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

A single-species population dynamics with dispersal in a spatially heterogeneous environment is modeled by a nonlinear reaction-diffusion equation with a potential term. To each nonlinear kinetics there corresponds a bifurcation curve that describes the relation between the growth rate and the central density of a steady-state population distribution. Our main concern is an inverse problem for this correspondence. The existence of nonlinear kinetics realizing a prescribed bifurcation curve is established. It is shown that the freedom of such kinetics is of degree finite and even, depending only on the heterogeneity of the environment, and conversely that any nonnegative even integer occurs as the degree of freedom in some environments. A discussion 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 equations.