Many biological populations reside in increasingly fragmented

landscapes, where habitat quality may change abruptly in space. A reaction-diffusion model

for a single species which grows and disperses in a one-dimensional

heterogeneous landscape is presented. The landscape is composed of two homogeneous adjacent patches with different diffusivities and net growth functions (monostable and bistable). An interface condition connects population density and flux between the two patches. We first classify all possible positive steady states using a phase plane approach. We continue by analyzing the stability properties of certain simple possible positive steady states. We end by applying bifurcations theory.

Numerical simulations reveal fold bifurcations.