Evolutionary foraging in grids: Intermittent search dynamics emerge in finite, depletable landscapes

Shailendra Bhandari; Alex Szorkovszky; Anis Yazidi; Pedro G. Lind

q-bio.PESep 30, 2026

Evolutionary foraging in grids: Intermittent search dynamics emerge in finite, depletable landscapes

Authors: Shailendra Bhandari, Alex Szorkovszky, Anis Yazidi, Pedro G. Lind

Organizations: Department of Computer Science OsloMet – Oslo Metropolitan University St. Olavs plass, N-0130 Oslo, Norway · Simula Research Laboratory Numerical Analysis and Scientific Computing Oslo, 0164, Norway · Department of Informatics University of Oslo Gaustadalléen 23B, 0373 Oslo, Norway · Department of Technology Kristiania University of Applied Sciences Oslo, Norway

Abstract

How search strategies evolve in finite, depletable landscapes remains a question in foraging theory. We study this problem with an evolutionary simulation in which agents forage on a two-dimensional toroidal lattice containing non-renewable resources distributed uniformly or as Lévy dust. Each agent carries a heritable genome encoding step lengths, velocities, and turning angles, and selection acts on a fitness function combining energetic gain, movement cost, and coverage efficiency. By allowing movement traits to evolve without imposing a prescribed power-law step-length distribution, we test whether evolved trajectories are better described by intermittent-search or Lévy-walk dynamics. Our results indicate that evolved search is more consistent with intermittent dynamics than with strict scale-free Lévy motion in the finite depletion-driven landscapes considered here. We characterize the dynamics by fitting second- and fourth-order displacement moments to intermittent-search and Lévy-walk models. While a Lévy-like random walk fits the evolutionary trajectories well (mean adjusted $R^2$ > 0.9 in most tested conditions), intermittent search achieves a closer fit (mean adjusted $R^2$ > 0.99) for all tested resource distributions. This preference holds across the tested grid sizes and resource densities. Five independent evolutionary runs per environment on a 503 x 503 grid at nominal resource density $ρ$ = 0.15 reproduce this preference for the uniform environment and five Lévy-dust environments. Evolution rapidly reshapes the movement genome toward short displacements while retaining a sparse tail of longer relocations, consistent with local exploitation punctuated by occasional transfer. The framework provides a controlled setting for studying how search rules emerge under resource limitation and may inform resource-constrained exploration in autonomous systems.

Figures & tables

Figure 1: Resource landscapes and best evolved trajectories for the $L=503$ torus. Top row : placement-step distributions used to generate the resource fields, shown for the uniform control and for Lévy-dust environments with $\mu\in\{1.0,1.5,2.0,2.5,3.0\}$ . Main panels : resource locations (gray points) overlaid with the trajectory of the best evolved agent in each environment (colored line). Changing $\mu$ changes the spatial correlation structure of the resource field by altering the frequency of long placement steps: smaller $\mu$ gives heavier-tailed placement steps, whereas larger $\mu$ suppresses long placement steps. The inset in the $\mu=3.0$ panel shows a magnified local patch.

Figure 2: Evolution of fitness and the step lengths on the

503\times 503

grid. Left: mean fitness over 1500 generations for each resource environment, with shaded standard errors across agents within a single evolutionary run at each generation. Right: distributions of step-length genome entries at generations 0, 200, and 1500. Selection rapidly concentrates the encoded step-length pool at short values while retaining a sparse tail of longer entries.

Figure 3: Moment-based comparison of evolved trajectories on the

503\times 503

grid. Empirical second moments

m_{2}

and fourth moments

m_{4}

are compared with best-fit IS and LW models. Columns correspond to the six resource environments. Across environments, IS gives a closer fit to the empirical moment curves.

Grid	Environment	$\Gamma$	$\bar{R}^{2}_{\mathrm{IS}}$	$\bar{R}^{2}_{\mathrm{LW}}$
$211\times 211$	uniform	$0.028\pm 0.004$	$0.9985\pm 0.0004$	$0.970\pm 0.004$
$211\times 211$	$\mu=1.0$	$0.019\pm 0.002$	$0.9984\pm 0.0002$	$0.980\pm 0.002$
$211\times 211$	$\mu=1.5$	$0.022\pm 0.003$	$0.9981\pm 0.0003$	$0.976\pm 0.003$
$211\times 211$	$\mu=2.0$	$0.024\pm 0.005$	$0.9993\pm 0.0002$	$0.975\pm 0.005$
$211\times 211$	$\mu=2.5$	$0.030\pm 0.007$	$0.9987\pm 0.0005$	$0.969\pm 0.007$
$211\times 211$	$\mu=3.0$	$\mathbf{0.08\pm 0.02}$	$0.9986\pm 0.0006$	$0.92\pm 0.02$

