The mathematical modeling of population is of wide interest to both applied mathematicians, biologists, and ecologists, as it allows us to answer fundamental questions about the conditions under which a population will persist. The study of differential equations models of this sort dates back to the 1930s, with the work of Robert Fischer, a statistician and biologist. While the basic model he proposed is now well understood, advances in biology continue to pose new and exciting mathematical questions.
At the core of this project is a desire to understand the conditions under which a population living in a patch will persist when the population exhibits nonlinear density dependent disperal either in the interior of the patch, on the boundary of the patch, or both.
We often talk about existence and uniqueness theorems for initial value problems in an introductory Ordinary Differential Equations class, and note that as long as the nonlinearity in the differential equation is relatively well behaved, you are guaranteed to have the existence of a unique solution. For boundary value problems, however, the results are quite different. Even with well behaved nonlinearities in your differential equation, you can still have nonexistence or multiplicity of solutions depending on your boundary conditions.
In this project, we would explore a particular example in which there is computation evidence to suggest the existence of multiple solutions for a problem. Furthermore, the choice of boundary condition seems to influence this multiplicity, as perturbations of the boundary condition will yield a problem which has a unique solution.
Preliminary Readings: [1],[2, Chapters 5&6]
The Fucik spectrum (first introduced by S. Fucik and E.N. Dancer in the late 1970s) is a nonlinear resonance set used to describe (in the ODE case) asymmetrical oscillating systems and (in the matrix case) numerical approximations to asymmetrical oscillating systems. In particular, we are interested in exploring conditions for the solvability of certain nonlinear ODE and matrix equations when parameters lie in the Fucik spectrum, which often turn out to be generalizations of the Fredholm Alternative for linear operators.
In the matrix case, the Fucik spectrum can be explicitly described in certain cases, and tools from linear algebra and multivariable calculus can be used to explore the solvability of certain nonlinear equations.
Preliminary Readings:[1], [2, Chapters 1&2], [3]
Paul Halmos