Dynamics of a reaction-diffusion-advection model for two competing species

Yuan Lou,Xinfu Chen,King-Yeung Lam

doi:10.3934/dcds.2012.32.3841

Abstract

We study the dynamics of a reaction-diffusion-advection model for two competing species in a spatially heterogeneous environment. The two species are assumed to have the same population dynamics but different dispersal strategies: both species disperse by random diffusion and advection along the environmental gradient, but with different random dispersal and/or advection rates. Given any advection rates, we show that three scenarios can occur: (i) If one random dispersal rate is small and the other is large, two competing species coexist; (ii) If both random dispersal rates are large, the species with much larger random dispersal rate is driven to extinction; (iii) If both random dispersal rates are small, the species with much smaller random dispersal rate goes to extinction. Our results suggest that if both advection rates are positive and equal, an intermediate random dispersal rate may evolve. This is in contrast to the case when both advection rates are zero, where the species with larger random dispersal rate is always driven to extinction.

Highlights

In this paper we consider ut= ∇ · (μ∇u − αu∇m) + u(m(x) − u − v) in Ω × (0, ∞), vt= ∇ · (ν∇v − βv∇m) + v(m(x) − u − v) μ ∂u ∂n − αu ∂m ∂n = ν∂v ∂n β v u(x, 0) = u0(x) ≥ 0, v(x, 0) = v0(x) ≥ 0, in Ω × (0, ∞), on ∂Ω × (0, ∞), (1)
Theorem 1.4 implies that if one of the random diffusion rates is suitably large and the other is sufficiently small, both species can coexist. This result is in the same spirit as Theorem 1.3, though the proofs are rather different
Combining with (9), we have proved φ1 ≥ θμ,α in Ω1 = Ω

Summary

In this paper we consider

Given any α > 0, β > 0, there exists ν2 such that if 0 < ν < ν2, for sufficiently large μ, both (θμ,α, 0) and (0, θν,β) are unstable, and (1) has at least one asymptotically stable coexistence steady state. Theorem 1.3 says that for any given advection rates, if one of the random diffusion rates is small and the other is sufficiently large, two species can coexist! Theorem 1.4 implies that if one of the random diffusion rates is suitably large and the other is sufficiently small, both species can coexist. This result is in the same spirit as Theorem 1.3, though the proofs are rather different.

Then we have

Now w