Nonmonotone Waves in a Three Species Reaction-diiusion Model

This paper establishes the existence of a nonmonotone travelling wave for a reaction-diiusion system modeling three competing species. General existence results for travelling waves in higher dimensional systems depend on monotonicity and therefore do not apply to the result obtained here. The proof demonstrates an application of the homotopy invariant, the connection index, to a higher dimensional ow where few explicit results are available. The result is obtained in a singular perturbation regime where the fast-slow structure can be exploited to construct a singular limit solution from the lower dimensional reduced ows. A priori estimates show the connecting solution to be uniformly approximated by the singular limit and these estimates make it possible to construct the higher dimensional isolating neighborhoods necessary for deening the index. The index is computed by continuing the equations to a system containing a lower dimensional invariant manifold. 1. Introduction Reaction-diiusion equations are used extensively as continuous space-time models for interacting and diiusing chemical and biological species, combustion, phase transitions, and neurophysiology. In mathematical ecology the interactions between individuals can take the form of competition for limited resources, predator-prey interactions, or mutualistic relationships, where the diiusion terms model the migration of a species (see 7], 20]). Though these equations are relatively simple, they can exhibit a variety of interesting spatial and spatio-temporal patterns, including wave fronts and wave pulses. The general system