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

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发表日期：

2024年2月14日

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Ningkui Sun,
Di Zhang

摘要

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 $c_0$ $c_{0}$ , 2 and $\beta ^*$ $β^{*}$ for the advection coefficient $-\beta $ $- β$ with $\beta ^*>2>c_0>0$ $β^{*} > 2 > c_{0} > 0$ , which play key roles in the dynamics, and prove that a spreading-vanishing dichotomy result holds when $-2<\beta \leqslant c_0$ $- 2 < β ⩽ c_{0}$ ; a small spreading-vanishing dichotomy result holds when $c_0<\beta <2$ $c_{0} < β < 2$ ; a virtual spreading-transition-vanishing trichotomy result holds when $2\leqslant \beta <\beta ^*$ $2 ⩽ β < β^{*}$ ; only vanishing happens when $\beta \geqslant \beta ^*$ $β ⩾ β^{*}$ ; a virtual vanishing-transition-vanishing trichotomy result holds when $\beta \leqslant -2$ $β ⩽ - 2$ . 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.