Abstract

This talk is concerned with some optimization problems that naturally arise in population biology. We consider the case where a biological species lives in an unfavorable habitat, and investigate an optimal policy to protect it with limited resource.

The first problem is about an optimal policy so that the species can survive, which can be formulated as the minimization of a principal eigenvalue for a linear eigenvalue problem. For an arbitrary domain, it is shown that any global minimizer must be of ``bang-bang" type.

When a biological species can survive, the next question is to maximize the total population, which is formulated as the maximization of the total mass of a positive steady state for a nonlinear parabolic PDE. It is shown that any global maximizer is again of ``bang-bang" type, but the maximizer is not necessarily the same as the minimizer of the first problem.