Expanding speed of the habitat for a species in an advective environment

Recently, Gu et al. [7,8] studied a reaction-diffusion-advection equation $u_t =u_{xx} -β u_x + f(u)$ in $(g(t), h(t))$, where $g(t)$ and $h(t)$ are two free boundaries satisfying Stefan conditions, $f(u)$ is a Fisher-KPP type of nonlinearity. When $β ∈ [0,c_0)$, where $c_0 := 2\sqrt{f'(0)}$, they found that for a spreading solution $(u,g,h)$, $h(t)/t \to c^*_r (β)$ and $g(t)/t \to c^*_l (β)$ as $t \to ∞$, and $c^*_r (β) > c^*_r(0) = - c^*_l (0) > - c^*_l (β) >0$. In this paper we study the expanding speed $C^*(β) :=c^*_r(β) - c^*_l (β)$ of the habitat $(g(t), h(t))$, and show that $C^*(β)$ is strictly increasing in $β ∈ [0,c_0)$. When $β ∈ [c_0, β^*)$ for some $β^*>c_0$, [8] also found a virtual spreading phenomena: $h(t)/t \to c^*_r(β)$ as $t\to∞$, and a back forms in the solution which moves rightward with a speed $β - c_0$. Hence the expanding speed of the main habitat for such a solution is $C^*(β) := c^*_r(β) -[β -c_0]$. In this paper we show that $C^*(β)$ is strictly decreasing in $β∈ [c_0, β^*)$ with $C^*(β^* -0)=0$, and so there exists a unique $β_0∈ (c_0, β^*)$ such that the advection is favorable to the expanding speed of the habitat if and only if $β∈ (0,β_0)$.