Spatial Colonization-Extinction Dynamics

• Spatial colonization-extinction dynamics is a framework that quantifies how local extinctions and site colonization, driven by dispersal and spatial structure, determine population persistence.
• Models like the Levins metapopulation and stochastic lattice frameworks rigorously characterize critical thresholds, scaling laws, and phase transitions in heterogeneous environments.
• Insights from these dynamics inform conservation by highlighting how connectivity, dispersal range, and environmental fluctuations influence biodiversity and guide management strategies.

Spatial colonization-extinction dynamics quantify the interplay between local population extinction events and colonization of unoccupied sites within a spatially structured landscape. This field unifies patch-occupancy frameworks such as the Levins model, stochastic contact processes, persistence-time distributions, and large-deviation analyses of rare extinction trajectories. Research systematically addresses how dispersal kernels, spatial coupling topology, environmental noise, and the geometry of initial populations modulate persistence, coexistence, and phase transitions in ecosystems, with direct implications for biodiversity, invasion biology, and conservation management.

1. Patch-Occupancy Models and Time-Dependent Colonization–Extinction Rates

The classical Levins metapopulation model frames colonization and extinction as mean-field processes across a network of identical habitat patches, tracking the occupancy fraction $x(t)$ via

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with constant per-patch colonization ($c$) and extinction ($e$) rates. Persistence occurs iff $c > e$ (Robledo et al., 29 Apr 2025). Robledo & Bustamante generalize this model to nonautonomous dynamics, introducing time-varying colonization $c(t)$ and extinction $e(t)$. The equation becomes

$$\dot x(t) = c(t)\, x(t)\, [1 - x(t)] - e(t)\, x(t),$$

with $c(t), e(t)$ bounded and $c(t)\geq c_* >0$.

Defining the net growth rate $r(t) = c(t) - e(t)$, persistence/extinction is controlled by the lower and upper Bohl exponents:

$$\mu^- = \liminf_{T\to\infty} \frac{1}{T} \int_0^T r(s)\, ds, \qquad \mu^+ = \limsup_{T\to\infty} \frac{1}{T} \int_0^T r(s)\, ds.$$

• Persistence: If $\mu^- > 0$, the system admits a globally attracting strictly positive bounded occupancy, even with fluctuations in $c(t),e(t)$.
• Extinction: If $\mu^+ < 0$, extinction is globally exponentially stable.
• Intermediate regime: If $\mu^- < 0 < \mu^+$, the system is driven arbitrarily close to extinction infinitely often, though weak persistence—rebounds to positive occupancy—may occur depending on the fine structure of $c(t),e(t)$.

This approach rigorously decouples instantaneous from long-term risk, anchoring management interventions to the time-averaged difference between colonization and extinction (Robledo et al., 29 Apr 2025).

2. Lattice Models: Scaling of Extinction Thresholds with Dispersal Range

Spatially structured stochastic lattice models, such as the contact process, characterize discrete extinction–colonization dynamics under finite dispersal (Juhász et al., 27 Dec 2025). On an infinite $d$-dimensional lattice with neighborhood (dispersal area) $A$, the critical colonization-to-extinction ratio for persistence is

$\lambda_E(A) = c/e\Big|_{\text{threshold}},$

which obeys universal scaling laws:

• Small-area regime $(A\lesssim 200)$: Shift $S(A) = \lambda_E(A) - 1 \sim A^{-0.74}$.
• Large-area regime $(A \gg 800)$: $S(A) \sim (6/\pi)\ln(A/3.5)/A$.

These scalings are robust to lattice geometry (diamond/square/circular/hexagonal), manifesting the directed-percolation universality class. Ecologically, larger dispersal areas dramatically lower the threshold for persistence at small $A$, with diminishing returns at large $A$. Evolutionary constraints thus typically situate real dispersal at the crossover between these regimes (Juhász et al., 27 Dec 2025).

3. Persistence-Time Distributions and Macroecological Scaling

Neutral models express species persistence in local patches as a renewal process with colonization and extinction transitions. The duration between colonization and extinction—persistence time $\tau$—follows a broad probability density

$p_{\tau}(t) = C t^{-\alpha} e^{-\nu t},$

with exponent $\alpha$ governed by dispersal network dimensionality and topology:

• 1D: $\alpha \approx 1.5$, 2D: $\alpha \approx 1.82$, Mean field: $\alpha=2$.

Importantly, the persistence-time statistics directly underlie spatial biodiversity patterns (species–area and endemics–area relationships), via

$$\langle S(A)\rangle = \nu(A)\, N(A)\, \langle \tau \rangle,$$

where $\nu(A)$ is the area-dependent diversification rate and $N(A)$ the number of individuals. This provides an explicit mapping between temporal turnover and static spatial scaling (Suweis et al., 2012).

