Propagation in multi-dimensional Fisher-KPP equations
- Advection-reaction-diffusion equations (ARD) are non-linear elliptic or parabolic equations that are used to model a variety of systems in natural sciences and engineering, ranging from biological (population dynamics) to chemical and astrophysical systems (reactions in a fluid). They describe the evolution of a normalized quantity, such as a local density of a population or a temperature, as a result of three processes: transport inside the domain (advection), creation or depletion due to an energy source (reaction), and spatial movement (diffusion). These equations exhibit interesting behaviors such as growth/decay, spreading and mixing. Over the past decades there has been significant progress towards understanding and quantifying the behavior of ARD equations. A great focus was shown in equations of biological invasions, more specifically in the phenomenon of spreading, which happens as a result of the invasion of an unstable equilibrium state by a positive stable one. There are two main threads of research in this field: (1) front-like spreading, which leads to the study of traveling fronts, their speeds of propagation and stability properties, and (2) multi-directional spreading, arising from initial data that have compact support or fast decay in every direction. In this thesis we study the problem of spreading rates for the Cauchy problem in multi-dimensional periodic Fisher-KPP equations. Localized initial data give rise to an invasion that will happen typically at different speeds and profiles in each unit direction, but independent on the size and distribution of the original mass. The main result will be on precise asymptotics for the location of level sets of solutions for these data. The trajectory of the thesis will be as follows: In the first part, after a brief introduction of the problem and past results, we study the linearized Dirichlet equation in a half space moving at a constant speed derived from the slowest traveling fronts. This provides intuition for the location of the fronts in the compactly supported case, as well as concrete bounds that can be compared to the solution of the original problem. In the second part we prove the main result by controlling the propagation of the Fisher-KPP solution using sub- and super-solutions constructed from the linearized fronts.
|Type of resource
|electronic; electronic resource; remote
|1 online resource.
|Stanford University, Department of Mathematics.
|Statement of responsibility
|Submitted to the Department of Mathematics.
|Thesis (Ph.D.)--Stanford University, 2017.
- © 2017 by Beniada Shabani
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).
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