Global well-posedness, pattern formation and spiky stationary solutions in a Beddington–DeAngelis competition system

Abstract

This paper investigates a reaction-advection-diffusion system that describes the evolution of population distributions of two competing species in an enclosed bounded habitat. Here the competition relationships are assumed to be of the Beddington–DeAngelis type. In particular, we consider a situation where first species disperses by a combination of random walk and directed movement along with the population distribution of the second species which disperse randomly within the habitat. We obtain a set of results regarding the qualitative properties of this advective competition system. First of all, we show that this system is globally well-posed and its solutions are classical and uniformly bounded in time. Then we study its steady states in a one-dimensional interval by examining the combined effects of diffusion and advection on the existence and stability of nonconstant positive steady states of the strongly coupled elliptic system. Our stability result of these nontrivial steady states provides a selection mechanism for stable wavemodes of the time-dependent system. Moreover, in the limit of diffusion rates, the steady states of this fully elliptic system can be approximated by nonconstant positive solutions of a shadow system that admits boundary spikes and layers. Furthermore, for the fully elliptic system, we construct solutions with a single boundary spike or an inverted boundary spike, i.e., the first species concentrates on a boundary point while the second species dominates the remaining habitat. These spatial structures model the spatial segregation phenomenon through interspecific competitions. Finally, we perform some numerical simulations to illustrate and support our theoretical findings.