The diffusive competition problem with a free boundary in heterogeneous time-periodic environment

In this paper, we consider the diffusive competition problem with a free boundary and sign-changing intrinsic growth rate in heterogeneous time-periodic environment, consisting of an invasive species with density u and a native species with density v. We assume that v   undergoes diffusion and growth in RNRN, and u   exists initially in a ball Bh0(0)Bh0(0), but invades into the environment with spreading front {r=h(t)}{r=h(t)}. The effect of the dispersal rate d1d1, the initial occupying habitat h0h0, the initial density u0u0 of invasive species u, and the parameter μ (see (1.3)) on the dynamics of this free boundary problem are studied. A spreading–vanishing dichotomy is obtained and some sufficient conditions for the invasive species spreading and vanishing are provided. Moreover, when spreading of u happens, some rough estimates of the spreading speed are also given.