4. Fundamental Mechanisms: Spatial Coupling, Stochasticity, and Initial Geometry

Spatial structure introduces nontrivial corrections to extinction thresholds and persistence (Soroka et al., 2016, Finkelshtein, 2020). The critical mortality rate $m_c(\varepsilon)$ for population persistence in spatial logistic models shifts with dispersal kernel length scale $\ell \sim 1/\varepsilon$:

• $d \geq 3$: $m_c(\varepsilon) = f - C \varepsilon^d$
• $d = 2$: $$m_c(\varepsilon) = f - C' \varepsilon^2 \ln(1/\varepsilon)$$
• $d = 1$: $m_c(\varepsilon) = f - C'' \varepsilon^3$

Spatial correlations generically depress the extinction threshold compared to well-mixed (mean-field) predictions.

In bistable (strong Allee-effect) media, the spatial geometry of the initial population—patch shape, size, and aspect-ratio—governs survival. Critical sizes (width, radius, patch area) in 1D/2D stochastic birth–death–movement models set sharp thresholds for extinction versus expansion, with diffusion peeling mass from high-perimeter regions and compact shapes enhancing persistence (Li et al., 2021, Meerson et al., 2010). The strong-Allee regime exhibits nucleation-dominated extinction where a critical nucleus, determined from reaction–diffusion saddle solutions, sets the spatial scale of spontaneous collapse.

5. Multi-Species and Cyclic Competition: Phase Structure and Extinction Pathways

For systems with cyclic or competitive interactions (May–Leonard, Lotka–Volterra), spatial structure and mobility engender a spectrum of persistence and extinction regimes (Rulands et al., 2010, Rulands et al., 2013):

• Domain Coarsening: Static spatial segregation with extinction via domain collision.
• Oscillating Traveling Waves: Intermediate mobility enables cyclic invasion fronts that prolong multi-species coexistence. Extinction is governed by front random walk and annihilation.
• Heteroclinic Orbits: High mobility washes out spatial structure, leading to globally synchronized predator–prey cycles terminated by demographic fluctuations.

Phase diagrams map transitions in extinction time scaling as a function of mobility, competition form, and dimensionality. The presence and stability of global attractors in the effective free-energy landscape correspond to empirical patterns of biodiversity and extinction rate (Rulands et al., 2013).

Competition–dispersal tradeoffs, stochastic immigration, and spatial dispersal heterogeneity further generate coexistence, bistability, and abrupt loss of specialist strategies, with sharp thresholds (cusp catastrophes) emerging from pair-approximation and lattice simulations (Martinez-Garcia et al., 2020).

6. Environmental Noise, Heterogeneity, and Nontrivial Extinction Dynamics

Temporal disorder (global environmental noise) fundamentally alters critical extinction behavior. In time-independent (quenched) environments, extinction–colonization transitions often fall into the directed-percolation class. With temporal fluctuations, infinite-noise fixed points emerge, generic densities decay logarithmically, and system lifetimes scale as power laws rather than exponentially (Barghathi et al., 2017). Spatial disorder (quenched random environments with drift) leads to anomalous critical exponents, Griffiths phases (broad distribution of local extinction events), and non-self-averaging survival times (Juhász, 2013).

In metapopulations with collapse events (catastrophes), dispersion into multiple colonies strictly increases survival probability compared to solitary populations. The critical survival threshold depends on birth rate, probability of post-collapse survival, and spatial neighborhood degree; spatial extension introduces wave-like patterns and persistent local refugia not present in non-spatial analogues (Machado et al., 2015, Roudenko et al., 2024).

7. Synthesis: Conservation, Management, and Evolutionary Implications

The core determinant of spatial population fate is the time-averaged, network-modulated balance between colonization and extinction, reshaped by dispersal scale, stochasticity, spatial correlations, and nonlinearity. Mathematical theorems and numerical simulations jointly demonstrate that:

• Increasing dispersal range generally promotes persistence, with sharply diminishing returns beyond a critical dispersal area (Juhász et al., 27 Dec 2025); real species are expected to evolve near this crossover.
• Highly fragmented or low-dimensional habitats amplify demographic stochasticity, reducing resilience to extinction (Finkelshtein, 2020, Soroka et al., 2016).
• Conservation planning should prioritize connectivity, temporal buffering of unfavorable periods, and identification of spatial thresholds that delineate weak persistence, metastability, and unavoidable collapse (Robledo et al., 29 Apr 2025, Li et al., 2021).

These principles are robust to ecological scenario—ranging from microbial rock–paper–scissors, interplanetary colonization, to landscape-scale plant metapopulations—and provide a quantitative framework for interpreting empirical macroecological scaling laws, designing spatial reserves, and predicting impact of environmental change on survival trajectories.