  In this paper, we propose a novel free boundary problem to model the movement
of single species with a range boundary. The spatial movement and birth/death
processes of the species found within the range boundary are assumed to be
governed by the classic Fisher-KPP reaction-diffusion equation, while the
movement of a free boundary describing the range limit is assumed to be
influenced by the weighted total population inside the range boundary and is
described by an integro-differential equation. Our free boundary equation is a
generalization of the classical Stefan problem that allows for nonlocal
influences on the boundary movement so that range expansion and shrinkage are
both possible. In this paper we prove that the new model is well posed and
possesses steady state. We show that the spreading speed of the range boundary
is smaller than that for the equivalent problem with a Stefan condition. This
implies that the nonlocal effect of the weighted total population on the
boundary movement slows down the spreading speed of the population. While the
classical Stefan condition categorizes asymptotic behavior via a
spreading-vanishing dichotomy, the new model extends this dichotomy to a
spreading-balancing-vanishing trichotomy. We specifically analyze how habitat
boundaries expand, balance or shrink. When the model is extended to have two
free boundaries, we observe the steady state scenario, asymmetric shifts, or
even boundaries moving synchronously in the same direction. These are newly
discovered phenomena in the free boundary problems for animal movement.
