Chase-Escape Models and Phase Transitions

• Chase-escape is a pursuit–evasion model family with asymmetric dynamics where red expands into unoccupied space while blue follows exclusively through red territory.
• The model employs graph-based stochastic processes on trees, lattices, and random graphs to determine critical thresholds and phase transitions using explicit analytic criteria.
• Variants incorporating spontaneous death, conversion, and multi-agent interactions extend its applications to biological systems, network dynamics, and combinatorial optimization.

Chase-escape is a family of pursuit-evasion models centered on an asymmetric competitive rule: one type expands into previously unoccupied territory, while the other can advance only through sites already occupied by the first type. In the graph-based stochastic formulation, red particles spread to adjacent uncolored sites and blue particles overtake adjacent red sites, killing them; the principal questions are whether red escapes to infinity, whether both types persist, and where the corresponding phase transitions occur (Durrett et al., 2018). The same name also covers variants with spontaneous death, conversion, or distance-dependent rates, as well as multi-agent pursuit systems and a search heuristic on combinatorial state spaces (Beckman et al., 2019, Bailey et al., 28 Jul 2025, Ohira, 2014).

1. Core stochastic formulation

On a connected rooted graph $G=(V,E)$, a standard initialization places red at the root $\rho$ and blue at an auxiliary vertex $\mathfrak b\notin G$ attached to $\rho$ by a special edge $\mathfrak e$. Each edge $e$ carries independent passage times $t_e^R\sim \mathrm{Exp}(\lambda)$ and $t_e^B\sim \mathrm{Exp}(1)$. Red spreads only to adjacent uncolored sites, blue spreads only to adjacent red sites, and a red site becomes blue when blue reaches it. The coexistence event is denoted $A$, and the critical parameter is

$\lambda_c(G):=\sup\{\lambda:P_\lambda(A)=0\}.$

This formulation makes chase-escape a first-passage-percolation-type competition with a one-sided accessibility constraint on blue (Durrett et al., 2018).

An equivalent lattice language treats sites as empty, prey, or predator. In that version prey spread to neighboring empty sites at rate $\rho$0, predators spread to neighboring prey sites at rate $\rho$1, and predators never colonize empty sites directly. The ratio $\rho$2 is then the natural control parameter. The paper on chase-escape percolation explicitly compares this setup to SIR dynamics, with empty, prey, and predator playing the roles of susceptible, infected, and removed, but with the crucial difference that removal can occur only adjacent to already removed sites (Kumar et al., 2020).

A recurring technical complication is that on graphs with cycles the monotonicity of $\rho$3 in $\rho$4 is not immediate. The graph-based literature therefore distinguishes exact critical definitions from more operational threshold notions on finite graph sequences, especially when the event of interest is that red reaches a positive fraction of vertices rather than merely surviving indefinitely (Durrett et al., 2018, Bernstein et al., 2021).

2. Critical behavior on trees, lattices, and random graphs

On the infinite $\rho$5-ary tree $\rho$6, chase-escape admits an explicit threshold: $\rho$7 with

$\rho$8

The same paper gives $\rho$9 with $\mathfrak b\notin G$0, and $\mathfrak b\notin G$1 on the ladder graph (Durrett et al., 2018). On the tree, the derivation reduces each ray to a one-dimensional red-blue gap process and yields the asymptotic survival-to-depth estimate

$\mathfrak b\notin G$2

so the criterion $\mathfrak b\notin G$3 produces the exact critical value (Durrett et al., 2018).

For $\mathfrak b\notin G$4, the picture is numerical rather than fully rigorous. Simulations on large boxes gave strong evidence for a threshold near $\mathfrak b\notin G$5, together with fractal occupied sets, rough boundaries, and empirical hole-size statistics

$\mathfrak b\notin G$6

near criticality (Tang et al., 2018). A later Monte Carlo study sharpened the square-lattice estimate to

$\mathfrak b\notin G$7

and reported critical exponents consistent with ordinary two-dimensional undirected percolation, including $\mathfrak b\notin G$8, $\mathfrak b\notin G$9, $\rho$0, and $\rho$1 (Kumar et al., 2020).

