Non-local reaction-diffusion equations with a barrier

Non-local reaction-diffusion equations arise naturally to account for diffusions involving jumps rather than local diffusions related to Brownian motion. In ecology, long distance dispersal require such frameworks. In this work we study a one-dimensional non-local reaction-diffusion equation with bistable and monostable type reactions. The heterogeneity here from due to the presence of a barrier outside of which the equation is a classical homogeneous reaction-diffusion equation and is motivated by applications in ecology, sociology, and biology. In some finite interval, which we refer to as the barrier zone, there is decay. For bistable equations we first establish the existence of a generalized traveling front that approaches a traveling wave solution as t → −∞, propagating in a heterogeneous environment. We then study the problem of obstructing the propagation of the generalized traveling front. As in the local diffusion case, we prove that obstruction is possible for bistable equations but not for monostable equations. An interesting difference between the local dispersal and the non-local dispersal is that in the latter the obstructing steady states are discontinuous. We characterize these jump discontinuities and discuss the scaling between the range of the dispersal and the critical length of the barrier. We further explore other differences between the local and the non-local dispersal cases and state some open problems.