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 $\Omega \subset {{\mathbb {R}}^2}$ with homogeneous Neumann boundary conditions, where the parameters $\mu$, $a_{i}$, $b_{i}$, $c_{i}$ ($i=1,2$) are positive constants, m(x) is the prey’s resource, and the dispersal rate function $d_{i}(w)$ satisfies the the following hypothesis:

• $d_{i}(w)\in C^2([0,\infty ))$, $d_{i}'(w)\le 0$ on $[0,\infty )$ and $d(w)>0$.

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 $W^{1,p}(\Omega )$ with $p>2$. 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 $d_1(w)$ or $d_2(w)$ is non-constant) while it still holds true if both d(w) and $d_2(w)$ are constant.