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

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Highlights

  • 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.

Abstract

We consider a lattice model of the annihilation process A+BB, 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.

Introduction

It has been shown recently [1], [2], [3], [4], [5], [6], [7], [8], [9] that the famous assertion of Blaise Pascal [10], that all misfortune of man comes from the fact that he does not stay peacefully in his room, can be given a scientific content in specific circumstances. This is the so-called “Pascal Principle”, which states that a prey A, chased by predators, whose initial locations are uniformly distributed around the prey and which are performing a certain class of random motions, optimizes its survival probability at any time t by staying immobile. This assertion, of course, may not hold for some realistic situations, when, for instance, one has only a finite number of predators which are initially non-uniformly distributed around the prey [11]. Clearly, this would also be not the best survival strategy in case when the prey can perceive the predators from far away, and move accordingly [7]. However, it does represent the best survival strategy if both the prey and the predators, initially uniformly distributed around the prey, are both unable to perform remote detection, are not knowledgeable of the current locations of each other, and if the predators move randomly until the first encounter of any of the predators with the prey. This statement was first conjectured and taken for granted for Brownian motion in 2002 [1], [2], although a similar property has been derived previously for a hopping motion with distance-dependent hopping probabilities over a d-dimensional disordered array of donor centers [3]. Further on, it has been proven for Brownian motion in one-dimensional systems [4] and simultaneously, for quite a general class of d-dimensional lattice random walks [5], [6]. Subsequently, the “Pascal principle” received some attention in the mathematical literature in which it has been refined, generalized, applied to some particular situations and rigorously proven for different types of random motion [12], [13], [14], [15], [16].

The main limiting assumption for the validity of Pascal principle is that at any time, the most probable position of any predator should coincide with its initial position. The purpose of this article is to show that the principle can be generalized to broader conditions. In Section 2, we first consider a simple generalization of Pascal Principle, which nevertheless requires some precautions. In Section 3, we address more complex situations, where the Pascal principle can be generalized into a “shadow principle”, which yields the optimal survival strategy of a prey in the presence of a predator whose most probable position changes with time, or of N independent identical predators. This case is discussed, and a simple example is given in Section 4. Section 5 is devoted to conclusions and perspectives. The Appendices present some complementary derivations and discussions. In particular, it is shown in Appendix B that the shadow principle holds in more general conditions than those which are studied in the main text.

Section snippets

Model

This model is similar to the model used to introduce the Pascal principle [4], [5], [6]. We consider a point particle A, representing the prey, (or a target), and N identical and independent particles B, representing the predators. These particles perform independent jump processes on a regular lattice consisting of M nodes. In many cases, but not always, we will focus on the thermodynamic limit when both N and M tend to infinity, while the ratio N/M is kept fixed and equal to a concentration b

Annihilation by predators performing Markov jump processes

We now assume that each predator performs a jump process on a general network, which allows for describing persistent random walks and, in some cases, non instantaneous annihilation of the prey.

Discussion and example: discrete Ornstein–Uhlenbeck process with drift

The conditions imposed on the process of the predators B, described in Section 3.1 are clearly restrictive: this is true, in particular, for condition (ii), which stipulates that the maximum n-steps transition probability of B from a given initial position is independent of this position. However this situation can be extended as discussed in Appendix B. It is shown, in particular, that condition (9) can be relaxed, which significantly increases the validity range of the shadow principle. In

Conclusion

We have considered a prey chased by one or several predators performing random displacements with a non-zero average motion. Then we have shown that a simple principle, the “shadow principle”, gives the trajectory that maximizes the survival probability of the prey, provided that the stochastic process representing the movements of the predator satisfies specific conditions. In particular, from each initial position y0 there should be a “preferred” trajectory Γ¯(k|y0), such that if a predator

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