Qualitative studies of advective competition system with   Beddington-DeAngelis functional response
  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. 
Ling Jin, Qi Wang, Zengyan Zhang
math.AP