Probability density functions.

<p>Different combinations of shape parameters (scale parameters are fixed) for the four different move length probability density functions here considered as reorientation strategies: (a) Lévy truncated (), (b) log-normal (), (c) stretched exponential (), and (d) gamma (). For all the distributions (except for the gamma), the smaller the shape parameter the heavier the tail, hence the larger the probability of large move lengths.</p

Role of diffusion and the close-to-distant encounter ratio in search efficiency.

<p>Comparison of key quantities across <i>optimal</i> reorientation strategies at (a,b), and for any initial condition (c,d). Panels a and b: r.m.s. behavior (a) and values for the diffusion constant, diffusion exponent, and close-to-distant encounter ratio (b) at the asymmetric search condition (, , and ). Parameter values as follows: Lévy truncated (, ), log-normal (, ), stretched exponential (, ) and gamma (, ). Solid brown lines show the r.m.s. behavior (a) and the diffusion exponent (b) for a non-truncated Lévy reorientation strategy with Lévy index . Panels c and d: search efficiency (c) and -ratio (d) for the <i>optimal</i> reorientation strategies at different initial conditions in the whole range up to . Solid brown lines indicate the results for a pure Lévy walk with . Note that, regardless , Lévy reorientation strategies show the largest search efficiency compared to the other reorientation strategies, though truncation decreases the efficiency when reaching the symmetric limit .</p

Root mean square behavior.

<p>R.m.s. behavior for the different reorientation strategies (different parametrization for each case) in the asymmetric search condition. In each case the fixed parameter is set at the search optimal. Note the switch in the spreading dynamics of the searcher at times (for some of the parametrization), coinciding with the parameter . Solid lines: numerical simulations. Symbols: analytical results for both the short-term first-passage-time regime, , defined in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0106373#pone.0106373.e119" target="_blank">Eq. (9)</a> (open symbols), and the long-term Brownian regime, defined in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0106373#pone.0106373.e157" target="_blank">Eq. (12)</a> (filled symbols). Notice the nice agreement between numerical and analytical results. We used parameters , , , and .</p

Factorized search efficiency.

<p>Factorized search efficiency, , analyzed to understand the relative contribution of the encounters of near (subscript ) and distant (subscript ) targets to the global search efficiency in the asymmetric search scenario. The behavior of the traveled distances (, , and ) and the partial quantities (, , , ) is shown for each pdf model. Except for the gamma model, the search dynamics goes from ballistic to Brownian with increasing shape parameters (compare across -axes with Fig. 2). The scale parameters are fixed at the search optimal. Note that the minimal is close to the minimal , suggesting that, in the asymmetric search condition, the encounter efficiency of distant targets is relevant to the global search efficiency. However, the precise optimal strategy in each case results from the subtle balance between exploring nearby areas and accessing faraway regions. Analytical (black lines) and numerical (symbols) results are displayed with nice agreement. We used parameters , , , and .</p

Diagrams showing the symmetric and asymmetric initial search conditions for the 1D stochastic search model.

<p>(a) The 1D searching environment with equally spaced point-like targets. The initial searcher location is . A detail of the searcher perceptual range (or radius of vision ) is shown. (b) An example of the walk movement dynamics where a target is found after four steps. (c) In the asymmetric condition, each time the searcher finds a target, the search is re-initialized by placing it a distance to the right or to the left of the target found. (d) In the symmetric starting condition, .</p

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

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