Self-organized criticality in complex model ecosystems

Abstract

We show that spatial extensions of many-species population dynamics models, such as the Lotka-Volterra model with random interactions we focus on in this work, generically exhibit scale-free correlation functions of population sizes in the limit of an infinite number of species. Using dynamical mean-field theory, we describe the many-species system in terms of single-species dynamics with demographic and environmental noises. We show that the single-species model features a random mass term, or equivalently a random space-time averaged growth rate, poising some species very close to extinction. This introduces a hierarchy of ever larger correlation times and lengths as the extinction threshold is approached. In turn, every species, even those far from extinction, are coupled to these near-critical fields which combine to make fluctuations of population sizes generically scale-free. We argue that these correlations are described by exponents derived from those of directed percolation in spatial dimension $d=3$, but not in lower dimensions.

Figures (6)

• Figure 1: Vertex associated with the environmental noise $\delta g(x,t)$. An outgoing leg with an arrow corresponds to a response field $\hat{n}$ and an outgoing leg with no arrow corresponds to a field $n$.
• Figure 2: Fourier-space representation of the contraction of the vertex arising from the environmental noise and the vertex with $(p,q)$ externel legs of the directed percolation theory at the renormalization scale $\ell$. A continuous line with an arrow denotes the Gaussian propagator $G_{\ell}(k,\omega)$ of the directed percolation theory at the same renormalization scale.
• Figure 3: A typical 1-particle irreducible graph where the dotted line is not part of the loop. One can go around the loop in the direction of the arrow, so the graph vanishes by causality.
• Figure 4: A second-order graph which introduces graphical corrections to the flow equation of the large-scale amplitude $A_\ell$.
• Figure 5: Two-time correlation function of the dynamics in Eq. (\ref{['eq:DP_fully_connected']}), with $\delta n(t) = n(t) - \bar{m}$. As the critical point is approached $\epsilon=\bar{g}-\bar{g}_{c}\to0$, the correlation function, rescaled by its value at equal time, converges to an exponentially decaying function with an $\epsilon$-independent characteristic timescale. This holds for $D-\bar{g}_{c}<0$ (left) and for $D-\bar{g}_{c}>0$ (right). We have followed dornic2005integrationweissmann2018simulation for simulating the demographic noise. Parameters: $D=2$ (left) and $D=0.5$ (right), $dt=10^{-3}$.