  We study a reaction-diffusion equation with a nonlocal reaction term that
models a population with variable motility. We establish a global supremum
bound for solutions of the equation. We investigate the asymptotic (long-time
and long-range) behavior of the population. We perform a certain rescaling and
prove that solutions of the rescaled problem converge locally uniformly to zero
in a certain region and stay positive (in some sense) in another region. These
regions are determined by two viscosity solutions of a related Hamilton-Jacobi
equation.
