RESEARCH
RESEARCH INTERESTS
I am interested in studying steady state reaction diffusion equations arising in mathematical biology and elliptic partial differential equations and systems.
Prof. Ratnasingham Shivaji was my advisor. I studied steady state reaction diffusion equations under his direction.
My current research focus is on studying the structure of positive solutions of steady state reaction diffusion equations which arise in ecology. Our studies include logistic growth models, weak Allee growth models, and grazing models where a parameter influences the equation as well as the boundary conditions. Many of the ecological research projects I worked on have been supported by my advisor's NSF grant. I am also interested in studying the existence multiplicity and uniqueness of positive solutions to elliptic partial differential equations and systems.
Mathematical Biology
Mathematical biology is the use of mathematical tools to study biological systems. Mathematical biology focuses on the mathematical representation and modeling of biological processes, using techniques and tools of applied mathematics. Describing systems in a quantitative manner means their behavior can be better studied, and hence properties can be predicted that might not be evident to the experimenter. This requires precise mathematical models. Mathematical biology employs many components of mathematics, and has contributed to the development of new techniques. In particular, elliptic partial differential equations can be used to model the population density of a certain species living in a habitat surrounded by a hostile environment. These models can be studied and make predictions on population persistence and understand the effects of the matrix hostility on the steady states.
A weak Allee model arising in population dynamics
A steady state reaction diffusion equation where a parameter influences the equation as well as the boundary conditions.
A grazing model arising in ecological systems
A recent research work: An establishment of a Sigma-shaped bifurcation curve for a class of reaction-diffusion equations.
Conference Talks
Presenting a talk at JMM, Boston, MA, January 2023.
University of Massachusetts-Amherst, Amherst, MA (Fall Eastern Sectional Meeting - October 2022)
Plenary Talks
UCM Scholars Symposium, University of Central Missouri, Warrensburg, MO, April 2024.
Colloquium/Seminar Talks
Appalachian State University, North Carolina, Boone, NC, November 2022.
Wake Forest University, North Carolina, Winston Salem, NC, September 2023.
Invited Talks
American Mathematical Society (AMS) Spring Southeastern Sectional Meeting, Auburn University, Auburn, AL, March 2019.
AMS Spring Southeastern Sectional Meeting, University of Florida, Gainesville, FL, November 2019.
AMS Annual Meeting, Colorado Convention Center, Denver, CO, January 2020.
AMS Fall Southeastern Sectional Meeting (virtual meeting), October 2020.
AMS Fall Western Sectional Meeting (virtual meeting), October 2020.
AMS Annual Meeting (virtual meeting), January 2021.
AMS Spring Eastern Sectional Meeting (virtual meeting), March 2021.
AMS Fall Central Virtual Sectional Meeting (virtual meeting), October 2021.
AMS Fall Eastern Sectional Meeting, University of Massachusetts-Amherst, Amherst, MA, October 2022.
AMS Fall Southeastern Sectional Meeting, the University of Tennessee at Chattanooga, Chattanooga, TN, October 2022.
AMS Annual Meeting, John B. Hynes Veterans Memorial Convention Center, Boston, MA, January 2023.
AMS Fall Central Sectional Meeting, Creighton University, Omaha, NE, October 2023.
AMS Annual Meeting, Moscone Center, San Francisco, CA, January 2024.
Contributed Talks
The 15th Annual UNCG Regional Mathematics and Statistics Conference, University of North Carolina at Greensboro, Greensboro, NC, November 2019.
The 39th Southeastern-Atlantic Regional Conference on Differential Equations, Embry-Riddle Aeronautical University, Daytona Beach, FL, October 2019.
The 14th Annual UNCG Regional Mathematics and Statistics Conference, University of North Carolina at Greensboro, Greensboro, NC, November 2018.
The 38th Southeastern-Atlantic Regional Conference on Differential Equations, University of North Georgia, Gainesville, GA, October 2018.
9th International Online Conference on Mathematical Analysis, Differential Equation \& Applications (MADEA 9), June 2021.
UNC Greensboro PDE Conference of 2021, University of North Carolina at Greensboro, Greensboro, NC, July 2021.
The 40th Southeastern-Atlantic Regional Conference on Differential Equations, NC State University, Raleigh, NA, November 2022.
MAA Southeastern Section Meeting, Coastal Carolina University, Conway, NC, March 2023.
MAA Missouri Section Meeting, William Jewell College, Liberty, MO, April 2024.
International Conference on Differential Equations (ICDE), Florida Gulf Coast University, Fort Myers, FL, June 2024.
International Conference Talks
9th International Online Conference on mathematical analysis, differential equation & applications (MADEA 9), June 2021.
10th international eurasian conference on mathematical sciences and applications (IECMSA), August 2021.
5th International Conference on Mathematics (ICOM 2021), December 2021.
III International Conference on Mathematics and its Applications in Science and Engineering (ICMASE 2022), July 2022.
SIMIODE EXPO 2024 online international conference, February 2024.
SOME RECORDED TALKS
AMS March 2021.mp4Presenting a talk at AMS Spring Eastern Sectional Meeting (virtual), March 2021.
20191103_093409.mp4Presenting a talk at AMS Spring Southeastern Sectional Meeting, University of Florida, Gainesville, FL, November 2019.
