Volume 7, Number 1, Spring 1999
A COMPETING REACTION-DIFFUSION SYSTEM
WITH SMALL CROSS-DIFFUSIONS
Abstract. This paper is concerned with a coupled system
of reaction-diffusion equations modeling the competition
of two species with cross diffusion. We discuss the existence of
positive steady-state solutions in relation to the intrinsic birth
rates a1, a2 of species when the cross-diffusion coefficients are
small. Our main goal is to determine the set Λ of ( a1, a2) so
that the coupled system possesses positive steady-state solutions.
It is shown that Λ is an unbounded, connected set in
R+2 whose boundary consists of two monotone nondecreasing
curves a1 = H1(a2) and a2 = H2(a1). For ( a1, a2 ) inside Λ,
the system has positive solutions, and for (a1,a2) outside Λ,
it has only the trivial and semi-trivial solutions.