Stochastic and Multiscale Modeling and Computation Seminar by Shalmali Bandyopadhyay: Bifurcation in Competing and Cooperating Species Models with Nonlinear Boundary Coupling

Speaker: Shalmali Bandyopadhyay, assistant professor of mathematics, University of Tennessee, Martin

 

Title: Bifurcation in Competing and Cooperating Species Models with Nonlinear Boundary Coupling

 

Abstract: We analyze a coupled system of parameter-dependent elliptic PDEs with Robin boundary conditions in which the boundary terms depend nonlinearly on the solutions. The system models two interacting species on a bounded domain, where the boundary interaction function takes two forms — competitive or cooperative — depending on a perturbation parameter. The qualitative behavior of solutions, including existence, nonexistence, and uniqueness, changes significantly depending on which interaction regime is in effect. Using sub-supersolution methods, Green's identities, and eigenvalue perturbation theory, we establish these results analytically. In the one-dimensional setting, we construct complete bifurcation diagrams via the quadrature method and a shooting algorithm, illuminating how coupling through the boundary drives qualitatively distinct solution behavior across the two regimes.