Sparse random graphs exhibit a tree-like but not identical threshold structure. For the configuration model with i.i.d. degrees of finite second moment, the mean excess degree parameter

$\rho$2

induces the tree-like quantity

$\rho$3

The density-survival threshold is proved to satisfy $\rho$4, and the expected total number of ever-red vertices undergoes a sharp phase transition exactly at $\rho$5 (Bernstein et al., 2021). On random geometric Gilbert graphs, chase-escape with susceptible nodes and white knights is well defined, and there are proved regimes of global survival and extinction, together with estimates for local survival; the proofs combine tree comparison, percolation arguments, and finite-degree approximations (Hinsen et al., 2019).

3. Death, distance dependence, and conversion

A major extension is chase-escape with death, whose state space is

$\rho$6

with spontaneous red death $\rho$7 at rate $\rho$8. On the infinite rooted $\rho$9-ary tree $\mathfrak e$0, this creates three phases: coexistence, escape of red without blue, and extinction. The red extinction threshold is explicit,

$\mathfrak e$1

while the coexistence threshold $\mathfrak e$2 is determined through weighted Catalan numbers and the radius $\mathfrak e$3 of a generating function $\mathfrak e$4. Coexistence is possible only when $\mathfrak e$5, where

$\mathfrak e$6

and the critical equation is $\mathfrak e$7. At $\mathfrak e$8, the paper proves $\mathfrak e$9, in contrast with ordinary chase-escape on trees (Beckman et al., 2019).

The distance-dependent tree model replaces constant host birth and death rates by sequences $e$0 and $e$1 indexed by the current distance to the nearest blue ancestor. On the augmented tree $e$2, each red vertex $e$3 spreads to a white neighbor at rate $e$4 and dies at rate $e$5, where $e$6 is the distance from $e$7 to the nearest blue vertex on its ancestral path. Under explicit regularity assumptions, expected coexistence is characterized exactly by the radius of convergence $e$8 of a weighted Catalan generating function: $e$9 Here $t_e^R\sim \mathrm{Exp}(\lambda)$0 is the set of sites that are blue at some time, so blue’s unbounded spread encodes joint persistence (Hernandez-Torres et al., 2022).

Conversion introduces spontaneous red-to-blue transition at rate $t_e^R\sim \mathrm{Exp}(\lambda)$1. In the multiple-sclerosis lesion model, the state space is $t_e^R\sim \mathrm{Exp}(\lambda)$2, red spreads at rate $t_e^R\sim \mathrm{Exp}(\lambda)$3, each red converts to blue at rate $t_e^R\sim \mathrm{Exp}(\lambda)$4, and blue chases red at rate $t_e^R\sim \mathrm{Exp}(\lambda)$5. The central observable is the total number of blue sites at fixation,

$t_e^R\sim \mathrm{Exp}(\lambda)$6

interpreted as aggregate damage. The paper proves that $t_e^R\sim \mathrm{Exp}(\lambda)$7 is monotone in $t_e^R\sim \mathrm{Exp}(\lambda)$8 for $t_e^R\sim \mathrm{Exp}(\lambda)$9, monotone in $t_e^B\sim \mathrm{Exp}(1)$0 for $t_e^B\sim \mathrm{Exp}(1)$1, and monotone in $t_e^B\sim \mathrm{Exp}(1)$2 for $t_e^B\sim \mathrm{Exp}(1)$3. On bounded-degree graphs with nontrivial site-percolation threshold,

$t_e^B\sim \mathrm{Exp}(1)$4

hence

$t_e^B\sim \mathrm{Exp}(1)$5

This identifies the asymptotic order of the conversion-driven phase transition (Bailey et al., 28 Jul 2025).

