Optimal Channel Networks accurately model ecologically-relevant geomorphological features of branching river networks

The paper addresses how the spatial structure of river networks influences ecological dynamics, particularly metapopulation stability and persistence. River networks are fractal systems shaped by stochastic and deterministic processes, with scaling described by Horton’s laws and power-law distributions of areas and lengths. While these properties are central in geosciences, ecological studies often approximate rivers with simplified synthetic networks: balanced binary trees (BBTs), random branching networks (RBNs), or geomorphologically grounded Optimal Channel Networks (OCNs). Prior work sometimes treated branching probability as an intrinsic, scale-invariant driver of ecological dynamics. The study aims to (i) test whether branching probability is scale-invariant, (ii) compare the topology and scaling of BBTs, RBNs, and OCNs to real rivers, and (iii) evaluate the consequences of using these analogues for metapopulation stability and persistence metrics.

Foundational geomorphology shows river networks follow Horton’s laws (bifurcation and length ratios) and power-law scaling of drainage areas and lengths. OCNs emerge from energy minimization and reproduce real rivers’ scaling. Ecological research has used: (1) BBTs with equal path lengths from sources to outlet for theoretical explorations of connectivity and stability; (2) RBNs assembled from random link lengths (often exponential/geometric), with several studies emphasizing branching probability as a key driver of richness, genetic diversity, and stability; and (3) OCN-derived analogues respecting geomorphological principles, used in some ecohydrological studies to capture realistic scaling and connectivity. Prior claims that branching probability inherently characterizes network complexity had not been rigorously tested across scales nor linked systematically to ecological metrics, motivating the present comparative analysis.

Synthetic networks: Generated 50 OCNs on 200×200 lattices (A = 40,000 pixels) with random outlet placement using the R package OCNers. Extracted each OCN at threshold drainage areas A_T = 20, 100, 500 pixels (and A_T = 1 for area-scaling analysis). For each OCN and A_T, computed number of nodes N and branching ratio p_r = N_ℓ/N, then generated matched RBNs and BBTs with the same N and p_r.

• RBNs: Sampled link lengths from a geometric distribution with mean 1/p_r, ensuring total network length equals N, imposed outdegree 1, indegree 0 or 2, loopless assembly (following Terui et al. 2018).
• BBTs: Grew trees by orders: for nodes of order i, assigned 1 or 2 upstream nodes with probability p (different from observed p_r), until N nodes allocated. Derived relation p ≈ p_r/(2 − p_r).

Real rivers: Extracted 50 basins (25 Europe, 25 North America) using open-source DEMs via the elevatr R package, selecting mountainous regions to ensure robust drainage extraction. Matched each basin to ~40,000 pixels (±20%) by tuning DEM resolution. Flow directions computed using D8 (TauDEM). Extracted channel networks at A_T = 20, 100, 500 pixels.

Scaling relations: Employed previously derived OCN relationships to approximate expected N and p_r as functions of A_T and A (pixels): E[N] ≈ 0.435 A_T^0.4 A; E[p_r] ≈ 1.531 A_T^−0.52 A^−0.032. Used these to interpret scale effects (visualized in Fig. 2a,b).

Topological and scaling analyses: Computed Horton’s bifurcation ratio R_b and length ratio R_l across network orders and assessed power-law scaling of drainage areas (β) for rivers, OCNs, RBNs, and BBTs. For Fig. 5c, excluded nodes with drainage area >2000 pixels for fitting.

Ecological metrics: Calculated two core measures:

• Metapopulation stability via coefficient of variation CV_M. Assumed local means and SDs scale with habitat size H_i; covariance between nodes i and j decays exponentially with along-stream distance d_ij: C_ij = σ_i σ_j exp(−d_ij/α). Evaluated two scenarios: • Uniform patch sizes (H_i constant): CV_M,U = sqrt((1 + 2∑∑ exp(−d_ij/α))/N). • Non-uniform patch sizes: H_i proportional to river width ∝ A_i^0.5; used number of upstream nodes U_i as a proxy for drainage area, normalized so total habitat sums to 1; CV_M,H computed with H_i weighting in both variance and covariance terms. Also evaluated CV_M,H in the α→0 limit to isolate patch-size effects (CV_M,0).
• Metapopulation capacity λ_M as the leading eigenvalue of M with off-diagonal entries H_i H_j exp(−d_ij/α) and zeros on the diagonal, for uniform (λ_M,U) and non-uniform (λ_M,H) patch-size scenarios. Sensitivity: Assessed α (mean dispersal distance) at α = 10, 20, 100, 200, 1000 (in pixel-length units). Compared metrics across network types and A_T scales (20, 100, 500).

