The Free Boundary Problem Of A Predator-prey Model With The Fear Effect

The predator-prey problem is one of the important phenomena in the population and is used to describe the relationship between species.The study of this biological phenomenon can help to better control the population and is of great significance in protecting species and maintaining the ecological balance.The free boundary problem of the predator-prey model describes the phenomenon that the activity space of species changes with time,and the predator-prey model with fear effect describes the situation that the presence of the predator will affect the birth rate of the prey.In this paper,we study the free boundary problem of the predator-prey model with fear effect,Beddington-De Angelis response function and common free boundary.What is unique is the fear effect and the introduction of the common freedom boundary.The difficulty is that the emergence of the fear effect will make it difficult to analyze the dynamic properties of the classical solution.In order to solve the influence of the fear effect,it is necessary to construct a new upper solution and then apply the comparison principle to prove the long-term behavior of the solution and the phenomenon of spreadvanishing.It is also more complicated to study the spread speed of free boundary in this complex model.Considering the actual situation of biology,the results obtained in this paper have more realistic value.The main contents include the use of Schauder theory,pL theory,maximum principle,Hopf boundary lemma in Nonlinear Parabolic Equations and contraction mapping principle to obtain the local existence uniqueness,boundedness and global existence of classical solutions.The spread-vanishing dichotomy between predator and prey is studied by using the comparison principle.In the case of spread,the boundedness of velocity of free boundary is discussed.The area of the initial active area is used as the critical condition to determine whether the species can continue to spread.