Neutral theory of cooperative dynamics

TL;DR

The paper develops a neutral theory for cooperative ecosystems where replication requires cooperation, introducing a two-rule dynamics with migration into a well-mixed population of fixed size $N$ and migration rate $\mu$. They derive a closed-form steady-state abundance distribution $P_n \propto \frac{(1-\mu)^n}{n} \exp\left(-\frac{(n-N\lambda^*)^2}{2N}\right)$, with a self-consistently determined Simpson index $\lambda^*$, and show how the distribution transitions between a Logseries-dominated regime at high migration and a bimodal distribution at low migration due to a cooperator core. The work also provides dynamical insights via the maximum-abundance distribution $Q_m$ and residence times, revealing a long-lived dynamical core and short-lived non-core species, with explicit formulas for $\tau_{core}$ and $\tau_{out}$. These results connect to microbiome patterns and origin-of-life questions, offering testable predictions for engineered microbial consortia and potential extensions to spatial networks. Overall, the study offers a minimal, analytically tractable baseline for understanding cooperative dynamics under neutral assumptions and highlights how cooperation can stabilize diversity against migration limitations.

Abstract

Mutualistic interactions are widespread in nature, from plant communities and microbiomes to human organizations. Along with competition for resources, cooperative interactions shape biodiversity and contribute to the robustness of complex ecosystems. We present a stochastic neutral theory of cooperator species. Our model shares with the classic neutral theory of biodiversity the assumption that all species are equivalent, but crucially differs in requiring cooperation between species for replication. With low migration, our model displays a bimodal species-abundance distribution, with a high-abundance mode associated with a core of cooperating species. This core is responsible for maintaining a diverse pool of long-lived species, which are present even at very small migration rates. We derive analytical expressions of the steady-state species abundance distribution, as well as scaling laws for diversity, number of species, and residence times. With high migration, our model recovers the results of classic neutral theory. We briefly discuss implications of our analysis for research on the microbiome, synthetic biology, and the origin of life.

Figures (7)

• Figure 1: Neutral model of cooperators The population evolves according to two rules: (a) during replication, individuals of different species (black and red balls) cooperate. The red individual replicates and replaces a randomly chosen individual (blue ball). (b) during migration with probability $\mu$, an individual from a new species (yellow ball) enters from an external pool and substitutes a randomly-chosen individual (blue ball).
• Figure 2: Schematic illustration of the species abundance distribution $P_n$ in steady state. This distribution depends on population size $N$, migration rate $\mu$, and the steady-state Simpson index ${\lambda^*}$. In Eq. \ref{['eq:Pn1']}, it is approximated as the product of Logseries distribution, arising from competition for space, and a Gaussian distribution, arising from cooperative interactions. At low migration, the combined distribution exhibits a bimodal shape, with local maxima at $n=1,n=N{\lambda^*}-1/{\lambda^*}$ and a local minimum at $n=1/{\lambda^*}$, see Eq. \ref{['eq:nminmax']}. Note that the two terms combine multiplicatively, not additively, so their areas do not add up to the area of the combined distribution.
• Figure 3: Scaling of Simpson index and species abundance distributions.(a) Scaling of Simpson index versus migration probability $\mu$ in a system with $N=10^5$ individuals. We compare mean and standard deviations from simulations of the full system (100 runs; black), numerical inversion of Eq. \ref{['eq:solA0']} (red), approximations in low migration \ref{['eq:ss-div']} (blue) and high migration \ref{['eq:simphighmig']} (orange) regimes. Solid vertical line indicates $\mu_B$\ref{['eq:bimodality']} where bimodality is lost; dotted vertical line indicates $\mu_L$\ref{['eq:transition']} where system transitions to Logseries regime. (b)-(d) Empirical histograms from simulations (across 1000 runs) versus predicted steady-state distributions \ref{['eq:Pn1']} for three migration probabilities. As in Fig. \ref{['fig:schematicss']}, shaded areas represent Logseries (orange) and Gaussian (blue) contributions. (b) Low-migration regime exhibits a bimodal distribution with a cooperator core. (c) Bimodality disappears once migration probability increases beyond $\mu_B$\ref{['eq:bimodality']}. (d) At higher migration probabilities, including the Logseries transition point where $\mu_L={\lambda^*}$\ref{['eq:transition']}, the distribution approaches the Logseries, as predicted by Hubbell's neutral theory.
• Figure 4: Scaling of number of species $R^*$ versus migration probability $\mu$ ($N=10^5$). We compare mean and standard deviations from simulations of the full system (100 runs; black); approximations in the low-migration regime of the number of total species (red), core species \ref{['eq:Rcore_estimate']} (blue), and non-core species \ref{['eq:Rout_estimate']} (orange); approximations in the high-migration regime of the total number of species (purple). Solid vertical line indicates $\mu_B$\ref{['eq:bimodality']} where bimodality is lost; dotted vertical line indicates $\mu_L$\ref{['eq:transition']} where system transitions to Logseries regime.
• Figure 5: Abundance trajectories from entry to extinction, illustrating that species that enter the dynamical core reside for much longer times. We sample two abundance trajectories from the stationary process ($N=10^5,\mu=10^{-5}$), one for a species that enters the dynamical core and one for a species that does not. Dashed gray line indicates the abundance position of the local minimum discussed in Sec. \ref{['sec:dyn']}\ref{['sec:infiltration-probability']}, stars indicate maximum abundances reached.