• Branching probability/ratio is scale dependent: The branching ratio p_r = N_ℓ/N varies with observational scale (A_T and pixel size l). The same drainage network can yield nearly any p_r in [0,1) depending on A_T and resolution. River rankings by p_r change across A_T values, demonstrating p_r does not capture inherent branching complexity.
• Horton’s laws: Real rivers, OCNs, and RBNs exhibit R_b and R_l within typical hydrological ranges (R_b ≈ 3–5; R_l ≈ 1.5–3.5), whereas BBTs show systematically lower ratios, failing to satisfy Horton’s laws.
• Area scaling: Power-law exponents for drainage area distributions: Rivers β = 0.46; OCNs β = 0.45; RBNs β = 0.51 (deviating from typical β = 0.43 ± 0.02); BBTs lack power-law scaling. River-specific β ranged 0.36–0.57, tending toward 0.43 ± 0.02 with greater A (more resolved catchments).
• Ecological metrics: OCNs closely match real rivers for metapopulation stability (lower CV_M) and capacity (λ_M), across A_T scales and for intermediate–high α. RBNs overestimate λ_M and yield higher CV_M (less stability); BBTs show the strongest biases (highest CV_M, overestimated λ_M), with some exceptions at extreme parameter values. • CV_M decreases as A_T decreases (finer resolution, larger N) for rivers, OCNs, and RBNs, consistent with portfolio effects; BBTs often violate this trend at moderate–high α. • Using coarser DEM resolution (larger l) at fixed A_T (km²) inflates p_r by reducing N (node count), artificially increasing CV_M due to smaller N. • λ_M,U and λ_M,H: OCNs ≈ real rivers; RBNs and BBTs generally overestimate. For very low α (e.g., 0.1) and low A_T (fine networks), OCN λ_M,U can be underestimated relative to real rivers, likely due to higher N in extracted rivers; this mismatch disappears under non-uniform patch sizes (λ_M,H).
• Mechanisms: Random analogues (BBT, RBN) produce overly compact connectivity (shorter mean pairwise distances) and misrepresent patch-size distributions, leading to overestimated dispersal effects (higher synchrony, instability) and inflated persistence capacity relative to real rivers/OCNs.

Findings demonstrate that branching probability/ratio is not an intrinsic, scale-invariant descriptor of river complexity; it depends on observation scale and species-relevant spatial resolution. Consequently, ecological inferences based on p_r or random analogues can be biased. OCNs, which embed geomorphological optimality and randomness, reproduce Horton ratios and drainage-area scaling and yield ecological metrics (stability and capacity) that align with real rivers. RBNs, though satisfying some topological laws, deviate in area scaling and produce biased ecological predictions; BBTs are particularly unrealistic. These results underscore that realistic network topology and scaling are essential for accurate projections of metapopulation stability, synchrony, and persistence in riverscapes and support integrating geophysical principles into ecological modeling.

The study establishes that branching probability/ratio is scale dependent and unsuitable as a universal driver of ecological dynamics in river networks. OCNs are superior synthetic landscapes, matching real rivers’ geomorphological scaling and producing accurate metapopulation stability and capacity estimates. Random analogues (especially BBTs, but also RBNs) bias ecological metrics by distorting distances and patch-size structure. The authors recommend OCNs for theoretical and applied ecohydrological studies and urge explicit reporting of observational scales (A, A_T, node size) when using synthetic networks. Future work should integrate additional ecological realism (e.g., Euclidean distances among non-flow-connected sites, fat-tailed and density-dependent dispersal, abiotic heterogeneity linked to drainage area) within OCN-based frameworks and broaden validation across diverse regions and resolutions.

• Scale dependence: Results hinge on chosen observational scales (A_T, DEM pixel size l) and on approximations E[N], E[p_r] derived from a limited set of OCNs.
• Network selection: Real basins were mainly mountainous to ensure extraction quality; findings may require testing in low-relief regions.
• Ecological assumptions: Dispersal modeled with exponential kernels; covariance based on along-stream distance only; Euclidean distances, fat-tailed and density-dependent dispersal were not fully incorporated into primary analyses.
• Proxy for drainage area in random networks: Used upstream node count as a surrogate for drainage area in RBNs and BBTs, which lack true area definitions.
• DEM and extraction method: D8 algorithm and threshold-based channel initiation may influence extracted network structure.
• Metric sensitivity: Some discrepancies at very low α and fine A_T suggest sensitivity to N differences between real rivers and OCNs.