Title: Deriving mesoscopic models of collective behaviour for finite populations,
Abstract: Animal groups exhibit emergent properties that are a consequence of local
interactions. Linking individual-level behaviour to coarse-grained descriptions
of animal groups has been a question of fundamental interest. Here, we present
two complementary approaches to deriving coarse-grained descriptions of
collective behaviour at so-called mesoscopic scales, which account for the
stochasticity arising from the finite sizes of animal groups. We construct
stochastic differential equations (SDEs) for a coarse-grained variable that
describes the order/consensus within a group. The first method of construction
is based on van Kampen's system-size expansion of transition rates. The second
method employs Gillespie's chemical Langevin equations. We apply these two
methods to two microscopic models from the literature, in which organisms
stochastically interact and choose between two directions/choices of foraging.
These `binary-choice' models differ only in the types of interactions between
individuals, with one assuming simple pair-wise interactions, and the other
incorporating higher-order effects. In both cases, the derived mesoscopic SDEs
have multiplicative, or state-dependent, noise. However, the different models
demonstrate the contrasting effects of noise: increasing order in the pair-wise
interaction model, whilst reducing order in the higher-order interaction model.
Although both methods yield identical SDEs for such binary-choice, or
one-dimensional, systems, the relative tractability of the chemical Langevin
approach is beneficial in generalizations to higher-dimensions. In summary,
this book chapter provides a pedagogical review of two complementary methods to
construct mesoscopic descriptions from microscopic rules and demonstrates how
resultant multiplicative noise can have counter-intuitive effects on shaping
collective behaviour.