Confinement-controlled chase-escape dynamics

TL;DR

This work examines chase–escape dynamics on a 2D lattice with randomly placed static obstacles, showing that geometric disorder, encoded by the obstacle density $\\phi$, governs the efficiency and temporal structure of pursuit. By combining BFS-based geometric accessibility analysis, survival statistics described by Weibull-like decay, and transport characterization via the mean-squared displacement, the study reveals a geometry-driven crossover: path elongation in connected environments and fragmentation-driven confinement compete to shape trapping times, capture costs, and survival. A key finding is that a finite-size precursory fragmentation occurs below the thermodynamic percolation threshold $\\phi_c \\approx 0.60$, leading to a regime near $\\phi_c$ where transport becomes strongly subdiffusive and dominated by geometry. The results provide a unified statistical–physics picture of cooperative pursuit in disordered media with potential implications for swarm robotics and navigation in cluttered environments.

Abstract

We investigate a minimal chase-and-escape model on a two-dimensional square lattice with randomly distributed static obstacles, focusing on how geometric disorder controls collective pursuit dynamics. Chasers and escapers move according to short-range sensing rules, while the density of obstacles tunes the connectivity of the accessible space. Using a combination of geometric analysis, dynamical observables, survival statistics, and transport characterization, we establish a direct link between lattice connectivity and pursuit efficiency. A Breadth-First Search analysis reveals that obstacle-induced fragmentation leads to a progressive loss of accessibility before the percolation threshold, defining the effective initial conditions for the dynamics. The trapping time and capture cost exhibit a non-monotonic dependence on obstacle density, reflecting a competition between path elongation in connected environments and geometric confinement near the percolation threshold. Survival analysis shows that the decay of the escaper population follows a Weibull form, with characteristic time and shape parameters displaying clear crossovers as a function of obstacle density, signaling the coexistence of cooperative capture and confinement-dominated trapping. Transport properties, quantified through the mean-squared displacement exponent, further support this picture, revealing sub-diffusive dynamics and a convergence toward a geometry-controlled regime near percolation. Overall, our results demonstrate that chase--and--escape dynamics in disordered environments are governed by a geometry-driven crossover, where percolation and connectivity act as unifying control parameters for spatial, temporal, and collective behavior.

Figures (10)

• Figure 1: Schematic illustration of the sensing radius (SR) defined using the Manhattan metric for a chaser located at the center of the lattice. Sites within SR = 1 and SR = 2 determine the regions where directed motion rules apply. This schematic illustrates the local decision rules used in the agent-based dynamics and is not drawn to scale.
• Figure 2: Fraction of topologically inaccessible escapers as a function of obstacle density $\phi$, obtained from the BFS analysis. The rapid increase observed for $\phi \gtrsim 0.50$ indicates the onset of connectivity loss prior to the percolation threshold $\phi_c$, highlighting finite-size fragmentation of the accessible space.
• Figure 3: Mean shortest path length $\langle d \rangle$ between chasers and escapers as a function of obstacle density $\phi$, computed using BFS and considering only topologically accessible pairs. The nonmonotonic behavior reflects a crossover from path elongation in connected lattices to co-confinement within isolated clusters at high $\phi$.
• Figure 4: Trapping time $TT$ as a function of obstacle density $\phi$ for different initial chaser populations. The nonmonotonic behavior reflects a competition between path elongation in connected lattices and geometric fragmentation near the percolation regime. The shaded region marks the effective percolation threshold $\phi_c$.
• Figure 5: Cost of capture as a function of obstacle density $\phi$ for different initial chaser populations. At low $\phi$, the cost depends strongly on the number of chasers, whereas for $\phi \gtrsim \phi_c$ the distributions converge, indicating that geometric confinement dominates the pursuit efficiency.