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Free boundary problems of the competition model with sign-changing coefficients in heterogeneous time-periodic environment

In this paper we consider the diffusive competition model with free boundary in the heterogeneous time-periodic environment, in which the variable intrinsic growth rates of invasive and native species may change signs and be "very negative" in a "suitable large region" . The main objective is to understand the asymptotic behavior of the two competing species spreading via a free boundary. We first establish a spreading-vanishing dichotomy, namely the two species either successfully spread to the right-half-space as time $t$ goes to infinity and survive in the new environment, or they fail to establish and will extinct in the long run. Then we study the long time behavior of solution, sharp criteria for spreading and vanishing, and estimate of the asymptotic spreading speed of the free boundary.