Theses and Dissertations

Sarath Sasi

Issuing Body

Mississippi State University

Advisor

Shivaji, Ratnasingham

Committee Member

Neumann, Michael M.

Committee Member

Miller, Len

Committee Member

Lim, Hyeona

Committee Member

Yarahmadian, Shantia

Date of Degree

8-11-2012

Document Type

Dissertation - Open Access

Major

Mathematical Sciences

Degree Name

Doctor of Philosophy

College

College of Arts and Sciences

Department

Department of Mathematics and Statistics

Abstract

In this thesis we study two reaction-diffusion models that have been used to analyze the existence of alternate stable states in ecosystems. The first model describes the steady states of a logistic growth model with grazing in a spatially homogeneous ecosystem. It also describes the dynamics of the fish population with natural predation. The second model describes phosphorus cycling in stratified lakes. The same equation has also been used to describe the colonization of barren soils in drylands by vegetation. In this study we discuss the existence of multiple positive solutions, leading to the occurrence of S-shaped bifurcation curves. We were able to show that both the models have alternate stable states for certain ranges of parameter values. We also introduce a constant yield harvesting term to the first model and discuss the existence of positive solutions including the occurrence of a Sigma-shaped bifurcation curve in the case of a one-dimensional model. Again we were able to establish that for certain ranges of parameter values the model has alternate stable states. Thus we establish analytically that the above models are capable of describing the phenomena of alternate stable states in ecological systems. We prove our results by the method of sub-super solutions and quadrature method.

URI

https://hdl.handle.net/11668/20175

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