A Hamilton-Jacobi Approach to Time-Delayed Nonlocal Diffusion Models in Shifting Habitats

This paper is concerned with the spatial propagation dynamics of time-delayed nonlocal diffusion equations in shifting habitats.

We employ the theory of viscosity solutions for Hamilton-Jacobi equations to provide a complete classification of the spreading speeds.

In particular, we derive variational formulas that explicitly characterize how these speeds depend on the decay rate of the initial data, the habitat shifting speed, and the maturation delay across three regimes: locally determined, nonlocally selected, and locked.

We reveal a distinct directional asymmetry in the nonlocal selection mechanism and a novel delay-insensitivity phenomenon.

We also establish the threshold conditions under which the habitat locking effect eliminates the classical decelerating role of time delay.