Heterogeneous search landscapes with representative trajectories of different strategies.

<p>Fragmented search landscapes containing <i>N</i>_<i>t</i> = 10⁴ targets placed in <i>N</i>_<i>p</i> = 10 heterogeneous patches (gray regions) with: (A) same average distance between inner targets, , and radii uniformly distributed in the range 0.03<i>M</i> ≤ <i>R</i>^(<i>p</i>) ≤ 0.3<i>M</i>, <i>M</i> = 10⁴; (B) same radius, <i>R</i>^(<i>p</i>) = 0.1<i>M</i>, and uniformly distributed in the range ; and (C) distinct sizes uniformly distributed in the range 0.03<i>M</i> ≤ <i>R</i>^(<i>p</i>) ≤ 0.3<i>M</i>, but fixed number (10³) of inner targets per patch, so that . The darker the patch, the higher its homogeneous density of inner targets. We also show typical paths of a searcher with power-law (Lévy-like) distributions of step lengths displaying different degrees of diffusivity: nearly ballistic (<i>μ</i> = 1.1), superdiffusive (<i>μ</i> = 2.0), and Brownian (<i>μ</i> = 3.0). In this illustrative example the search ends upon the finding of only 10 targets.</p

Output distribution of step lengths for a <i>μ</i> = 2.0 Lévy searcher in a <i>super-dense</i> landscape.

<p>In the simulations, <i>l</i>_<i>t</i> = 2.5 with <i>N</i>_<i>t</i> = 50000 targets homogeneously placed. The distribution takes into account the first 10⁴ search steps, including non-truncated moves that end up without detecting a target and also a relatively large number of truncated steps due to targets encounters. Numerical simulation data are represented by circles. Dashed and dotted lines are, respectively, best fits to Brownian-like exponential and truncated power-law pdfs. The inset details the large-steps regime. Statistical data inference (MLE and AIC methods) indicates that the output distribution of step lengths in the super-dense regime is not properly described by a superdiffusive power-law (Lévy-like) pdf. Instead, it shows the signature of a Brownian motion.</p

Lévy dust distribution of targets.

<p>Search landscapes containing Lévy dust distributions of <i>N</i>_<i>t</i> = 10⁴ targets (see main text), drawn from <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005774#pcbi.1005774.e001" target="_blank">Eq (1)</a> with <i>d</i>₀ = 1, <i>d</i>_max = <i>M</i> = 10⁴, and (A) <i>β</i> = 1.1, (B) <i>β</i> = 2.0, (C) <i>β</i> = 2.5, and (D) <i>β</i> = 3.0. Larger values of <i>β</i> increase the degree of clustering of targets. The bouncing of coordinates technique applied to the <i>β</i> = 1.1 case results in a nearly homogeneous targets distribution.</p

Construction of a fractal patch environment.

<p>Illustration of a search landscape with <i>N</i>_<i>p</i> = 3 patches and <i>N</i>_<i>t</i> = 15000 targets (5000 targets per patch), forming Lévy dust distributions (see main text). Here, <i>β</i> = 2.5, <i>d</i>₀ = 2, <i>d</i>_max = <i>M</i>/10 in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005774#pcbi.1005774.e001" target="_blank">Eq (1)</a>, and <i>γ</i> = 2.0, <i>r</i>₀ = 500, <i>r</i>_max = <i>M</i> = 10⁴ in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005774#pcbi.1005774.e002" target="_blank">Eq (2)</a>. The parameters are chosen so that the patches do not overlap. Dotted lines are only a guide to visually delimit the patches regions.</p

Search efficiency <i>η</i> vs. power-law exponent <i>μ</i> in Lévy dust distributions.

<p>The searcher detected 10⁴ targets in a landscape with Lévy dust distributions of <i>N</i>_<i>t</i> = 10⁴ targets (see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005774#sec007" target="_blank">Methods</a> section). Parameters are set as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005774#pcbi.1005774.g002" target="_blank">Fig 2</a>. High clustering of targets and nearly homogeneous landscapes correspond to <i>β</i> = 3 and <i>β</i> = 1.1, respectively. In all cases, <i>η</i> is maximized for <i>μ</i>_opt ≈ 2, with a slight decrease in the optimal value (i.e. enhanced superdiffusion) observed as <i>β</i> → 3.</p

Fractal patches obtained by combining two Lévy dust distributions.

<p>Search landscapes containing Lévy dust distributions located in <i>N</i>_<i>p</i> = 50 patches. Here, <i>N</i>_<i>t</i> = 50000 (1000 targets per patch), <i>β</i> = 3.0, <i>d</i>₀ = 2, <i>d</i>_max = <i>M</i>/10 in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005774#pcbi.1005774.e001" target="_blank">Eq (1)</a>, and <i>r</i>₀ = 100, <i>r</i>_max = <i>M</i> = 10⁴, (A) <i>γ</i> = 1.1, (B) <i>γ</i> = 2.0, (C) <i>γ</i> = 2.5, (D) <i>γ</i> = 3.0, in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005774#pcbi.1005774.e002" target="_blank">Eq (2)</a>. For large <i>γ</i> the patches are so close that one cannot distinguish them only by visual inspection.</p

Output distribution of step lengths for a <i>μ</i> = 2.0 Lévy searcher in a <i>low-dense</i> landscape (<i>l</i>_<i>t</i> = 100).

<p>In the sparse regime, the number of truncated moves due to targets encounters is relatively low and long steps are much more frequent, if compared to the super-dense limit (see inset). Statistical data inference (MLE and AIC methods) indicates that the output distribution of step lengths in the low-dense regime is actually a power-law (Lévy-like), with best-fit exponent <i>μ</i> = 2.19 close to the original one.</p

Search efficiency <i>η</i> vs. power-law exponent <i>μ</i> of Lévy searches for 10⁴ targets in fragmented landscapes.

<p>In the simulations, <i>N</i>_<i>p</i> = 10 (circles) and <i>N</i>_<i>p</i> = 5 (squares) heterogeneous patches contain a total of <i>N</i>_<i>t</i> = 10⁴ inner targets (see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005774#sec007" target="_blank">Methods</a> section). In (A)-(C) the parameters determining the radii and average distances between inner targets are respectively set as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005774#pcbi.1005774.g001" target="_blank">Fig 1(A)–1(C)</a>. Ballistic and Brownian limits correspond to <i>μ</i> → 1 and <i>μ</i> = 3, respectively. In all cases, the efficiency <i>η</i> is maximized for the superdiffusive dynamics with <i>μ</i>_opt ≈ 2.</p