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

How search strategies evolve in finite, depletable landscapes remains a central question in foraging theory.

The gist: Evolved search dynamics in finite, depletable landscapes are more consistent with intermittent dynamics than with strict scale-free Lévy motion.

Evolutionary Framework and Setup

The study employs an evolutionary simulation where agents forage on a two-dimensional toroidal lattice containing non-renewable resources distributed either uniformly or as Lévy dust. Each agent carries a heritable genome encoding step lengths, velocities, and turning angles, which are represented as finite lists of admissible trait values rather than explicit parametric distributions. Selection acts on a fitness function that combines energetic gain, movement cost, and coverage efficiency. The agents are initialized at a resource-bearing lattice site if available; otherwise, the initial position is drawn uniformly from the lattice.

Resource Landscapes and Depletion Modeling

Resource landscapes are modeled in two ways:

1. Clustered resource landscapes generated as Lévy-dust fields, where successive resource placements are separated by power-law-distributed distances defined by a specific probability distribution (Equation 1). The exponent µ controls the spatial correlation of resource placement.

2. A uniform environment generated by an independent Bernoulli occupation of lattice sites with probability ρ.

Resources are non-renewable: once encountered by an agent, the resource at that site is removed for the remainder of that lifetime. This depletion shapes the realized trajectory jointly with the initial spatial resource field.

Agent Movement and Fitness Evaluation

Each movement decision involves sampling intended step length and velocity from heritable genomes. The movement update is governed by a rule where, with probability pcue = 0.5, the agent searches a square window of half-width rcue = 60; if an undepleted resource is found, the heading is set directly toward it; otherwise, a uniformly random heading is selected without falling through to the genome rule. The intended displacement is realized along the current heading and subject to periodic wrapping. The lifespan Tlife = 8000 counts movement updates, and the accumulated movement cost Q(t) is calculated from scalar increments ∆j (where cmove = 0.5).

Fitness F(t) is defined as the product of an energetic-efficiency term ηE(t) and a coverage-efficiency term ηC(t):

1. Energetic efficiency: ηE(t) = Eres(t)/(Eres(t) + Q(t)) = 1/(1 + Q(t)/Eres(t)).

2. Coverage efficiency: ηC (t) = N(t)/(N(t) + R(t)) = 1/(1 + R(t)/N(t)), where N is the number of distinct lattice sites visited and R is the number of revisits to already visited sites.

Classification of Evolved Trajectories

To distinguish between Lévy walk (LW) and intermittent search (IS) dynamics, researchers fitted theoretical displacement moments to empirical moments computed from simulated paths. The LW model is defined by straight flights with flight durations drawn from a heavy-tailed distribution (Equation 5). The IS model alternates between a diffusive phase of diffusivity D and a ballistic phase of speed vB, governed by switching rates λBD and λDB. Trajectories are classified using the log-moment classification framework, minimizing the mean squared discrepancy in logarithmic moment space to determine the model-comparison score Γ = R¯2IS − R¯2LW.

Key Findings on Evolved Dynamics

The analysis of evolved trajectories after 1500 generations revealed several consistent patterns:

. Step-length genomes shifted toward short displacements while retaining a sparse tail of longer entries. This suggests local exploitation punctuated by occasional transfer.

. Moment-based comparison consistently favored IS over LW dynamics, with the model-comparison score Γ > 0 across all environments and system sizes. For instance, at L = 503 and ρ = 0.15, five independent evolutionary runs per environment reproduced this preference for IS.

. Temporal structure analysis showed that Return intervals and resource-hit intervals have broader upper tails than new-site discovery intervals, indicating that the strongest intermittency appears in recurrence and resource-encounter timing rather than in long pauses in spatial discovery. This supports the interpretation of IS, combining local exploitation with occasional relocation.

. The results suggest a possible design principle for search: Movement rules dominated by short displacements but punctuated by occasional relocations may be useful in heterogeneous objective landscapes.

Robustness and Further Analysis

The preference for the IS model was maintained across different grid sizes (211 × 211, 503 × 503, and 1009 × 1009) and varying resource densities.

Based on this research, here are specific improvements for AI systems:

1. Enhanced Search Strategy for Resource-Constrained Environments: The core finding is that in finite, depletable landscapes, agents evolve a strategy closer to an Intermittent Search (IS) model rather than a strict Lévy Walk (LW).

2. Adaptive Movement Genome Generation: AI agents should utilize genome-encoded movement traits (step lengths, velocities, turning angles) that are not fixed by external power-law distributions but evolve based on fitness criteria combining energetic gain and spatial coverage efficiency.

3. Optimized Exploration/Exploitation Balance: The system should be designed to favor a strategy of short displacements punctuated by occasional relocation. This means the AI should prioritize local exploitation when resources are abundant or near, but be programmed to initiate long-range transfers when local search becomes inefficient due to depletion or high redundancy in the current area.

4. Resource-Aware Trajectory Planning: The system must incorporate a reactive stopping rule: movement segments terminate immediately upon encountering a resource. This allows the AI to make real-time decisions based on local sensing, effectively combining local scanning with relocation when necessary, mimicking observed biological behavior in heterogeneous landscapes.

5. Dynamic Search Parameter Tuning: The system can be designed to dynamically adjust its search parameters (e.g., step length distribution) based on the perceived state of the environment (e.g., resource density or local coverage metrics), rather than relying on a single, fixed power-law assumption for movement rules.

6. Improved Temporal Structure Modeling: The AI system can use waiting-time statistics to better understand its search dynamics:

7. Explicitly differentiate between:

8. Recurrence intervals (returning to visited sites) and resource-hit intervals (successful foraging events), as these are the primary drivers of temporal intermittency in optimal search strategies.

9. New-site discovery intervals, which can be treated as a comparatively regular timescale, suggesting that spatial discovery proceeds more predictably than recurrence or foraging events.

These improvements result in AI systems capable of:

1. Operating efficiently in complex, resource-limited environments (e.g., robotic exploration, autonomous vehicle navigation in dynamic infrastructure).

2. Developing adaptive search patterns that balance immediate reward (exploitation) with long-term knowledge acquisition (exploration) under strict energy or time constraints.

3. Creating search algorithms that are robust to environmental heterogeneity by evolving movement rules suited to the specific resource distribution (uniform vs. clustered Lévy dust).

4. Performing intelligent, reactive decision-making where movement is truncated or redirected based on immediate sensory feedback (resource encounter), leading to highly efficient energy management and targeted exploration paths.

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 squared > 0.9 in most tested conditions), intermittent search achieves a closer fit (mean adjusted R squared > 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.