SEARCDE Talk.MOVPresenting a talk at 38th SEARCDE, University of North Georgia, Gainesville, October 2018
AMS Talk (1).mp4Presenting a talk at AMS Fall Southeastern Sectional Meeting (virtual), October 2020.
Talk.mp4Presenting a talk at AMS Fall Western Sectional Meeting (virtual), October 2020.
COLLABORATORS
LIST OF PUBLICATIONS
Refereed Publications
N. Fonseka, R. Shivaji, B. Son and K. Spetzer, Classes of reaction-diffusion equations where a parameter influences the equation as well as the boundary condition, J. Math. Anal. Appl., 476 (2019), no. 2, 480-494.
N. Fonseka, J. Goddard II, Q. Morris, R. Shivaji and B. Son, On the effects of the exterior matrix hostility and a U-shaped density-dependent dispersal on a diffusive logistic growth model, Discrete Contin. Dyn. Syst. Ser. S, 2020, doi: 10.3934/dcdss.2020245.
J. T. Cronin, N. Fonseka, J. Goddard II, J.Leonard and R. Shivaji, Modeling the effects of density-dependent emigration, weak Allee effects, and matrix hostility on patch-level population persistence, Mathematical Biosciences and Engineering (2019), 17(2) : 1718-1742.
N. Fonseka, J. Machado and R. Shivaji, A study of logistic growth models influenced by the exterior matrix hostility and grazing in an interior patch, Electron J. Qual. Theory Differ. Equ. (2020), No. 17, 1-11.
N. Fonseka, A. Muthunayake, R. Shivaji and B. Son, Singular reaction-diffusion equations where a parameter influences the reaction term and the boundary condition, Topological Methods in Nonlinear Analysis, Vol 57, No 1 (2021), 221-242.
N. Fonseka, J. Goddard, R. Shivaji and B. Son, A diffusive weak Allee effect model with U-shaped emigration and matrix hostility, Discrete Contin. Dyn. Syst. Ser. B, 2020, doi: 10.3934/dcdsb.2020356.
A. Acharya, N. Fonseka and R. Shivaji, Analysis of reaction-diffusion systems where a parameter influences both the reaction terms as well as the boundary, Boundary Value Problems, 2021 (1), 1-8.
A. Acharya, N. Fonseka, J. Quiroa, and R. Shivaji, Sigma-shaped bifurcation curves, Advances in Nonlinear Analysis, Volume 10 (2021), https://doi.org/10.1515/anona-2020-0180.
A. Acharya, N. Fonseka, and R. Shivaji, Sigma-shaped bifurcation curves for classes of elliptic systems, Discrete Contin. Dyn. Syst. Ser. S, Accepted 2022.
N. Fonseka, D. Nichols, J. Goddard, K. Henderson, and R. Shivaji, Modeling effects of matrix heterogeneity on population persistence at the patch-level, Mathematical Biosciences and Engineering, Accepted 2022.
A. Acharya, N. Fonseka, J. Goddard, K. Henderson, R. Shivaji, On the effects of density-dependent dispersal on ecological models with logistic and Allee effect type growth terms, Discrete Contin. Dyn. Syst. Ser. B., Doi: 10.3934/dcdsb.2023142.
Manuscripts Submitted
A. Acharya, N. Fonseka, K. Henderson, and R. Shivaji, Sigma-shaped bifurcation curves influenced by nonlinear boundary conditions for classes of reaction diffusion systems, submitted 2023.
N. Fonseka, S.H. Rasouli, and A. Shabanpour, On the existence and multiplicity of positive solutions to classes of steady state reaction diffusion systems with multiple parameters, submitted 2023.
Manuscripts in Preparation
A. Acharya, N. Fonseka, J. Goddard, R. Shivaji, and B. Son, On the effects of the exterior matrix hostility and a hump-shaped density-dependent dispersal on a diffusive logistic growth model.
N. Fonseka, R. Shivaji, On the existence of $2n+1$ and $2n+2$ solutions for classes of steady state reaction diffusion equations.
N. Fonseka, A $2n+1$ solution theorem for classes of reaction diffusion systems with linear boundary conditions.
N. Fonseka, On the existence of an odd multiplicity of solutions for classes of p-Laplacian problems with Dirichlet boundary conditions.
N. Fonseka, On the existence of an odd multiplicity of solutions for classes of p-Laplacian problems with Robin boundary conditions.
A. Acharya, N. Fonseka, J. Goddard, K. Henderson, V. Munoz, and R. Shivaji, On the effects of density dependent dispersal on a diffusive logistic growth model involving grazing.
A. Acharya, N. Fonseka, K. Henderson, and R. Shivaji, Sigma-shaped bifurcation curves for classes of reaction diffusion systems with non-linear boundary conditions.
Electronic files of some of the publications
Fonseka_1_JMAA_.pdfFonseka_2_DCDS_S_.pdfFonseka_3_MBE_.pdfFonseka_4_EJQTDE_.pdfFonseka_5_TMNA_.pdfFonseka_6_DCDS_B_.pdfFonseka_7_BVP_.pdfFonseka_8_ANONA_.pdf
Poster presentation SIAM – PD, March 2022.
