Moving-habitat models: A numerical approach.

As the global climate changes, biological populations have to adapt in place or move in space to stay within their preferred temperature regime. Empirical evidence suggests that shifting speeds of temperature isoclines are location and elevation dependent and may accelerate over time. We present a mathematical tool to study transient behaviour of population dynamics within such moving habitats to discern between populations at high and low risk of extinction. We introduce a system of reaction-diffusion equations to study the impact of varying shifting speeds on the persistence and distribution of a single species. Our model includes habitat dependent movement behaviour and habitat preference of individuals. These assumptions result in a jump in density across habitat types and generalize previous studies. We build and validate a numerical finite difference scheme to solve the resulting equations. Our numerical scheme uses a coordinate system where the location of the moving suitable habitat is fixed in space and a modification of a finite difference scheme to capture the jump in density. We explore a variety of shifting-speed scenarios and contribute insights into the mechanisms that support population persistence through time in shifting habitats. One common finding is that a strong bias for the suitable habitat helps the population persist at faster shifting speeds, yet sustains a smaller total population at slower shifting speeds.