Chase-Escape with Conversion Dynamics

• Chase-escape with conversion is a stochastic process where captured prey is converted into chasers, altering extinction probabilities and driving phase transitions.
• The conversion mechanism, governed by probability P_V, creates a feedback loop that results in non-monotonic lifetimes and optimal survival regimes on various graphs.
• Analytical, simulation, and percolation methods elucidate critical thresholds and scaling laws, with applications in ecology, epidemiology, and network security.

The chase-escape with conversion process describes a class of stochastic dynamics in which predator (chaser) and prey (escapee) agents interact on graphs or lattices, and caught escapees can be converted into new chasers rather than simply being removed. This framework generalizes classical epidemic and competitive growth models by introducing an additional feedback mechanism, enabling rich phenomena such as non-monotonic lifetimes and phase transitions. Conversion mechanisms—where caught targets or infected nodes become new agents of the antagonist species—alter extinction probabilities, critical thresholds, and spatial-temporal patterns, with applications in ecology, epidemiology, and modeling inflammatory diseases.

1. Conversion Dynamics and Feedback Mechanisms

In the chase-escape with conversion paradigm, the essential innovation is the conversion event: when a chaser catches a target, the latter is not simply removed but transformed (converted) into a new chaser with probability $P_V$ (Nishi et al., 2011). This feedback amplifies the population of antagonists and produces a chain reaction; each new chaser increases future catch rates, catalyzing further conversions. Formally, the event is realized as:

• Chaser adjacent to target:
  • With probability $P_V$: $\text{target} \rightarrow \text{new chaser}$
  • With probability $1-P_V$: $\text{target is removed}$

The interaction rules on periodic lattices, along with exclusion volume constraints and spatial measures, drive the resulting dynamics. In process terms, conversion transforms typical removal events into generative ones.

Such mechanisms have been extended to various contexts, including models for inflammatory damage (where "conversion" mimics regulatory T-cell arrival suppressing active inflammation) and device-to-device networks where "patches" are spread as conversion analogues.

2. Phase Transitions and Critical Thresholds

Conversion dynamics lead to sharply defined phase transitions, often characterized by critical thresholds for escape or survival probabilities. On complete graphs, the chase-escape model with conversion shows a phase transition at "equal fitness": when the infection rate $\lambda$ matches the chasing rate (Junge et al., 2 Oct 2025). Specifically:

• For $\lambda < 1$: The process almost surely survives, i.e., not all white sites are eliminated.
• For $\lambda = 1$: The extinction probability is $1/2^\alpha$.
• For $\lambda > 1$: Extinction (complete conversion) occurs with probability 1.

More generally, on trees and bounded degree graphs, the existence of a nontrivial site percolation threshold introduces further structure. The critical value for macroscopic damage is given by (Bailey et al., 28 Jul 2025):

$$\lambda_c(\alpha) = \sup\{ \lambda : P_{\lambda, \alpha}(X < \infty) = 1 \}$$

For bounded degree $d$ and site percolation threshold $p_c(G)$:

$$\alpha(d-2) \leq \lambda_c(\alpha) \leq \frac{d+\alpha}{1-p_c(G)^{1/d}}$$

$\lambda_c(\alpha) = \Theta(\alpha)$

Thus, the interplay between conversion rates $\alpha$, network topology, and percolation thresholds sets the critical regime for survival or extinction.

3. Lifetimes, Population Dynamics, and Monotonicity

Conversion modifies statistical properties such as aggregate damage, lifetimes, and the total number of converted sites. Key findings across models include:

• Non-monotonic dependence on initial target number: Numerical simulations reveal a "maximum lifetime" as a function of initial targets; too many targets triggers rapid conversion chains, shortening lifetimes, while too few limits conversion impact (Nishi et al., 2011).
• Aggregate damage is monotone in conversion rate $\alpha$: Increasing $\alpha$ reduces total damage; more efficient conversion suppresses spread and enhances containment (Bailey et al., 28 Jul 2025).
• For the complete graph, the total number of converted sites grows logarithmically with system size, scaling as $C(K_{n+1}, 1, \alpha) / \log n \rightarrow \alpha$ as $n \to \infty$ (Junge et al., 2 Oct 2025). The expected number of surviving (white) sites trends to $2 \alpha$.
• On one-dimensional chains, star graphs, and complete graphs, monotonicity also holds for the infection rate $\lambda$ and the number of nodes.

