Nonlocal effects in a coupled road-field population system

Understanding when a population persists or dies out in environments intersected by transportation corridors requires capturing the influence of fast movement along the road and the resulting nonlocal interactions. We investigate a road-field reaction-diffusion framework that links a rapidly diffusing one-dimensional road to a two-dimensional half-plane, where the population follows a Kolmogorov-Petrovsky-Piskunov (KPP) type growth law subject to both local competition and a nonlocal crowding effect. By constructing appropriately coupled lower and upper solutions in this mixed-dimensional domain and applying a comparison principle tailored to the exchange between road and field, we obtain boundedness as well as global existence and uniqueness. To analyze long-term behavior under nonlocal competition, we extend classical persistence techniques and demonstrate that positive solutions remain uniformly away from extinction under suitable hypotheses. Concerning spatial invasion, we determine the propagation speed along the road: When the diffusion rate on the road is at most twice that in the field, the front travels more slowly than the classical KPP wave; once this ratio is exceeded, the spreading speed increases like the square root of the road diffusion parameter. Numerical simulations are consistent with these analytical conclusions and further show that intensifying nonlocal competition tends to reduce the eventual size of the positive equilibrium, thereby increasing the likelihood of extinction. These results highlight the joint impact of road-mediated transport and nonlocal effects on invasion outcomes in coupled habitats.