Species on a network

A live simulation of the generalized random Lotka–Volterra model on heterogeneous interaction networks. Each dot is a species, placed by degree with hubs at the centre. Dashed curves in the plots below are the heterogeneous dynamical mean-field prediction for the same network.

live
Abundance 0.01λ10λ extinct Dot size grows with degree
Runningt = 0
SimulationTheory
Survival fraction φ––
Mean abundance ⟨x⟩––
Survivors' mean degree K′/K––
Weighted mean m––
Degree assortativity rinitial → survivors–
Stability index≥ 1 suggests multiple attractors––

Survival by degree, φ(k)

Points: fraction of species in each degree bin still alive. Dashed: theory. In cooperative communities with α = β = 0, the marked degree k* is the least survivable.

Mean abundance by degree, ⟨x | k⟩

Average abundance within each degree bin, extinct species counted as zero.

Abundance distribution, ρ(x)

Histogram of surviving abundances, normalised over all species so the area equals φ. The theory is a mixture of truncated Gaussians, one per degree.

Trajectories

Forty species sampled across the degree range; lines closer to the text colour have higher degree. Lines that fall to the floor have gone extinct. Click the plot to jump the network view and the other plots to that time.

The model

S species with abundances xi follow

dxi/dt = xi ( λ − xi − Σj Jij Aij xj )

where A is a symmetric adjacency matrix with degrees ki and mean degree K. The strength Jij (effect of j on i) is Gaussian and independent of Jji, with

mean = (μ/K)(ki/K)α(kj/K)β,   variance = (σ²/K)(ki/K)α(kj/K)β

Positive J suppresses growth, so μ > 0 is a competitive community and μ < 0 a cooperative one. With α = β = 0 this is the model of Park, Lee, Lee and Park; nonzero α (receiver) and β (exerter) add the degree-dependent strengths of Lee et al. Scale-free networks come from the static model, with node weights i−1/(γ−1).

Integration uses fourth-order Runge–Kutta on log-abundance with an adaptive step. A species below 10−10 is extinct. The run stops when every species above 10−3 has |dx/dt| < 10−7 and every rarer species is either settled or still declining (declining ones are then removed), when abundances exceed 105, or at t = 10000.

The theory

The mean-field solution gives the stationary abundance of a species with degree k as a truncated Gaussian,

x(k) = max(0, λ − μ m ck − σ √(q ck) z),   ck = (k/K)α+1,   z ~ N(0,1)

with m = Σk P(k)(k/K)1+β⟨x(k)⟩ and q = Σk P(k)(k/K)1+β⟨x(k)²⟩ solved self-consistently. Here P(k) is the degree sequence of the network you are simulating (annealed approximation). The stability index σ²⟨(k/K)2+α+βΘ(x)⟩ reduces to the paper's σ²⟨k²Θ(x)⟩/K² at α = β = 0; the general form is a straightforward extension made for this demo.

Expect agreement in the unique-fixed-point phase when 1 ≪ K ≪ S. Deviations grow for small K, large |μ| or σ, strong heterogeneity at small S, and beyond the unbounded-growth or multiple-attractor boundaries.

J. I. Park, D.-S. Lee, S. H. Lee, H. J. Park, Phys. Rev. Lett. 133, 198402 (2024).

H. S. Lee, D.-S. Lee, S. H. Lee, S. Suweis, H. J. Park, arXiv:2607.18809.