Stability of Cluster Solutions in a Tritrophic Food Chain Model

We study a tritrophic food chain model which is based on Schnakenberg type kinetics. This model is realistic in a predator-prey context if cooperation of predators is prevalent in the system. The system also serves as a model for a sequence of irreversible autocatalytic reactions in a container which is in contact with a well-stirred reservoir. It is an extension of the Schnakenberg model suggested in [11, 27] for which there is only one prey and one predator. In this food chain model there is one predator feeding on the prey and a second predator feeding on the first predator. This means that the first predator plays a hybrid role: it acts as both predator and prey. It is assumed that the prey diffuses much faster than the first predator and that the first predator diffuses much faster than the second predator. It is further assumed that the feed rates are large. We construct single cluster solutions on an interval for which the profile of the second predator is that of the commonly observed spike in the Schnakenberg model. However, for the first predator a new cluster-type profile is observed which comes from the fact that it acts as predator and prey simultaneously. Its profile in leading order is that of two partial spikes connected by a thin transition layer. For this profile different scales are involved: The spikes for the first predator are on the small scale, the spike for the second on the very small scale, the width of the transition layer for the first predator is between the small and the very small scale. This type of interaction is a new effect for spike layer solutions. It is shown that these patterns exist if the feed rates are small enough. We study the stability properties of this solution in terms of the system parameters. We use a rigorous analysis for the linearized operator around single cluster solutions based on nonlocal eigenvalue problems. The following result is established: There is coexistence of a stable and an unstable cluster solution if the time-relaxation constants are small enough. Otherwise both solutions are unstable. Pattern Formation, Food Chain, Predator Prey Model, Reaction-Diffusion System, Cluster Solutions, Stability. Primary 35B35, 92C40; Secondary 35B40