Effect of the search space dimensionality for finding close and faraway targets in random searches,Physical Review E

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Effect of the search space dimensionality for finding close and faraway targets in random searches
Physical Review E ( IF 2.2 ) Pub Date : 2022-09-15 , DOI: 10.1103/physreve.106.034124
J R Colaço ₁ , H A Araújo _{1,

2} , M G E da Luz ₃ , G M Viswanathan ₄ , F Bartumeus _{5,

6,

7} , E P Raposo ₁

Affiliation

We investigate the dependence on the search space dimension of statistical properties of random searches with Lévy

α

-stable and power-law distributions of step lengths. We find that the probabilities to return to the last target found

(P_{0})

and to encounter faraway targets

(P_{L})

, as well as the associated Shannon entropy

S

, behave as a function of

α

quite differently in one (1D) and two (2D) dimensions, a somewhat surprising result not reported until now. While in 1D one always has

P_{0} \geq P_{L}

, an interesting crossover takes place in 2D that separates the search regimes with

P_{0} > P_{L}

for higher

α

and

P_{0} < P_{L}

for lower

α

, depending on the initial distance to the last target found. We also obtain in 2D a maximum in the entropy

S

for

α \in (0, 2]

, not observed in 1D apart from the trivial

α \to 0

ballistic limit. Improving the understanding of the role of dimensionality in random searches is relevant in diverse contexts, as in the problem of encounter rates in biology and ecology.

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