  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.
