Analysis of steady states to classes of reaction diffusion equations and systems
- UNCG Author/Contributor (non-UNCG co-authors, if there are any, appear on document)
- Ananta Acharya (Creator)
- Institution
- The University of North Carolina at Greensboro (UNCG )
- Web Site: http://library.uncg.edu/
- Advisor
- Shivaji Ratnasingham
Abstract: The focus of this dissertation is to study positive solutions for classes of nonlinear steady state reaction diffusion equations and systems. In particular, we consider four focuses. In Focus 1, we establish sufficient conditions on the reaction term for which the bifurcation diagram for positive solutions for a nonlinear reaction diffusion equation is S-shaped. In Focus 2, we extend the study in Focus 1 for classes of coupled reaction diffusion equations. In Focus 3, we analyze the classes of diffusive Lotka-Volterra competition models in fragmented patches. Finally, in Focus 4, we use the finite element method for the numerical computation of bifurcation diagrams in dimension N = 2 for examples in Focus 1 and Focus 3. We establish analytical results in any dimension, namely, we establish existence, nonexistence, multiplicity, and uniqueness results. Our existence and multiplicity results are achieved by the method of sub-supersolutions. Via computational methods we also obtain approximate bifurcation diagrams describing the structure of the steady states. Namely, we obtain these bifurcation diagrams via a 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 of steady states to classes of reaction diffusion equations and systems.
Analysis of steady states to classes of reaction diffusion equations and systems
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Created on 8/1/2023
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Additional Information
- Publication
- Dissertation
- Language: English
- Date: 2023
- Keywords
- Ecological modeling, Lotka Volterra, Sigma shaped, Steady states
- Subjects
- Ecology $x Mathematical models
- Reaction-diffusion equations
- Lotka-Volterra equations