Table 1: Model-comparison summary across environments and grid sizes at

\rho=0.15

. For each case, we report

\Gamma=\bar{R}^{2}_{\mathrm{IS}}-\bar{R}^{2}_{\mathrm{LW}}

, together with the adjusted coefficients of determination. For

L=211

and

L=1009

, values are mean

\pm

standard deviation over the final 500 best-agent trajectories from one evolutionary run. For

L=503

, values are mean

\pm

standard deviation across five independent run-level means, each computed from the final 500 best-agent trajectories. The runs used independent resource maps and GA initializations. Positive

\Gamma

indicates preference for the IS model. All results are reported with significant figures in accordance with their respective standard deviations.

Figure 4: Movement traits and return-interval statistics of the best evolved agents across grid sizes after 1500 generations. Columns correspond to resource environments. Top row: velocity-genome entries of the best evolved agents for

L=211

503

, and

1009

. Middle row: turn-angle increments recorded along the best evolved trajectories. Bottom row: running maximum of return intervals

M_{j}^{\mathrm{return}}

versus return-event index

j

Appendix figures & tables10 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 5: Evolutionary stabilization on the

503\times 503

grid. Left: best fitness as a function of generation. Middle: JS divergence

\mathrm{JS}(P_{g}(s),P_{0}(s))

between the population-level step-length distribution at generation

g

and the initial distribution. Right: final energetic and coverage efficiencies,

\eta_{E}

and

\eta_{C}

, for the best evolved agent in each environment. Vertical markers denote the operational plateau generation estimated from a rolling-slope criterion applied to fitness and genome redistribution.

Environment	$\rho=0.15$	$\rho=0.30$	$\rho=0.45$
Uniform	0.150	0.299	0.449
$\mu=1.0$	0.138	0.256	0.359
$\mu=1.5$	0.131	0.245	0.344
$\mu=2.0$	0.117	0.221	0.310
$\mu=2.5$	0.097	0.184	0.263
$\mu=3.0$	0.066	0.145	0.194

Appendix

Table 3: Realized resource occupancy

\rho_{\mathrm{real}}

, the fraction of distinct resource-bearing sites before depletion, for the saved

503\times 503

maps at the three original nominal densities. Each entry describes the corresponding resource map.

Panel of Figure 6 — Figure 6: Additional-size results for the $211\times 211$ torus. Top left : mean fitness over 1500 generations for each resource environment, with shaded standard errors across agents within the single evolutionary run at each generation. Top right : distributions of genome-encoded step-length entries at generations 0, 200, and 1500. Middle : resource maps overlaid with the trajectory of the best evolved agent in each environment. Bottom : empirical second and fourth displacement moments compared with best-fit IS and LW models.

Panel of Figure 7 — Figure 7: Additional-size results for the $1009\times 1009$ torus. Top left : mean fitness over 1500 generations for each resource environment, with shaded standard errors across agents within the single evolutionary run at each generation. Top right : distributions of genome-encoded step-length entries at generations 0, 200, and 1500. Middle : resource maps overlaid with the trajectory of the best evolved agent in each environment. Bottom : empirical second and fourth displacement moments compared with best-fit IS and LW models.

Panel of Figure 8 — Figure 8: Robustness of evolved search dynamics across resource densities on the $503\times 503$ torus. Top : best evolved trajectories overlaid on the corresponding resource maps for the highest tested density, $\rho=0.45$ . Bottom left : final-population distributions of genome-encoded step lengths for $\rho\in\{0.005,0.010,0.15,0.30,0.45\}$ , shown separately for each resource environment. Bottom right : model-selection score $\Gamma$ across densities, with error bars.

Environment	$\eta_{C}$ , baseline $F=\eta_{E}\eta_{C}$	$\eta_{C}$ , ablation $F=\eta_{E}$	Retained $\eta_{C}$ (%)	$\Gamma$ , ablation $F=\eta_{E}$
Uniform	$0.9278\pm 0.0065$	$0.9283\pm 0.0048$	$100.1\%$	$0.0190\pm 0.0013$
$\mu=1.0$	$0.9282\pm 0.0057$	$0.9232\pm 0.0055$	$99.5\%$	$0.0180\pm 0.0021$
$\mu=1.5$	$0.8997\pm 0.0053$	$0.8920\pm 0.0103$	$99.1\%$	$0.0216\pm 0.0034$
$\mu=2.0$	$0.8585\pm 0.0083$	$0.8489\pm 0.0073$	$98.9\%$	$0.0313\pm 0.0049$
$\mu=2.5$	$0.8129\pm 0.0076$	$0.8105\pm 0.0112$	$99.7\%$	$0.0373\pm 0.0100$
$\mu=3.0$	$0.7834\pm 0.0118$	$0.7422\pm 0.0154$	$94.7\%$	$0.0509\pm 0.0097$

Appendix

Table 4: Coverage-term ablation at

L=503

and

\rho=0.15

. Values are mean

\pm

standard deviation over the final 20 best-agent generations. They represent correlated late-generation best-agent evaluations from one evolutionary run per objective, rather than independent evolutionary replicates.

