# Department of Mathematical Sciences

## Colloquium Friday, November 11: Nalin Fonseka, Carolina University

Join us this Friday at 1 PM in 103A Walker Hall to hear Dr. Nalin Fonseka speak on....

Modeling Effects of Matrix Heterogeneity on Population Persistence at the Patch Level

Abstract: We study the structure of positive solutions to steady state reaction diffusion equation of the form:

$\dpi{300} \bg_white \left\{ \begin{array}{r} -u''=\lambda u (1-u); \; (0,1) \\ -u'(0)+\sqrt{\lambda} \gamma_1 u(0)=0 \\ u'(1)+\sqrt{\lambda} \gamma_2 u(1)=0 \end{array} \right.$

where lambda > 0 is a parameter which encompasses patch size and gamma 1 and 2 are positive parameters related to the hostility at the boundaries 0 and 1, respectively. Note here that the parameter lambda influences both the equation and the boundary conditions. In this paper, we establish existence, nonexistence, and uniqueness results for this model. In particular, we establish exact bifurcation diagrams for this model, first when gamma 2 is fixed and gamma 1 is evolving, and then when the Dirichlet boundary condition is satisfied at x = 0 (u(0)=0) and gamma 2 is evolving. In each case, our results are established combining a quadrature method and the method of sub-super solutions. Finally, we present some numerical results that we obtained for this model. Here, we numerically simulate how the gamma parameters affect the minimum size for  lambda beyond which a positive solution exists. This model arises in the study of steady states for a population satisfying a logistic growth reaction and diffusing in a region surrounded by two exterior hostile matrices, and lambda is related to the minimum patch size for existence of a positive steady state.

Dr. Fonseka earned his doctorate at the University of North Carolina at Greensboro in 2020 under the direction of Dr. Ratnasingham Shivaji. Prior to joining UNC-Greensboro, Dr. Fonseka earned his master’s degree in mathematics at Eastern Illinois University in 2015, and his bachelor’s degree in 2009, at the University of Peradeniya, Sri Lanka. His research focuses on Differential Equations (Nonlinear Elliptic Boundary Value Problems) and Mathematical Ecology (Steady State Reaction-Diffusion Equations modeling population dynamics, including the density-dependent dispersal on the boundary and effects of exterior matrix hostility). In particular, he studies positive solutions to classes of steady-state reaction-diffusion equations where a parameter influences the equation as well as the boundary conditions. To date, he has collaborated with over 8 researchers including an ecologist.  His research work has resulted in articles published in several journals including the Journal of Mathematical Analysis and Applications, Discrete and Continuous Dynamical Systems, Series S, Topological Methods in Nonlinear Analysis, Advances in Nonlinear Analysis, and Mathematical Biosciences and Engineering.

 Published: Nov 7, 2022 10:34am