This paper examines spatial propagation in nonlocal dispersal Fisher–KPP equations within shifting habitats, a scenario relevant to ecological invasions in the context of climate change. By employing the theory of viscosity solutions for Hamilton–Jacobi equations, we provide a complete classification of linear spreading speeds and characterize asymptotic states of solutions across the entire spatial domain, thereby resolving an open problem concerning nonlocal dispersal equations. Additionally, our findings reveal that environmental heterogeneity induces two distinct nonlocal determinacy mechanisms for spreading speeds, one of which occurs exclusively when the initial decay rate is appropriately small—a potentially novel finding in the literature. This work also represents the first application of the theory of viscosity solutions for Hamilton–Jacobi equations to analyze spreading speeds in nonlocal dispersal Fisher–KPP equations within shifting habitats.
Tao et al. (Thu,) studied this question.