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. • Obstacle-induced fragmentation controls chase–escape dynamics on lattices. • Trapping time and capture cost show a non-monotonic dependence on obstacle density. • Survival dynamics follows Weibull statistics with geometry-dependent crossovers. • Transport becomes geometry-dominated near percolation, leading to subdiffusion.
Rossatto et al. (Sun,) studied this question.