Dynamics of a two-state reversible population model

The effect of the diffusion rate on the persistence of a nematode population with two reversible states in a spatially heterogeneous environment is investigated. In the absence of advection, it is shown that when the toxin distribution is not identically maximal, the system admits a unique positive equilibrium. As the diffusion rate tends to zero, this equilibrium converges to that of the corresponding non-spatial kinetic system, concentrating near locations where resources are abundant and toxin levels are low. As the diffusion rate tends to infinity, the equilibrium becomes spatially homogeneous and approaches a constant determined by the spatial averages of the resource, toxin, and competition coefficients. These results show that slow diffusion promotes exploitation of local favorable habitats, whereas fast diffusion smooths spatial heterogeneity, highlighting the joint influence of diffusion and environmental heterogeneity on persistence.