Spatio-Temporal Patterns for a Generalized Innovation Diffusion Model

TL;DR

This paper introduces an exactly solvable spatio-temporal model of innovation diffusion based on Bass's framework, capturing how innovations spread across space and time with explicit analytical solutions.

Contribution

It extends the classical Bass model by incorporating spatial components, resulting in a nonlinear, exactly solvable field dynamics that reveals spatio-temporal patterns of innovation diffusion.

Findings

Derivation of explicit analytical solutions for the spatio-temporal diffusion patterns.

Identification of nonlinear collective dynamics leading to pattern formation.

A simplified yet comprehensive theoretical framework for spatial innovation spread.

Abstract

We construct a model of innovation diffusion that incorporates a spatial component into a classical imitation-innovation dynamics first introduced by F. Bass. Relevant for situations where the imitation process explicitly depends on the spatial proximity between agents, the resulting nonlinear field dynamics is exactly solvable. As expected for nonlinear collective dynamics, the imitation mechanism generates spatio-temporal patterns, possessing here the remarkable feature that they can be explicitly and analytically discussed. The simplicity of the model, its intimate connection with the original Bass' modeling framework and the exact transient solutions offer a rather unique theoretical stylized framework to describe how innovation jointly develops in space and time.