Inverse bifurcation problem, singular Wiener-Hopf equations, and mathematical models in ecology

A single-species population dynamics with dispersal in a spatially heteroge neous environment is modeled by a nonlinear reaction-diffusion equation wit h a potential term. To each nonlinear kinetics there corresponds a bifurcat ion curve that describes the relation between the growth rate and the centr al density of a steady-state population distribution. Our main concern is a n inverse problem for this correspondence. The existence of nonlinear kinet ics realizing a prescribed bifurcation curve is established. It is shown th at the freedom of such kinetics is of degree finite and even, depending onl y on the heterogeneity of the environment, and conversely that any nonnegat ive even integer occurs as the degree of freedom in some environments. A di scussion is also made on under what kind of environment the degree is equal to zero or is positive. The mathematical analysis involves the development of a general theory for singular multiplicative Wiener-Hopf integral equat ions.