Traveling waves for a reaction-diffusion model with a cyclic structure

Abstract

In this paper, a reaction-diffusion model with a cyclic structure is studied, which includes the SIS disease-transmission model and the nutrientphytoplankton model. The minimal wave speed c * of traveling wave solutions is given. The existence of traveling semi-fronts with c > c * is proved by Schauder's fixed-point theorem. The traveling semi-fronts are shown to be bounded by rescaling method and comparison principle. The existence of traveling semi-front with c = c * is obtained by limit arguments. Finally, the traveling semi-fronts are shown to connect to the positive equilibrium by a Lyapunov function.2010 Mathematics Subject Classification. Primary: 35K57; Secondary: 35C07, 92B05.

Traveling wave solutions of a class of multi-species non-cooperative reaction–diffusion systems *

“…Subsequently, Ai et al [1] obtained a similar general existence result by an approach involving upper and lower solutions and Schauder's fixed point theorem. This and several other related works (see, for example, [21,35,36] and references therein) demonstrate that such an approach could be rather powerful to yield sharp existence results for certain non-cooperative systems, when combined with certain delicate constructions of upper and lower solutions.…”

“…These conditions cover the classical SIS systems with B(u) = b(K − u) or B(u) = bu(K − u) and the mass-action transmission rate f(u)v = αuv; one may refer to [22][23][24] for the epidemiological meanings of the parameters. The existence of traveling wave solutions of (1.1) in the case B(u) = b(K − u) has been established in [36]. One of the main difficulties in the existence proof lies in proving the boundedness of weak traveling wave solutions.…”

“…The aforementioned existence results in [1,11,19,35,36] on weak traveling wave solutions were established for two species prey-predator systems, and the system in [21] is one with several predators and a single prey species. In this paper, we further develop the approach in [1,21,35,36] for a more general class of prey-predator systems, which allow for multiple prey species in the system. The new existence result is given in theorem 2.1.…”