Existence and stability of travelling wave solutions of competition models: A degree theoretic approach

Abstract

arise in various aspects of the scierices and engineering. In particular, they have been used to describe the evolution of interacting and diffusing species in mathematical ecology. This paper is concerned with competitive interactions; the purpose is to obtain large-amplitude, stable travelling wave solutions of (1) with arbitrary constant diffusion coefficients, and which are consistent with the principle of competitive exclusion. The existence and stability of large-amplitude solutions of systems is an important problem in nonlinear diffusion; see, for example, Fife [7]. We obtain such solutions for a robust class of nonlinear terms. This extends previous results of a similar nature which were proved with more restrictive hypotheses on f and g; see Gardner [ 11 ]. We also obtain the C O stability of such solutions. The method of proof employs topological degree. We construct a homotopy from a given field (f, g) to a new field, (f , ~) which is the gradient of a real-valued function. The results of [11] are then applied to obtain a travelling wave when the field is sufficiently near (f , ~). Next, we prove an a priori comparison theorem. In particular, we show that the components of the solution of (1) can be wedged between translates of the components of the travelling wave (modulo an exponentially decaying term in t), provided that the initial data are wavelike, (for a precise definition, see Theorem 2.3), and that a travelling front with monotone components exists. To this end, we