On the complete graph $t_e^B\sim \mathrm{Exp}(1)$6, chase-escape with conversion becomes a count process for $t_e^B\sim \mathrm{Exp}(1)$7. The critical point remains equal fitness, $t_e^B\sim \mathrm{Exp}(1)$8, but the critical formulas change: $t_e^B\sim \mathrm{Exp}(1)$9 This mean-field result shows that conversion modifies critical constants while leaving the threshold location unchanged (Junge et al., 2 Oct 2025).

4. Group chase-and-escape and collective geometry

A separate literature uses “group chase and escape” for many-body pursuit on a two-dimensional square lattice with periodic boundary conditions. In the original model, $A$0 chasers and $A$1 targets are placed randomly, each chaser moves one lattice step toward its nearest target, and each target moves one step away from its nearest chaser. Capture occurs when a chaser hops onto a target’s site. The principal macroscopic observables are the total catch time $A$2, the typical target lifetime

$A$3

and the unit cost

$A$4

For fixed $A$5, the paper reports that $A$6 has a minimum at an intermediate chaser number $A$7, so there is an optimal number of chasers; by contrast, several random-walk comparison models do not exhibit this interior optimum (0912.4327).

Local geometry was later quantified by two directional-correlation parameters. For a target $A$8 chased by multiple chasers,

$A$9

and for a chaser $\lambda_c(G):=\sup\{\lambda:P_\lambda(A)=0\}.$0 from which multiple targets escape,

$\lambda_c(G):=\sup\{\lambda:P_\lambda(A)=0\}.$1

The values $\lambda_c(G):=\sup\{\lambda:P_\lambda(A)=0\}.$2 distinguish, respectively, one-directional pursuit, surrounding, and pincer-like “drive into a group” geometries; analogous interpretations hold for $\lambda_c(G):=\sup\{\lambda:P_\lambda(A)=0\}.$3 on the escape side. These parameters connect capture-time crossovers to concrete group configurations rather than to agent counts alone (Matsumoto et al., 2012).

Two later modifications make the collective geometry explicit. The laziness model introduces random-walking chasers either uniformly, with each chaser lazy with probability $\lambda_c(G):=\sup\{\lambda:P_\lambda(A)=0\}.$4 at each step, or via division of labor, with a fixed lazy fraction $\lambda_c(G):=\sup\{\lambda:P_\lambda(A)=0\}.$5. Uniform laziness always worsens performance, but a moderate permanently lazy subpopulation can reduce $\lambda_c(G):=\sup\{\lambda:P_\lambda(A)=0\}.$6 by suppressing rare long-lived clustered-pursuit episodes and helping create pincer attack configurations (Masuko et al., 2016). A conversion model on the periodic lattice adds probability $\lambda_c(G):=\sup\{\lambda:P_\lambda(A)=0\}.$7 that a caught target becomes a new chaser and proliferation probability $\lambda_c(G):=\sup\{\lambda:P_\lambda(A)=0\}.$8 that a moving target leaves a new target behind. It reports non-monotonic target lifetimes, including maxima of the extinction time $\lambda_c(G):=\sup\{\lambda:P_\lambda(A)=0\}.$9 as a function of initial target number $\rho$00 for $\rho$01, and intermediate optimal $\rho$02 when conversion is not too strong (Nishi et al., 2011).

5. Continuous-space, disordered, and mobile-network generalizations

Continuous-space pursuit models replace graph growth by explicitly kinematic chasers and escapers. In a bio-inspired model with bounded acceleration, time delay, external noise, and soft closed boundaries, chasers pursue a faster prey in 2D and 3D. The model couples direct attraction to the prey with repulsion among chasers and, optionally, prediction of the prey’s future position. A single chaser cannot catch the faster prey, and one or two chasers have essentially no chance, but suitably tuned local interaction among a small group can produce route blocking, caging, and emergent encircling; prediction can compensate for delay and improve capture, especially in 3D (Janosov et al., 2017).

