The selection for dispersal: a diffusive competition model with a free boundary

This paper considers the population dynamics of an invasive species and a resident species, using a diffusive competition model in a radially symmetric heterogeneous environment with a free boundary. We assume that the resident species diffuses and expands in Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}^n}$$\end{document} , and the invasive species initially resides in a finite ball, but invades the environment with a spreading front that evolves as the free boundary. Our investigation aims to understand how the model dynamics are affected by the dispersal rate du\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${d_u}$$\end{document} , expansion capacity μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mu}$$\end{document} and initial number u0 of the invasive species. We show that a spreading–vanishing dichotomy exists and obtain the sharp criteria for spreading and vanishing by varying the parameters du, μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mu}$$\end{document} and u0. For the invasive species, we found an unconditional selection for slow dispersal rate, but a conditional selection for fast dispersal rate, that is, the selection for fast dispersal depends on the expansion capacity and initial number of the invasive species.