On the Lotka-Volterra competition system with dynamical resources and density-dependent diffusion.

ABSTRACT

In this paper, we consider the following Lotka-Volterra competition system with dynamical resources and density-dependent diffusion in a bounded smooth domain [Formula see text] with homogeneous Neumann boundary conditions, where the parameters [Formula see text], [Formula see text], [Formula see text], [Formula see text] ([Formula see text]) are positive constants, m(x) is the prey's resource, and the dispersal rate function [Formula see text] satisfies the the following hypothesis: [Formula see text], [Formula see text] on [Formula see text] and [Formula see text]. When m(x) is constant, we show that the system (*) with has a unique global classical solution when the initial datum is in functional space [Formula see text] with [Formula see text]. By constructing appropriate Lyapunov functionals and using LaSalle's invariant principle, we further prove that the solution of (*) converges to the co-existence steady state exponentially or competitive exclusion steady state algebraically as time tends to infinity in different parameter regimes. Our results reveal that once the resource w has temporal dynamics, two competitors may coexist in the case of weak competition regardless of their dispersal rates and initial values no matter whether there is explicit dependence in dispersal or not. When the prey's resource is spatially heterogeneous (i.e. m(x) is non-constant), we use numerical simulations to demonstrate that the striking phenomenon "slower diffuser always prevails" (cf. Dockery et al. in J Math Biol 37(1)61-83, 1998; Lou in J Differ Equ 223(2)400-426, 2006) fails to appear if the non-random dispersal strategy is employed by competing species (i.e. either [Formula see text] or [Formula see text] is non-constant) while it still holds true if both d(w) and [Formula see text] are constant.