How many more is different?

TL;DR

The paper addresses how collective behaviors change abruptly with the number of participants by introducing the concept of critical numerosities. It shows that a subtle modification to bifurcation analysis—grounded in a birth–death framework and focusing on the extrema of the stationary distribution—unifies the identification of critical numbers across discrete stochastic and continuous deterministic models. Through three insect- and ant-inspired models, the authors demonstrate how these numerosity-driven transitions emerge and why standard ODE approximations can fail near the critical points, highlighting how different models can predict opposing transitions for the same phenomenon. The work provides a framework to reinterpret finite-group behaviors, guide experiments that vary group size, and potentially generalize to higher-dimensional collective states, with implications for both natural and engineered collectives.

Abstract

From the formation of ice in small clusters of water molecules to the mass raids of army ant colonies, the emergent behavior of collectives depends critically on their size. At the same time, common wisdom holds that such behaviors are robust to the loss of individuals. This tension points to the need for a more systematic study of how number influences collective behavior. We initiate this study by focusing on collective behaviors that change abruptly at certain critical numbers of individuals. We show that a subtle modification of standard bifurcation analysis identifies such critical numbers, including those associated with discreteness- and noise-induced transitions. By treating them as instances of the same phenomenon, we show that critical numbers across physical scales and scientific domains commonly arise from competing feedbacks that scale differently with number. We then use this idea to find overlooked critical numbers in past studies of collective behavior and explore the implications for their conclusions. In particular, we highlight how deterministic approximations of stochastic models can fail near critical numbers. We close by distinguishing these qualitative changes from density-dependent phase transitions and by discussing how our approach could generalize to broader classes of collective behaviors.

Figures (7)

• Figure 1: Collectives exhibit qualitatively different behavior as they become more numerous.
• Figure 2: (a) Many models of collective behavior are birth-death Markov chains. (b) A standard bifurcation analysis, which treats the dynamics as deterministic, balances $b_n(x)$ and $d_n(x)$ to identify equilibria. (c) The formula (\ref{['eq: birth death']}) for the stationary distribution of the Markov chain in (a) indicates that $b_n(x-1)$ and $d_n(x)$ should be balanced instead.
• Figure 3: Critical numerosity in path selection. The Föllmer--Kirman (FK) model describes ants choosing between two identical paths to a food source kirman_ants_1993. ( a) Representative timeseries of the FK model from \ref{['eq: rates of fk model']} with parameters $r = 1$ and $s = 0.02$. ( b) The stationary distribution $\pi_n$ transitions from bimodal to unimodal as the number of individuals $n$ increases from below to above the critical numerosity $n_c \approx 50$. ( c) This reflects a change in collective behavior, from alternating consensuses to a lack of consensus.
• Figure 4: Critical numerosity in trail formation. The Beekman--Ratnieks--Sumpter (BRS) trail formation model describes a colony of ants attempting to forage using a pheromone trail beekman_phase_2001. ( a) Representative timeseries of the BRS model with parameters $q = 0.03$, $r = 0.002$, and $s = 2$. ( b) As $n$ increases beyond the critical numerosity $n_c \approx 50$, the stationary distribution forms a new peak at a positive value of $x$. ( c) This reflects a change in collective behavior from the absence of a trail to a trail that the majority of ants follow.
• Figure 5: Critical numerosity in task allocation. Pacala et al. model the allocation of social insects to one of two tasks. ( a) The model has one modified equilibrium, $x_{\mathrm{eq}}$, which is unstable (dashed line) and lies outside the interval $[1,n-1]$ (shaded region) when $n$ is small. As $n$ increases, $x_{\mathrm{eq}}$ becomes stable (solid line) and enters the interval at $n_c = 50$. ( b) The entry of $x_{\mathrm{eq}}$ into the interval $[1,n-1]$ produces a peak of $\pi_n (x)$ at a strictly positive value of $x$. ( c) This peak reflects a change in collective behavior from ants concentrating on the second task to increasingly working on the first.