# Analysis of a Diffusion Triopoly

• Gary M. Erickson
Part of the International Series in Quantitative Marketing book series (ISQM, volume 13)

## Abstract

In this chapter, we consider a triopoly with sales dynamics that involve diffusion:
$$\begin{gathered} {{\dot{S}}_1} = \left( {{\beta_2}A_2^{{{\alpha_2}}} + \varepsilon 2{S_2}} \right)\left( {N - \sum\limits_{{i = 1}}^3 {{S_i}} } \right) - {\delta_1}{S_1} \hfill \\ {{\dot{S}}_2} = \left( {{\beta_1}A_1^{{{\alpha_1}}} + {\varepsilon_1}{S_1}} \right)\left( {N - \sum\limits_{{i = 1}}^3 {{S_i}} } \right) - {\delta_2}{S_2} \hfill \\ {{\dot{S}}_3} = \left( {{\beta_3}A_3^{{{\alpha_3}}} + {\varepsilon_3}{S_3}} \right)\left( {N - \sum\limits_{{i = 1}}^3 {{S_i}} } \right) - {\delta_3}{S_3} \hfill \\ \end{gathered}$$
(8.1)
The ε i , i=1, 2, 3, in (8.1), are coefficients of “internal influence” that indicate that a competitor’s existing customers are assumed to be a positive factor in attracting new customers from the part of the market that are not currently customers of any of the competitors. Advertising is assumed to exert an “external influence” on currently uncommitted customers.

## Keywords

Diffusion Model Costate Variable External Influence Internal Influence Advertising Strategy
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