Environment	$\Gamma_{\mathrm{cue\,on}}$	$\Gamma_{\mathrm{cue\,off}}$	$\bar{R}^{2}_{\mathrm{LW,cue\,off}}$
Uniform	$0.0175\pm 0.0014$	$0.0010\pm 0.0001$	$0.9988$
$\mu=1.0$	$0.0185\pm 0.0017$	$0.0029\pm 0.0008$	$0.9969$
$\mu=1.5$	$0.0204\pm 0.0024$	$0.0059\pm 0.0024$	$0.9933$
$\mu=2.0$	$0.0326\pm 0.0046$	$0.0145\pm 0.0062$	$0.9818$
$\mu=2.5$	$0.0251\pm 0.0060$	$0.0267\pm 0.0175$	$0.9720$
$\mu=3.0$	$0.0424\pm 0.0085$	$0.0797\pm 0.0483$	$0.9181$

Appendix

Table 5: Fixed-genome cue-off evaluation at

L=503

and

\rho=0.15

. Genomes evolved under the standard

p_{\mathrm{cue}}=0.5

condition were re-evaluated with

p_{\mathrm{cue}}=0

, without further evolution. The cue-on and cue-off

\Gamma

values are the mean

\pm

standard deviation over the final

500

matched best-agent generations. The final column reports the mean adjusted LW coefficient of determination in the cue-off condition. These results are separate from the five-run aggregates in Table 1 .

Figure 9: Synthetic validation of the moment-based classifier and intermittent-search parameter estimation. Left: distributions of the model-comparison score

\Gamma=\bar{R}^{2}{\mathrm{IS}}-\bar{R}^{2}{\mathrm{LW}}

for 1000 synthetic L’evy-walk trajectories and 1000 synthetic intermittent-search trajectories. Positive values favor IS, whereas negative values favor LW. The confusion matrix shows that 999 of 1000 trajectories from each generating model were classified correctly. Right: estimated versus generating values of

\lambda_{BD}

for synthetic IS trajectories. The dashed line marks exact recovery. The inset summarizes the regression slopes and coefficients of determination for

D

v_{B}

\lambda_{BD}

, and

\lambda_{DB}

Env.	${\mathcal{F}}$	$\eta_{E}$	$\eta_{C}$	Return	Hit	New-site
				$q_{0.95}/\mathrm{mdn.}$	$q_{0.95}/\mathrm{mdn.}$	$q_{0.95}/\mathrm{mdn.}$
$\mu=1.0$	0.40 $\pm$ 0.02	0.43 $\pm$ 0.01	0.92 $\pm$ 0.02	25.4	3.0	2.0
$\mu=1.5$	0.39 $\pm$ 0.01	0.43 $\pm$ 0.01	0.90 $\pm$ 0.01	18.6	3.5	2.0
$\mu=2.0$	0.40 $\pm$ 0.01	0.46 $\pm$ 0.01	0.86 $\pm$ 0.01	81.5	6.0	2.0
$\mu=2.5$	0.37 $\pm$ 0.02	0.48 $\pm$ 0.02	0.81 $\pm$ 0.01	55.6	6.0	2.1
$\mu=3.0$	0.37 $\pm$ 0.05	0.49 $\pm$ 0.04	0.76 $\pm$ 0.06	42.9	5.0	3.0

Appendix

Table 6: Waiting-time statistics for the best evolved agents. Fitness and efficiency values - cf. Eqs. ( 2 ) and ( 3 ) - are reported as mean

\pm

standard deviation across grid sizes . Upper-tail spread is summarized by the median across grid sizes of

q_{0.95}/\mathrm{median}

. Return intervals quantify recurrence to previously visited sites, resource-hit intervals quantify the temporal spacing between resource encounters, and new-site discovery intervals quantify the timing of first visits to previously unvisited sites.

Figure 10: Empirical survival distributions of waiting-time observables in evolved search trajectories. Left : return intervals

\Delta t^{\mathrm{return}}

, measuring the time between successive visits to the same previously visited site. Middle : resource-hit intervals

\Delta t^{\mathrm{hit}}

, measuring the time between successive resource encounters. Right : new-site discovery intervals

\tau_{n}

, measuring the waiting time between first visits to successive previously unvisited sites.

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