Two-Horizon Coexistence in Complex Systems

• Two-Horizon Coexistence Region is the parameter space where two distinct subsystems coexist without one dominating, defined by analytical inequalities and overlapping spatial or parametric scales.
• Mathematical models such as multitype contact processes and Richardson models quantify coexistence using renewal structures, probabilistic controls, and two-scale dynamics.
• The region’s significance spans ecology, disease dynamics, and black hole thermodynamics, offering insights on stability, phase transitions, and microphysical signatures.

The Two-Horizon Coexistence Region encompasses a set of states or parameter regimes in spatial, ecological, and thermodynamic systems characterized by the simultaneous persistence, interaction, or overlap of two distinct subsystems—often termed “horizons.” In population dynamics, ecology, and stochastic processes, such regions quantify robust multitype coexistence due to spatial scale separation or graph geometry. In black hole thermodynamics and quantum gravity, they index the interface between classically disjoint event horizons, with analytic order parameters and statistical properties closely resembling two-level mixtures. The fundamental property is that in the Two-Horizon Coexistence Region, neither subsystem dominates or extinguishes the other, and the underlying system displays distinctive stability, phase structure, or microphysical signatures.

1. Mathematical Models of Two-Horizon Coexistence

Stochastic, ecological, and thermodynamic models formalize two-horizon coexistence using domain-specific notation and dynamics.

• Multitype Contact Processes: On spatial graphs with two scales (microscopic and mesoscopic connectivity), the two-scale multitype contact process defines states via vertex occupation $\eta_t(x)\in\{0,1,2\}$, transition rates for birth and death, and generator

$$\mathcal{L} f(\eta) = \sum_{x}\sum_i [\lambda_i(x,\eta)\ (f(\eta^{x,i}) - f(\eta)) + \delta_i\,\mathbf{1}_{\{\eta(x)=i\}}\ (f(\eta^{x,0}) - f(\eta))]$$

where births occur along short and long edges at rates $\beta_i$ and $B_i$ respectively (Lanchier, 2010).

• Richardson Models: In the two-type Richardson model, coexistence is defined by simultaneous infinite growth of both infection types, with the parameter space for coexistence (Coex(G)) often exhibiting non-monotonic, multi-horizon structure (Deijfen et al., 2015).
• Ecotonal Metacommunity Models: Deterministic Lotka–Volterra dynamics with block-structured interactions and mixing strength $\mu$ yield equilibrium abundances for each community. The critical coexistence region is characterized by

$(R_i^{(1)})^2 < 4 R_i^{(2)}\quad \forall i$

and by overlap in penetration depths $x_{L,i}(\mu), x_{R,j}(\mu)$, quantifying spatial range extension (Heidelman et al., 10 Jan 2025).

• Black Hole Thermodynamics: In the coexistence region of AdS or de Sitter black holes, two order parameters, the horizon-radius ratio $\alpha=r_s/r_l$ and molecule-number ratio $\beta$, characterize the mixed phase of small and large black holes. Analytic expressions for $P(\alpha)$ and $T(\alpha)$ define the phase diagram (Wei et al., 2023).

2. Geometry, Scale Separation, and Phase Structure

The defining feature of two-horizon coexistence is the presence of distinct spatial or parametric “horizons” enabling persistent non-exclusive occupation or activity.

• In two-scale spatial processes on graphs, coexistence region geometry is dictated by the partitioning of the lattice into patches (size $N$), each connected internally at short range but only via centers at long range. For $N\gg1$, the coexistence region acquires positive Lebesgue measure in parameter space, yielding a robust two-dimensional region

$$\{ (\beta_1,\beta_2,B_1,B_2) : R_1>1\ge R_2,\ r_2>1\ge r_1 \}$$

in contrast to measure-zero “neutral” coexistence on single-scale lattices (Lanchier, 2010).

• Richardson model constructions generate finely tuned graphs—for instance, with two spines and sparse bridges—where infinite coexistence only occurs for isolated points (e.g., $\lambda=2$), intervals (e.g., $[2,5]$), or further, countable sets, breaking monotonicity and symmetry constraints typical in Euclidean graphs (Deijfen et al., 2015).
• In ecotonal models, the spatial horizon is the transition zone between two communities. The coexistence region is characterized by analytic overlap in penetration depths, maximizing local species richness when ranges from both horizons intersect (Heidelman et al., 10 Jan 2025).
• In black hole thermodynamics, the coexistence region is demarcated in $(T,P)$ space by coexistence of two horizon radii. The order parameter $\alpha$ undergoes critical scaling $\alpha_c - \alpha\propto(T_c-T)^{1/2}$, and the coexistence region is analytically parameterized via reduced variables.

3. Analytical Criteria and Order Parameters

Robust identification of the coexistence region relies on precise inequalities, probabilistic arguments, and analytic order parameters.

