Existence for a Diffusing Population with Discrete Returns to the Interior

We examine a one-dimensional steady-state diffusion model for a biological population in which individuals collected either throughout the domain or from one endpoint of the domain are returned to the population at a discrete set of locations. Specifically, the modeling equations are u Žx. fŽx, uŽx.. ŽÝni 1 i iŽx..gŽH01 Žs.uŽs. ds. 0 in the former case and u Žx. fŽx, uŽx.. ŽÝni 1 i iŽx..gŽu Ž0.. in the latter case, where the function g specifies the rate of return and f is the usual net population growth term. With Dirichlet boundary conditions imposed at both ends of the domain, the topological transversality theorem is used to show that a positive solution exists provided g does not increase too rapidly.