A Class Of The Prey–Predator Model With A Free Boundary In Time–Periodic Environment

In the nature world, the environment changes with the seasons, so some ecological phenomena change with time periodically. Hence, considering the time–periodic phenomenon has a strong theoretical and practical significance. When the environment is bad for the species’ survival and reproduction, their intrinsic growth rate will be negative. So it is also important to consider that the intrinsic growth rate changes sign. In the process of biological invasion, the species’ boundary varies with the time. This motivates previous researchers to introduce the free boundary problem. Hence, studying a class of the prey–predator model with a free boundary and sign–changing coefficient in time–periodic environment is more realistic than the corresponding periodic problem on the fixed region.We consider the prey–predator model with sign–changing periodic intrinsic growth rate in heterogeneous time–periodic environment, where the prey species lives in the whole space but the predator species lives in a small region enclosed by a changeable boundary. Firstly, we get the global existence and uniqueness of the solutions. Then, we prove a spreading and vanishing dichotomy. Sharp criteria for spreading and vanishing with h0, d1, μ and u0(x) as variable factors are obtained, where h0, d1, μ and u0(x) are the initial habitat size, the dispersal rate,expansion capacity and initial population size of the predator species, respectively.Moreover, some rough estimates of the asymptotic spreading speed are also given when spreading occurs.The results show that, if the initial region is large enough, the predators will spread successfully. If the initial region is small, only the one with strong expansion capacity or large initial function will successfully spread. In order to control some pest species, we always put one kind of their natural enemies. The results of this paper have guiding significance to pest control.