Two-dimensional intermittent search processes: An alternative to Lévy flight strategies

Lévy flights are known to be optimal search strategies in the particular case of revisitable targets. In the relevant situation of nonrevisitable targets, we propose an alternative model of two-dimensional (2D) search processes, which explicitly relies on the widely observed intermittent behavior of foraging animals. We show analytically that intermittent strategies can minimize the search time, and therefore do constitute real optimal strategies. We study two representative modes of target detection and determine which features of the search time are robust and do not depend on the specific characteristics of detection mechanisms. In particular, both modes lead to a global minimum of the search time as a function of the typical times spent in each state, for the same optimal duration of the ballistic phase. This last quantity could be a universal feature of 2D intermittent search strategies.

Figure 1

(Color online) Two models of intermittent search: The searcher alternates slow reactive phases (regime 1) of mean duration ${\tau }_1$, and fast nonreactive ballistic phases (regime 2) of mean duration ${\tau }_2$. Top: the slow reactive phase is diffusive and detection is infinitely efficient. Bottom: the slow reactive phase is static and detection takes place with finite rate $k$.

Figure 2

(Color online) Simulations (points) versus analytical approximate (line) of the search time in the dynamic mode: the search time rescaled by the value in absence of intermittence as a function of ${\tau }_2$ (left) and $\ln {\tau }_1$ (right) (the logarithmic scale has been used due to the flatness of the minimum), for $D=1$, $v=1$, $b=451$. Left: $a=10$, ${\tau }_1=1.88$ (green squares); $a=1$, ${\tau }_1=50$ (blue circles); $a=0.1$, ${\tau }_1=250$ (red crosses). Right: $a=10$, ${\tau }_2=21.6$ (green squares); $a=1$, ${\tau }_2=22.5$ (blue circles); $a=0.1$, ${\tau }_2=41.6$ (red crosses).

Figure 3

(Color online) Simulations (points) versus analytical approximate (line) of the search time in the static mode: the search time rescaled by the optimal value as a function of ${\tau }_2∕a$ (left) and ${\tau }_1$ (right) for $k=1$, $v=1$, $b=28$. Left: $a=10$, ${\tau }_1=2.14$ (green squares); $a=1$, ${\tau }_1=0.9$ (blue circles); $a=0.1$, ${\tau }_1=0.3$ (red crosses). Right: $a=10$, ${\tau }_2=8$ (green squares); $a=1$, ${\tau }_2=1.7$ (blue circles); $a=0.1$, ${\tau }_2=0.22$ (red crosses).