Intrinsic Noise in Collective Dynamics

Abstract

Animal groups exhibit many emergent properties that are a consequence of local interactions. Linking individual-level behaviour to group-level dynamics has been a question of fundamental interest from both biological and mathematical perspectives. However, most empirical studies have focussed on average behaviours ignoring stochasticity at the level of individuals. On the other hand, conclusions from theoretical models are often derived in the limit of in nite systems, in turn neglecting stochastic e ects due to nite group sizes. In our study, we use a stochastic framework that accounts for intrinsic-noise in collective dynamics arising due to (a) inherently probabilistic interactions and (b) nite number of group members. We derive equations of group dynamics starting from individual-level probabilistic rules as well as from real data to understand the e ects of such intrinsic noise and the mechanisms underlying collective behaviour. First, using the chemical Langevin method, we analytically derive models (stochastic di erential equations) for group dynamics for a variable m that describes the order/ consensus within a group. We assume that organisms stochastically interact and choose between two/four directions. We nd that simple pairwise interactions between individuals lead to intrinsic-noise that depends on the current state of the system (i.e. a multiplicative or state-dependent noise). Surprisingly, this noise creates a new ordered state that is absent in the deterministic analogue. Next, focussing on small-to-intermediate sized groups (10-100), we empirically demonstrate intrinsic-noise induced schooling (polarized or highly coherent motion) in sh groups. The fewer the sh, the greater the intrinsic-noise and therefore the likelihood of alignment. Such empirical evidence is rare, and tightly constrains the possible underlying interactions between sh. Our model simulations indicate that sh align with each other one at a time, ruling out other complex higher-order interactions. Further, we analyze the method to derive the group-level dynamical equation using simulated data from two different models of collective behaviour. In doing so we resolve important time-scale related issues with deriving the deterministic and stochastic components of the mesoscopic description from the data. Broadly, our results demonstrate that rather than simply obscuring otherwise deterministic dynamics, intrinsic-noise is fundamental to the characterisation of emergent collective behaviours, suggesting a need to re-appraise aspects of both collective motion and behavioural inference