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 infinite systems, in turn neglecting stochastic effects due to finite 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) finite number of group members. We derive equations of group dynamics starting from individual-level probabilistic rules as well as from real data.

First, using the chemical Langevin method, we analytically derive models (stochastic differential 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 find 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, we develop a method to derive the group-level dynamical equation directly from the data of collective dynamics. We assert that such an equation extracted from the data encodes important information about the underlying interactions. Therefore, we derive this equation describing the dynamics of order in two real systems- Fish (Etroplus suratensis) and Whirligig Beetles (Gyrinidae dineutes) in my next two chapters.

Focussing on small-to-intermediate sized groups (10-100), we demonstrate that intrinsic-noise induces schooling (polarized or highly coherent motion) in fish groups. The fewer the fish, the greater the intrinsic-noise and therefore the likelihood of alignment. Such empirical evidence is rare, and tightly constrains the possible underlying interactions between fish. Our model simulations indicate that E. suratensis align with each other one at a time (positive-pairwise), ruling out other complex interactions.

Finally, we apply the same method to swarms of Whirligig Beetles which shows contrasting rotational order. We find that a different set of interactions - negative-pairwise and positive-three-body interactions between individuals are required to explain the observed group dynamics. Whilst the three-body interactions can explain the structure of the deterministic part of the equation, negative-pairwise explains the stochastic counterpart.

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.