After studying the stability of large ecological networks, we wanted to try to describe the transient dynamics following a small perturbation of the equilibrium. We thus studied “reactivity”, which tells us whether perturbations of a stable equilibrium are going to be amplified before decaying.
which we published in frontiers in Ecology and Evolution, a new journal with an interesting peer-review scheme (a topic dear to my heart!). In fact, I am so happy journals are trying new ways of doing peer review I decided to join the editorial board.
I always joke in the lab that anybody who wants to propose a new measure in ecology (or index, etc.) should pay $1,000, $2,000 if the new measure has an acronym. These funds could pay for graduate students to go to some conference.
The only exception to the rule is for studies showing that two seemingly different measures are in fact the same thing. This is the case of our recent study on nestedness, published today in Nature Communications:
The ghost of nestedness in ecological networks
Phillip P. A. Staniczenko, Jason C. Kopp & Stefano Allesina Ecologists are fascinated by the prevalence of nestedness in biogeographic and community data, where it is thought to promote biodiversity in mutualistic systems. Traditionally, nestedness has been treated in a binary sense: species and their interactions are either present or absent, neglecting information on abundances and interaction frequencies. Extending nestedness to quantitative data facilitates the study of species preferences, and we propose a new detection method that follows from a basic property of bipartite networks: large dominant eigenvalues are associated with highly nested configurations. We show that complex ecological networks are binary nested, but quantitative preferences are non-nested, indicating limited consumer overlap of favoured resources. The spectral graph approach provides a formal link to local dynamical stability analysis, where we demonstrate that nested mutualistic structures are minimally stable. We conclude that, within the binary constraint of interaction plausibility, species preferences are partitioned to avoid competition, thereby benefiting system-wide resource allocation.
We uploaded the code needed for the analysis here.
Back in 2001, when I was studying for my PhD entrance exam, I read May’s 1972 paper for the first time, and fell in love with this problem.
May’s result is very simple to explain: take S species and suppose that each species interacts with any other with probability C. Then, the expected number of connections for each species is SC. Now assume that the ecosystem is at a steady state: species do not change in density through time if not perturbed. If two species interact, the effect of species a on species b is taken from a normal distribution with mean 0 and variance σ². Finally, the effect of a species on itself is assumed to be -1 (i.e., there is some sort of self-regulation).
May showed that such networks are almost surely stable (i.e., persist in time despite small perturbations) whenever σ√SC<1, while whenever this threshold is crossed the system is almost surely unstable, and thus should not persist through time. This result sparked a fierce “complexity-stability” debated that lasts to this day. In fact, May’s results are difficult to reconcile with the staggering diversity observed in nature: many species (large S) interact in complex networks (large C) of ecological interactions, despite mathematical considerations would tell us this is not possible.
Si and I tackled this problem in a new article just published by Nature:
Stability criteria for complex ecosystems
Forty years ago, May proved that sufficiently large or complex ecological networks have a probability of persisting that is close to zero, contrary to previous expectations. May analysed large networks in which species interact at random. However, in natural systems pairs of species have well-defined interactions (for example predator–prey, mutualistic or competitive). Here we extend May’s results to these relationships and find remarkable differences between predator–prey interactions, which are stabilizing, and mutualistic and competitive interactions, which are destabilizing. We provide analytic stability criteria for all cases. We use the criteria to prove that, counterintuitively, the probability of stability for predator–prey networks decreases when a realistic food web structure is imposed or if there is a large preponderance of weak interactions. Similarly, stability is negatively affected by nestedness in bipartite mutualistic networks. These results are found by separating the contribution of network structure and interaction strengths to stability. Stable predator–prey networks can be arbitrarily large and complex, provided that predator–prey pairs are tightly coupled. The stability criteria are widely applicable, because they hold for any system of differential equations.
A new episode of WBEZ Clever Apes covers the work of the Allesina lab on rock-paper-scissor, featuring Jason and Anna playing rock-paper-scissor-bull!
“Charles Darwin ushered in modern biology with his explanation of how different species evolve. But his work leaves us with a paradox: Why should dozens or even thousands of species coexist in a single habitat? The theory suggests they ought to duke it out until just a few winners dominate. And yet we have such magnificent biodiversity all over. More than 2,000 species of trees share a single acre of rainforest in the Amazon. So what gives?”