The shadow principle: An optimal survival strategy for a prey chased by random predators

• We model a prey chased by independent predators uniformly distributed on a lattice.
• The prey optimizes its survival probability if it follows the “shadow” of the predators.
• This optimal trajectory mimics the preferred trajectory of the predators.

We consider a lattice model of the annihilation process A+B→B, when a mobile prey A is chased by identical, independent predators B performing random motions until one of them finds A and destroys it. It is assumed that each predator follows some “most probable” trajectory around which it performs a random motion. It is shown that, if the random motion of the predators satisfies certain conditions, the prey A can maximize its survival probability by following a specific trajectory which mimics the preferred trajectories of the predators: we call this optimal trajectory as the “shadow” of the predator. This is an extension of the so-called “Pascal Principle”, studied in the recent literature. We discuss the conditions which allow for such extensions, and give examples where they are realized.