Qualitative studies of advective competition system with Beddington-DeAngelis functional response

TL;DR

This paper analyzes a reaction-advection-diffusion model of two competing species with Beddington-DeAngelis functional responses, establishing existence, stability, and spatial pattern formation of solutions, supported by numerical simulations.

Contribution

It provides new theoretical insights into the existence, stability, and boundary spike solutions of a coupled competition system with advective movement, extending understanding of spatial segregation phenomena.

Findings

Global existence and boundedness of solutions in multi-dimensional domains.

Stability criteria for nonconstant steady states and wave mode selection.

Construction of boundary spike solutions and analysis of spatial segregation patterns.

Abstract

This paper investigates a reaction-advection-diffusion system modeling interspecific competition between two species over bounded domains. The kinetic terms are assumed to satisfy the Beddington-DeAngelis functional responses. We consider the situation that first species disperse by a combination of random walk and directed movement along the population density of the second species which disperse randomly within the habitat. For multi-dimensional bounded domains, we prove the global existence and boundedness of time-dependent solutions. For one-dimensional finite domains, we study the effect of diffusion and advection on the existence and stability of nonconstant positive steady states to the strongly coupled elliptic system. In particular, our stability result of these nontrivial steady states provides a selection mechanism for stable wavemodes of the time-dependent system. In the limit of diffusion rates, we show that the steady states of this full elliptic system can be approximated by nonconstant positive solutions of a shadow system and then we construct boundary spike solutions to this shadow system. For the full elliptic system, we also investigate solutions with a single boundary spike or an inverted boundary spike, i.e., the first species concentrate on the boundary point while the second species dominate the whole habitat except the boundary point. These spatial structures can be used to model the spatial segregation phenomenon through interspecific competitions. Some numerical studies are performed to illustrate and support our theoretical findings.