Imitation, Proximity, and Growth — A Collective Swarm Dynamics Approach

Abstract

This paper is based on the premise that economic growth is driven by an interplay between innovation and imitation in an economy composed of interacting firms operating in a stochastic environment. A novel approach to modeling imitation is presented based on range-dependent processes that describe how firms consider proximity when imitating peers who are found in a given neighborhood in terms of productivity. Using a particularly tractable approach, we are able to analyze how drastically different economic growth scenarios emerge from different imitation strategies. These emerging scenarios range from diffusive growth where the variance of productivity grows indefinitely, to balanced growth described by a traveling wave with fixed variance. The latter scenario is sustained only when imitation strength among firms exceeds a critical bifurcation threshold.

Cauchy Processes, Dissipative Benjamin–Ono Dynamics and Fat-Tail Decaying Solitons

“…Inspired by [19], let us now introduce the "follow-the-leader" (FLA) interaction kernel and discuss the corresponding Equation (20). The FLA algorithm assumes that agents permanently observe the relative positions of R of their fellows.…”

“…For Equation (3), the diffusive dynamics are used to model Brownian agents. These enable us to stylise the emergence of macroscopic spatiotemporal patterns for microscopic agents driven by sources of Gaussian white noise [2, 18,19]. In this context, the nonlinearity Au∂ x u describes mutual interactions via a follow-the-leader algorithm.…”

“…For long-range interactions λ → ∞, the solution U(x, t) in Equation ( 29) is yet unknown. One might infer whether the λ → ∞ regime sustains a steady-amplitude soliton, which does actually occur for Brownian agents [19].…”

Brownian Swarm Dynamics and Burgers’ Equation with Higher Order Dispersion

“…Inspired from [14,15], we now further specialise the dynamics and consider two specific types of mutual interactions [13]:…”

“…Adopting a mean-field approach, the collective swarm dynamics can be stylised by hydrodynamic equations that possibly allow for exact solutions. Developed in Section 2, we recall (see also [13]) that, for simple mutual interactions of the type "catch the leader" or conversely "catch the laggards", the resulting hydrodynamic evolution matches Equation (1). Such interactions can be viewed as special cases (i.e., limited to large swarms populations) of the more general class of dynamics introduced in [14,15].…”