producer and the second consumer vanish, the large amplitude spike solution is stable and the small-amplitude spike solution is unstable. We also derive results on the stability of solutions when these two time-relaxation constants are positive. We show a new effect that if the time-relaxation constant of the second consumer is bounded, the large-amplitude spike solution is still stable while it is unstable in the one-dimensional case.