  Moving-habitat models track the density of a population whose suitable
habitat shifts as a consequence of climate change. Whereas most previous
studies in this area consider 1-dimensional space, we derive and study a
spatially 2-dimensional moving-habitat model via reaction-diffusion equations.
The population inhabits the whole space. The suitable habitat is a bounded
region where population growth is positive; the unbounded complement of its
closure is unsuitable with negative growth. The interface between the two
habitat types moves, depicting the movement of the suitable habitat poleward.
Detailed modelling of individual movement behaviour induces a nonstandard
discontinuity in the density across the interface. For the corresponding
semi-discretised system we prove well-posedness for a constant shifting
velocity before constructing an implicit-explicit hybrid finite element method.
In this method, a Lagrange multiplier weakly imposes the jump discontinuity
across the interface. For a stationary interface, we derive optimal a priori
error estimates over a conformal mesh with nonconformal discretisation. We
demonstrate with numerical convergence tests that these results hold for the
moving interface. Finally, we demonstrate the strength of our hybrid finite
element method with two biologically motivated cases, one for a domain with a
curved boundary and the other for non-constant shifting velocity.
