The focus of this thesis is to study long term solutions for classes of steady state reaction diffusion equations. In particular, we study reaction diffusion models arising in mathematical ecology. We study how the patch size affects the existence, nonexistence, multiplicity, and uniqueness of the steady states. Our focus is also to study how various forms of density dependent emigrations at the boundary, and the effective matrix hostility, affect steady states. These considerations lead to the study of various forms of nonlinear boundary conditions. Further, they lead to the study of reaction diffusion models where a parameter (related to the patch size) gets involved in the differential equation as well as the boundary conditions.
We establish analytical results in any dimension, namely, establish existence, nonexistence, multiplicity, and uniqueness results. Our existence and multiplicity results are achieved by a method of sub-supersolutions and uniqueness results via comparison principles and a-priori estimates.
Via computational methods, we also obtain exact bifurcation diagrams describing the structure of the steady states. Namely, we obtain these bifurcation diagrams via a modified quadrature method and Mathematica computations in the one-dimensional case, and via the use of finite element methods and nonlinear solvers in Matlab in the two-dimensional case.
This dissertation aims to significantly enrich the mathematical and computational analysis literature on reaction diffusion models arising in ecology.
|Commitee:||Chhetri, Maya, Goddard, Jerome, II, Lewis, Tomas, Robinson, Stephen B., Zhang, Yi|
|School:||The University of North Carolina at Greensboro|
|Department:||College of Arts & Sciences: Mathematics and Statistics|
|School Location:||United States -- North Carolina|
|Source:||DAI-B 82/3(E), Dissertation Abstracts International|
|Keywords:||Boundary value problems, Density dependent emigration, Ecology, Mathematical biology, PDE, Reaction diffusion equations|
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