Deriving Mesoscopic Models of Collective Behavior for Finite Populations

Abstract

Animal groups exhibit many emergent properties that are a consequence of local interactions. Linking individual-level behaviour, which is often stochastic and local, to coarse-grained descriptions of animal groups has been a question of fundamental interest from both biological and mathematical perspectives. In this book chapter, we present two complementary approaches to derive 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 pairwise interactions, and the other incorporating ternary effects. In both cases, the derived mesoscopic SDEs have multiplicative/state-dependent noise, i.e., the strength of the noise depends on the current state of the system. However, the different models demonstrate the contrasting effects of noise: increasing the order/consensus in the pairwise interaction model, whilst reducing the order/consensus in the higher-order interaction model. We verify the validity of such mesoscopic behaviour by numerical simulations of the underlying microscopic models. Although both methods yield identical SDEs for binary-choice systems that are effectively one-dimensional, the relative tractability of the chemical Langevin approach is beneficial in generalizations to higher-dimensions. We hope that this book chapter provides a pedagogical review of two complementary methods to construct mesoscopic descriptions from microscopic rules, how the noise in mesoscopic models is often multiplicative/state-dependent, and finally, how such noise can have counter-intuitive effects on shaping collective behaviour.

Noise-induced effects in collective dynamics and inferring local interactions from data

“…Additionally, animal groups are finite in size, and in many taxa, groups are often relatively small. In such systems, the resulting group-level stochasticity, also called the intrinsic noise, can produce nontrivial collective dynamics [8,[11][12][13].…”

“…This consensus is possible, intriguingly, because smaller groups exhibit more fluctuations. Therefore, in the physics literature, the collective order or consensus in this simple system is also called (intrinsic-) noise-induced order [11,12].…”

“…In formal terms, mesoscopic dynamics of a collective state of the group, denoted by m, may be written in terms of a stochastic differential equation (SDE) [11,12,18]…”

“…Let us now consider the dynamics of the collective state given by the stochastic differential equation [11,12,14],…”

Fission-fusion dynamics and group-size-dependent composition in heterogeneous populations

“…In finite groups where individuals probabilistically interact among themselves, the noise at the group-level, also called intrinsic noise, increases with decreasing group size [51]. Intrinsic noise, in some cases, can cause bimodal states for small groups [52][53][54]. It may be worth investigating a plausible connection between our results, where stochasticity of merge and split events for small groups sizes plays an important role, with the phenomenon of noise-induced bimodality.…”