The Study Of A Predator-prey Model With Holling-Ⅱ Type Interactions And Mixed Boundary Conditions

This paper investigates a class of Holling-Ⅱtype predator-prey models with diffusions andmixed boundary conditions，in which the prey is subject to the homogeneous Neumann conditionand the predator to the homogeneous Robin boundary condition． According to theenvironment which the species survival in is homogeneous or not，namely，whether theparameter in the model is dependent on the spatial environment，we can divide the model intotwo types: the model in the homogeneous environment is a general predator-prey model,while the model in the inhomogeneous environment is refer to a control problem to the alienspecies u．In this paper, our work include：For the case that the environment is homogeneous，namely，the parameter in the model is not dependent on the spatial environment，by using thedegree theory， we proved that, under certain conditions, the model exists a positivesteady-state solution．While when the environment is inhomogeneous，namely，the parameterin the model is dependent on the spatial environment，we studied the local stability,uniqueness of the positive steady-state solution， and the asymptotic behavior of thenon-negative solution to the model．The contents of the paper are as follows．Chapter1introduces the research significance of biomathematical model， currentresearch situation，and main tasks of this paper．Chapter2，under the condition of the homogeneous environment， we studied theexistence of a positive steady-state solution to the model．Firstly，we discuss the necessarycondition of the existence of positive steady-state solution．Next, by using the index theoryand the topologic degree theory，we show that the model at least possesses a positivesteady-state solution under certain conditions．Further, we show the asymptotic behavior ofthe non-negative solution to the model．In Chapter3， we assume that the species u lives in a heterogeneous environment andthe growth rate of the prey satisfies that λ∈（0,λ₁^D）. By taking the birth rate of thepredator as the bifurcation parameter，the local bifurcation of its positive solutions whichemanates from the semi-trivial solution is analyzed．The stability and uniqueness of thesemi-trivial solution and the positive solution are studied and global bifurcation is discussed．In Chapter4， we assume that the species u lives in a heterogeneous environment， and the growth rate of the prey satisfies thatλ∈（λ₁^D,+∞）. We discuss the nonexistenceof the positive solution of the model in a heterogeneous environment and the dynamicalbehavior of the corresponding paroblic system．The last part is a summary of the paper，and there are also some open problems whichneed to be discussed in the future．...