Double-Jump migration and diffusive instability.

In order to look into the stability consequences of a particular migration process in which individuals choose to settle, we formulated a continuous-time multi-species multi-patch model in which individuals migrate by one or two instantaneous jumps while making the second jump with a certain probability that possibly depends on the conditions at the end point of the first jump. It turned out that a second jump has some quantitative effects on diffusive instability even when it occurs in the absence of density-dependent mechanisms. When a second jump happens as a natural interspecific response of individuals, and such a response is sufficiently strong, it has crucial effects on diffusive instability: it leads to diffusive instability in the case of competitive interactions, whereas it annihilates diffusive instability in the case of prey-predator interactions. We demonstrated these results in the context of two specific examples.