• Lotka-Volterra (Ecotonal) Criterion: Full coexistence for all $\mu$ requires $(R_i^{(1)})^2 < 4 R_i^{(2)}$ for all species, implying second-order feedback dominates extinction risk. The boundaries of the region are mathematically explicit, and the overlap of species’ penetration depths quantifies ecological mixing (Heidelman et al., 10 Jan 2025).
• Contact Process Coexistence Theorem: For two-scale graphs, there exists an open, positive measure set of parameters for which both types persist, shown via block-oriented percolation arguments and renewal structure. On single-scale lattices, coexistence collapses to a measure-zero set (Lanchier, 2010).
• Nonmonotonic Coexistence Set: For constructed graphs in the Richardson model, Chebyshev, LLN, and CLT provide sharp probabilistic control of infection race outcomes across spines and bridges, explicitly determining the coexistence region Coex(G) (Deijfen et al., 2015).
• Black Hole Dual Ratio Parameters: The coexistence region’s thermodynamic state is fully specified by $(\alpha, \beta) \in [0,1]^2$, where $\alpha$ is the horizon-radius ratio and $\beta$ the molecule fraction. Maxwell construction ensures $G_s(T,P)=G_l(T,P)$, with $\alpha$ serving as an order parameter for the first-order phase transition (Wei et al., 2023).

4. Microphysical and Thermodynamic Signatures

The two-horizon coexistence region encodes distinctive microphysical or macrostate properties.

• Quantum Black Holes: In charged 4D-EGB-de Sitter spacetime, effective thermodynamic quantities are constructed for the inter-horizon region. The heat capacity $C_{Q,l,\alpha}(T_{\rm eff})$ displays a Schottky-like peak, characteristic of a two-level system. The number of micro-degrees of freedom $N$ is estimated by

$N \simeq \frac{C_{Q,l,\alpha}(T_{\max})}{0.439}$

with $C_{\max}/(N\,k_B)\simeq0.439$, matching paramagnetic systems and indicating a gapped spectrum with two dominant levels (Liu et al., 10 Mar 2025). The region is therefore a quantum interface with quantifiable spectrum and order parameters.

• Contact Process Renewal Structure: Survival mechanisms in two-scale graphs rely on renewal blocks and translation-invariant random walks, yielding i.i.d. increments interpreted as percolation in time-space (Lanchier, 2010).
• Species Penetration (Ecotones): Analytical formulas for the penetration depth

$$x_{L,i}(\mu) = \frac{n^L_{1,i} - \mu (M_1 n^L_2)_i}{\Delta n_{1,i} + \mu (M_1 \Delta n_2)_i}\,L$$

quantify spatial overlap and thus enhanced biodiversity at the interface (Heidelman et al., 10 Jan 2025).

5. Biological, Ecological, and Physical Interpretation

The two-horizon coexistence region appears in both biological and physical contexts and carries rich interpretive significance.

• Metapopulation Ecology: Two-scale contact processes model patchy landscapes: local dispersal (microscopic edges) supports superior competitors (type 2); rare long-range dispersal (mesoscopic edges) favors invaders (type 1). Both types persist in open coexistence regions when their respective strategies align with the scale of resource (Lanchier, 2010).
• Disease Dynamics: Analogous to metapopulations, seasonal flu (local contacts) and H1N1 (global airline flights) persist in two-scale graphs, simulating robust coexistence of diseases exploiting different spatial transmission modes.
• Community Ecology (Ecotones): The mathematical structure of the two-horizon coexistence region in ecotonal models supports increased species richness in transitional zones, with coexistence sustained except at critical mixing strengths which trigger extinction thresholds (Heidelman et al., 10 Jan 2025).
• Quantum Gravity: In spacetimes with multiple horizons, the coexistence region reflects a mixture of quantum states associated with the two horizons. The two-level Schottky anomaly identifies quantum microstates and degrees of freedom, with implications for black hole entropy and quantum corrections (Liu et al., 10 Mar 2025).

6. Geometric, Probabilistic, and Ecological Implications

The emergence and structure of two-horizon coexistence regions depends strongly on system geometry, graph topology, and feedback strength.

• Graph Geometry: Non-symmetric, non-transitive graph designs enable coexistence at isolated ratios, intervals, or countable sets, violating monotonicity seen in symmetric lattices (Deijfen et al., 2015).
• Stability and Robustness: Two-scale or multi-horizon systems are robust to perturbations in parameter values, retaining coexistence across open regions of parameter space. By contrast, single-scale models offer coexistence only on neutral, measure-zero subspaces.
• Boundary Structure: The mathematical boundary of the coexistence region (e.g., $\partial \text{Coex}(G)$) is always at most countable, reflecting underlying monotonicity in rate dependence (Deijfen et al., 2015).
• Ecological Inference: The overlap criterion of penetration depths quantifies biodiversity enhancement, offering analytic conditions for ecotonal zones to support greater species richness than either parent community (Heidelman et al., 10 Jan 2025).

7. Unified Definition and Synthesis

The Two-Horizon Coexistence Region should be defined as the open subset of parameter space (contact rate ratios, mixing strengths, horizon radii, etc) for which both constituent subsystems persist with non-zero densities or abundances, neither is driven extinct, and interaction overlap (spatial or parametric) produces emergent stability, enriched diversity, or distinctive thermodynamic/statistical signatures. Analytical inequalities, renewal structures, graph geometry, order parameters, and heat capacity anomalies all serve to precisely characterize this region within diverse mathematical and physical frameworks (Lanchier, 2010, Deijfen et al., 2015, Heidelman et al., 10 Jan 2025, Wei et al., 2023, Liu et al., 10 Mar 2025).