A more force-based two-dimensional realization introduces a “dynamical trap” made of many target-tracking chasers. Each chaser experiences a tracking force

$\rho$03

combining moving-direction synchronization with steering toward assigned destination regions near the target. The paper’s principal result is geometric: one undifferentiated chaser group fails, while four groups assigned to a cross-like arrangement around the target can create a stable enclosing configuration. Velocity synchronization is indispensable, prediction of one-step future target position improves efficiency but is not necessary, and the offset parameter $\rho$04 must be large enough to avoid merely pushing the target away (Gao, 27 Dec 2025).

At the two-agent level, intelligent active-particle pursuit treats the pursuer as a deterministic self-steering particle and the evader as a cognitive run-and-tumble particle. Capture occurs when

$\rho$05

the pursuer has constant speed ratio $\rho$06 and maneuverability $\rho$07, and the evader tumbles with distance-dependent hazard

$\rho$08

The simulations identify two escape regimes: forward-biased tumbling with continuous slight directional adjustments is best against less athletic pursuers because it impedes alignment while preserving persistence, whereas narrow backward tumbles at short alert distance are the effective high-risk strategy against dominant pursuers (Goh et al., 14 Aug 2025).

Environmental geometry can itself control chase-escape efficiency. On a $\rho$09 square lattice with randomly placed static obstacles, breadth-first search reveals progressive loss of accessibility before the finite-size percolation threshold $\rho$10. The trapping time $\rho$11 and capture cost

$\rho$12

vary non-monotonically with obstacle density, the escaper population decays in Weibull form

$\rho$13

and the chaser mean-squared displacement remains subdiffusive, supporting a geometry-driven crossover from cooperative capture to confinement-dominated trapping (Rossatto et al., 9 Jan 2026).

At a larger scale, chase-escape has also been formulated for malware and patch propagation in dynamic device-to-device networks on random street systems. Devices and white knights are Cox point processes moving by a street-constrained random waypoint model, and infection or patch transmission requires proximity on the same street for a sufficiently long time. The resulting process has proved regimes of global survival and extinction, and, most notably, an “in-and-out” dependence on speed: malware can die out at very low and very high device speeds while surviving at intermediate speeds (Cali et al., 2022).

6. Optimization metaphor and conceptual scope

The term also appears in combinatorial optimization as a pursuit process on state space rather than physical space. In the two-state metaheuristic for the traveling salesman problem, one current solution is the evader and performs a local improving move, while the other is the chaser and moves closer to the evader in state space without regard to cost. If the chaser reaches lower cost, the roles are switched; if the two states coincide, one copy is perturbed and local search restarts from the perturbed neighborhood. In the TSP implementation, the evader uses a random two-city swap accepted only if path length decreases, while the chaser aligns a randomly chosen city with its position in the evader’s permutation. On berlin52.tsp, the reported average tour length improved from $\rho$14 to $\rho$15 at a budget of $\rho$16 permutations and from $\rho$17 to $\rho$18 at $\rho$19, which the authors describe as marginal (Ohira, 2014).

Across these literatures, chase-escape is not a single model but a structural asymmetry. In graph-growth models, blue can move only through red territory (Durrett et al., 2018). In tree variants with death or distance dependence, the frontier is encoded by renewal events, weighted Catalan numbers, and continued fractions (Beckman et al., 2019, Hernandez-Torres et al., 2022). In group pursuit, capture depends on encirclement, pincer geometry, or group partition rather than on simple nearest-direction pursuit (Matsumoto et al., 2012, Gao, 27 Dec 2025). In optimization, the chaser is constrained by the evader’s current configuration rather than the objective itself (Ohira, 2014). This suggests that “chase-escape” now denotes a class of asymmetric competitive dynamics in which pursuit is mediated by the evader’s footprint, trajectory, or state, and in which phase transitions, survival thresholds, and capture efficiencies are governed as much by geometry and accessibility as by nominal speed.