Stability of New Spike Solutions in a Supply Chain Model

We study a supply 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 and if there is practically an unlimited pool of prey, i.e. the predators will not feel the impact of the limited amount of prey due to its large quantity. 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 [12, 28] for which there is only one prey and one predator. In this supply 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 both the prey and the second predator diffuse much faster than the first predator. We construct new single spike solutions on an interval for which the profile of the first predator (middle component) is that of a spike. The profile of the prey and the second predator only varies on a large spatial scale which comes from the faster diffusion of these components. They both interact with the middle component in a novel way. It is shown that there exist two different single spike solutions if the feedrates are small enough, a large-amplitude and a small-amplitude spike. 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 spike solutions based on nonlocal eigenvalue problems. The following result is established: The large-amplitude spike solution is stable if the time-relaxation constants for both predators are small enough. The small-amplitude spike solution is always unstable. Dedicated to the memory of Professor Klaus Kirchgässner, with deep gratitude. Pattern Formation, Supply Chain Model, Predator-Prey Model, Reaction-Diffusion System, Spiky Solutions, Stability. Primary 35B35, 92C40; Secondary 35B40