Dynamical behavior of solutions of a reaction-diffusion-advection model with a free boundary

This paper is devoted to study the population dynamics of a single species in a one-dimensional environment which is modeled by a reaction-diffusion-advection equation with free boundary condition. We find three critical values c(0), 2 and beta(& lowast;) for the advection coefficient -beta with beta & lowast; > 2 > c(0) > 0, which play key roles in the dynamics, and prove that a spreading -vanishing dichotomy result holds when -2 <= beta <= c(0); a small spreading -vanishing dichotomy result holds when c(0) < beta < 2; a virtual spreading -transition -vanishing trichotomy result holds when 2 <= beta < beta(& lowast;); only vanishing happens when beta >= beta(& lowast;); a virtual vanishing -transition -vanishing trichotomy result holds when beta <= -2. When spreading or small spreading or virtual spreading happens for a solution, we make use of the traveling semi -wave solutions to give a estimate for the asymptotic spreading speed and asymptotic profile of the right front.