Robbert Fokkink

Ambush frequency should increase over time during optimal predator search for prey

We advance and apply the mathematical theory of search games to model the problem faced by a predator searching for prey. Two search modes are available: ambush and cruising search. Some species can adopt either mode, with their choice at a given time traditionally explained in terms of varying habitat and physiological conditions. We present an additional explanation of the observed predator alternation between these search modes, which is based on the dynamical nature of the search game they are playing: the possibility of ambush decreases the propensity of the prey to frequently change locations and thereby renders it more susceptible to the systematic cruising search portion of the strategy. This heuristic explanation is supported by showing that in a new idealized search game where the predator is allowed to ambush or search at any time, and the prey can change locations at intermittent times, optimal predator play requires an alternation (or mixture) over time of ambush and cruise search. Thus, our game is an extension of the well-studied ‘Princess and Monster’ search game. Search games are zero sum games, where the pay-off is the capture time and neither the Searcher nor the Hider knows the location of the other. We are able to determine the optimal mixture of the search modes when the predator uses a mixture which is constant over time, and also to determine how the mode mixture changes over time when dynamic strategies are allowed (the ambush probability increases over time). In particular, we establish the ‘square root law of search predation’: the optimal proportion of active search equals the square root of the fraction of the region that has not yet been explored.

Chaotic homeomorphisms are generic

Abstract The set of all chaotic measure-preserving homeomorphisms on a compact n -dimensional manifold ( n≥2 ) is a residual set in the space of all measure-preserving homeomorphisms.

Variations on a theorem of lusternik and schnirelmann

Abstract A fixed-point free map f: X → X is said to be colorable with k colors if there exists a closed cover β of X consisting of k elements such that C∩f(C) = φ for every C in β. It is shown that each fixed-point free involution of a paracompact Hausdorff space X with dim X ≤ n can be colored with n + 2 colors. Each fixed-point free homeomorphism of a metrizable space X with dim X ≤ n is colorable with n + 3 colors. Every fixed-point free continuous selfmap of a compact metrizable space X with dim X ≤ n can be colored with n + 3 colors

The “Princess and Monster” Game on an Interval

A minimizing searcher

On Ruckle's Conjecture on Accumulation Games

S

On Search Games That Include Ambush

and a maximizing hider

A search game model of the scatter hoarder's problem

H

Bihomogeneity of solenoids

move at unit speed on a closed interval until the first (capture, or payoff) time

How to hide information for later use on networks

T=\min \{ t:S(t)=H(t)\}

An addition theorem for the color number

that they meet. This zero-sum princess and monster game or less colorfully search game with mobile hider was proposed by Rufus Isaacs for general networks