The Influence Of Refuge And Harvesting For Four Kinds Of Population Systems

This paper consists of four parts:Firstly, a two species Lotka-Volterra competitive system incorporating a constant proportion of refuge for one species is studied. If the Lotka-Volterra competitive system is globally stable (in the absence of refuge), then we show that the refuge only changes the position of the unique positive equilibrium and retains the stable property; If the Lotka-Volterra competitive system is extinct or bistable (in the absence of the refuge), by choosing enough large refuge, we can make the system become globally stable, however, if refuge size is small enough, the system is still extinct or bistable.Secondly, a Lotka-Volterra competitive system incorporating a constant proportion of refuge for one species and independent harvesting in either species is studied. We show that the unique positive equilibrium of the system is locally stable and globally stable, which means that enough large refuge could make two species coexist; After that, detailed analysis about the influence of harvesting is carried out, our study shows that, under some suitable restriction, harvesting has no influence on the final density of one species; The possibility of existence of a bionomic equilibrium is discussed, and the different influence of refuge for bionomic equilibrium of two kinds of systems is studied; At last, the optimal harvesting policy is obtained by using the Pontryagin’s maximal principle.Thirdly, a nonautonomous discrete modified Leslie-Gower predator-prey system with a prey refuge is studied. Sufficient conditions which guarantee the permanence and global attractivity of the system are obtained. Our results show that enough large prey refuge could keep the system persistent, and prey species will be driven to extinction if the refuge is too small, however predator species still keep persistent due to the affordable of other food resource.Finally, we study a nonautonomous ratio-dependent Leslie-Gower system incorporating a prey refuge. By applying the comparison theorem of differential equations and constructing a suitable Lyapunov function, a set of sufficient conditions which ensure the persistent property and global attractivity of the system is obtained. Also, by applying Fluctuation Lemma, a set of sufficient conditions which ensure the extinction of the prey species is obtained. Our study shows that due to the alternative food resource predator species is always permanent, which means that prey refuge has no influence on the persistent property of predator species. However, refuge plays important role on the persistent property of the prey species:enough large prey refuge could keep the prey species persistent, while small refuge could lead to the extinction of prey species.