| Studying the mathematical problems arising from the complex biological phenomena is an important approach to study ecology and epidemiology.Mathematicians have been sparing no efforts to find appropriate ways to describe various biological phenomena and have achieved fruitful results.In this paper,a free boundary problem with local-nonlocal diffusions and different free boundaries and an age-structured hepatitis B virus infection model are studied,the dynamical behaviors are analyzed to better explain the biological phenomena in ecology and epidemiology.Chapter 1 introduces the research background and current status,as well as the main contents of this paper.Chapter 2 pays attention to a free boundary problem with local-nonlocal diffusions and different free boundaries.By Contraction Mapping Theorem、Fixed Points Theorem、the theory of ODEs and the parabolic theory,the existence,uniqueness,regularity and estimates of global solution is obtained.Chapter 3 continues to discuss the model proposed in Chapter 2.For the classi-cal L-V competition and predation models,with the help of eigenvalue theory of nonlo-cal diffusion operator,upper and lower solutions method and comparison principle,the spreading-vanishing dichotomy is established,the criteria governing spreading and van-ishing is given,and the long-time behaviors of the global solution is obtained.Moreover,when spreading occurs,the estimation of spreading speeds is obtained under some specific conditions.In Chapter 4,we focus on an age-structured hepatitis B virus infection model.Firstly,we obtain the well-posedness of the model by reformulating it as an abstract initial value problem.Then,the basic reproduction number?0is determined.In addition,the local stability of each steady states is established by linearizing the system and analyzing the corresponding characteristic equations.At last,we investigate the uniform persistence of the system when?0>1 and prove the global stability of each steady states by constructing appropriate Lyapunov functionals. |