Crucially, the feedback inherent in conversion (target-to-chaser amplification) produces collective hunting efficiencies and can invert intuitive expectations about survivor lifetimes.

4. Analytical, Simulation, and Percolation Methodologies

Analyses of chase-escape with conversion employ explicit couplings (matching passage and conversion times), branching processes, birth-death chain reductions, and percolation theory. Representative methods:

• Independent exponential passage times are assigned to edges (spread) and vertices (conversion), enabling comparison between processes of varying $\lambda$, $\alpha$, or graph size (Bailey et al., 28 Jul 2025).
• On trees, the expected number of red (active) sites is bounded using branching process methods, with transition probabilities given by

$p_k^+ = \frac{\lambda}{1+\lambda+\alpha k}$

• For the complete graph, extinction probabilities are derived by analyzing the competition between an exponential random variable (death) and a Gamma random variable (conversion), yielding $2^{-\alpha}$ at criticality (Junge et al., 2 Oct 2025).
• Percolation techniques and stabilization arguments establish macroscopic survival or extinction, particularly in random geometric or device mobility models (Hinsen et al., 2019, Cali et al., 2022), with regimes determined by the underlying connectivity and local rates.

Monte Carlo simulations corroborate analytic thresholds and scaling relations, notably identifying universality classes (e.g., undirected percolation for the 2D chase-escape percolation model) (Kumar et al., 2020).

5. Applications: Biological Damage, Network Epidemics, and Universal Behaviors

The chase-escape with conversion framework finds direct applications in modeling:

• Inflammatory lesion development in multiple sclerosis, where "conversion" maps to regulatory suppression, and damage corresponds to the aggregate of converted (damaged) sites (Bailey et al., 28 Jul 2025).
• Malware and patch dynamics in device-to-device wireless networks; the conversion mechanism represents propagation of defense (patches) versus infection, with survival thresholds tied to agent speed, patch/infection delay, and street network geometry (Cali et al., 2022).
• Ecological models, such as rabies spread, predator-prey encounters, and social contagion, where conversion encapsulates recruitment or conversion of individuals into spreaders or defenders (Nishi et al., 2011, Tang et al., 2018).

Near criticality, these models display universal features: fractal spatial patterns, power-law distributions of holes (voids) in the colonized region $N(S) \approx S^{-1}$ (Tang et al., 2018), and stretched-exponential cluster size tails (Kumar et al., 2020).

6. Parameter Regimes and Optimization of Survival

A major theme is the identification of optimal parameter regimes for maximum target survival or containment:

• Lifetime optimization is achieved by balancing conversion probability $P_V$ and target self-multiplication probability $P_T$; too much proliferation triggers chain conversion, too little leads to rapid extinction (Nishi et al., 2011).
• On device networks, global survival of infection is possible within specific speed windows; too slow or too fast device movement precludes sustainable outbreak propagation (Cali et al., 2022).
• On random and structured graphs, phase transitions demarcate regions of local and global survival based on rate ratios and percolation properties (Hinsen et al., 2019).

A plausible implication is that in systems with feedback conversion, survival or containment is maximized not by simply minimizing spread or maximizing defense, but by careful tuning of competing rates within the constraints imposed by network structure and agent interaction topology.

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Setting	Phase Transition Location	Key Scaling/Formula
Complete graph	$\lambda = 1$	Extinction: $2^{-\alpha}$; Survivors: $2\alpha$; Converted: $\alpha\log n$
Bounded degree graph	$\lambda_c(\alpha) = \Theta(\alpha)$	Finite/infinite damage based on site percolation threshold
2D lattices	$p_c \approx 0.5$	Fractal region, power-law holes
D2D network	Window in device speed	Global survival/extinction by percolation arguments

7. Connections to Broader Statistical Physics and Stochastic Systems

Chase-escape with conversion models reveal deep connections to universality in percolation, KPZ interface fluctuation scaling, and interacting particle systems. Critical exponents—such as roughness $\alpha \approx 1/2$, dynamical $z \approx 1.5$, and cluster growth $\gamma = 43/18$—identify robust scaling behavior, offering quantitative links to classical models such as Eden growth or undirected bond percolation (Kumar et al., 2020).

The theoretical approaches and results in this domain inform not only the understanding of biological and network phenomena but also offer methods for analyzing general classes of spatially extended, feedback-driven stochastic systems.