Reaction-Diffusion Equations With Free Boundary And Their Application

Many natural phenomena, such as heat conduction in Physics, change of sub-stance concentration in chemical reaction and biological invasion in Biology, could lead to reaction-diffusion equation, which is a typical class of semi-linear parabolic partial differential equations. The studies about the reaction-diffusion equations with free boundary are just the one of important aspect in reaction-diffusion equa-tions researches.The current dissertation focuses on the theoretical applications of the reaction-diffusion equations with free boundary in population ecology (containing epidemi-ology), and tries to apply the theoretical framework established by some pioneering works to investigate and analyse three typical and widely used models, i.e., the L-V competition system with two similar species population, the L-V competition sys-tem in a radially symmetric setting and the partially degenerate reaction-diffusion system.Firstly, we study the asymptotic dynamics of a L-V competition system with two similar species, of which one is invasive and the other is resident and free bound-ary. By considering the population dynamics of the invasive species, we show that a spreading-vanishing dichotomy holds, and obtain the sharp criteria for the spread-ing and vanishing. The sharp criteria mean that a positive threshold D* exists such that if 0< d1≤D*, where d1 is the dispersal rate of the invasive species, then the spreading of the invasive species always occurs, however, if d1>D*, only the one with strong expansion capacity or large initial function could successfully spread, or else it will finally vanish. Moreover, the well-posedness of the two species population L-V competition system with free boundary and the associated principal eigenvalue problem are also be discussed.Secondly, we deals with the population dynamics of an invasive species and a resident species in the third chapter, which are modeled as a diffusive L-V competi- tion system in a radially symmetric setting with a free boundary. We first distinguish the problem according to the characteristics of the invasive species to the inferior case, the superior case and the weak competition case, and then investigate the pop-ulation dynamics of each case. A spreading-vanishing dichotomy is established and some criteria are defined to distinguish the dichotomy, while some rough estimates are also given of the asymptotic spreading speed when spreading occurs. Our results show that the invasive species inevitably vanishes if it is inferior; if it is superior, then only the one with strong expansion capacity or large initial function will suc-cessfully spread and wipe out the resident species, or else it will vanish; lastly, if it is weak competition, then only the one with strong expansion capacity or large initial function will could successfully spread and coexist with the resident species, or else it will vanish.Finally, the fourth chapter is contributed to the spreading frontiers in the par-tially degenerate reaction-diffusion systems with a mobile and a stationary com-ponent, which are described by a class free boundary condition. To investigate the dynamics of the spreading frontiers, we first present the well-posedness, dis-cuss the associated principal eigenvalue problems and obtain some needful results. Furthermore, by using some constructive methods, a spreading-vanishing dichoto-my is established and some sharp criteria are defined to distinguish the dichotomy. At last, as applications, we consider a man-environment-man epidemic model, a reaction-diffusion model with a quiescent stage in population dynamics and a spa-tial propagation model of West Nile virus, and obtain their asymptotic dynamics.