Some Free Boundary Problems From Ecology And Epidemiology

Free boundary problems have been widely used in the study of ecology and infectious diseases.This doctoral dissertation summarizes the research methods of these problems and explains the phenomena in real life and production by studying several free boundary problems in ecology and infectious diseases.The main concerns of free boundary problems are the suitability of the global solution,the spreading and vanishing of long time behavior,the spreading and vanishing dichotomy,the criteria for spreading and vanishing and the spreading speed of free boundary and the shape of propagation profile when the spread occurs.This paper takes the concerned problems as the mainline and takes the free boundary problem with degenerate diffusion term and the free boundary problem with general diffusion term as examples.At first,we study the well posedness of the solution to the free boundary problem with degenerate diffusion term.We mainly discuss two kinds of problems,one has different regions,and the other shares the same region.For the first one,we mainly discuss an SI model with the logistic term,in which the distribution area of the infected I has free boundary boundaries.At the initial time,the susceptible S is distributed in the whole space,and the distribution area of the infected I is bounded.With the increase of time,the distribution area of the infected will be enlarged.For the later one,we study a general form.Although they all have degenerate diffusion term,because of the different distribution area,the proof of demonstration differ.For these two kinds of problems,we obtain the existence,uniqueness and regularity of global solutionsSecondly,we continue to discuss the SI model with degenerate diffusion term and logistic term,which has been talked about in the last chapter,and study its dynamic properties.We give the spreading and vanishing dichotomy,the criteria for spreading and vanishing and the long-time behavior of the solution when the spreading and vanishing happens.Finally,we use numerical simulation to explain our results and estimate the range of the critical value of the moving parameter through numerical simulation.Thirdly,the free boundary problem of predator-prey model with degenerate diffusion term and the age structure is studied.The slow moving of prey relative to predator can be ignored,so the diffusion rate of prey is 0,and its distribution area is the whole space.The predator has age structure,which consist of the young predator and the adultpredator.Because the young rely on the adult,we get an ordinary differential equation and two partial differential equations with the same free boundary.We get the existence,uniqueness and regularity of the global solution,the spreading and vanishing dichotomy,the criteria for spreading and vanishing and the long time behavior in vanishing case.Finally,the spreading speed of the free boundary to weak competition model of the Lotka Volterra type is studied.The two competings share the same left boundary,but the right is different,and the two different free boundaries move in the same direction.It is well known that the two competings will coexist under the weak competition.In this part,we use the comparison principle to construct the fine upper and lower solutions,and get the spreading speed of the fast propagation species and the shape of the propagation profile after complicated and detailed calculation and estimation.