Innovation diffusion model in patch environment.
type: Item
instance of: scholarly article
title: Innovation diffusion model in patch environment.
zbMATH Open document ID: 1052.91076
DOI: 10.1016/S0096-3003(01)00268-5
author: Wang Wendi
author: P. Fergola
author: C. Tenneriello
published in: Applied Mathematics and Computation
publication date: 2003-01-28T00:00:00Z
review text: A mathematical model is proposed to describe the dynamics of users of one product in two different patches. Advertisement force, contact rate between users and non-users, population dispersal rate and returning rate from user class to non-user class are chosen as key parameters. For the product with a long life-span, it is assumed that the returning rate is proportional to the number of users. It is shown that this model has a unique positive equilibrium which is a globally stable. For the product with a short life-span, time delays are introduced to represent the duration of the product in two different markets and the stability of a positive equilibrium is analyzed in one reasonable case. Periodic advertisements are also incorporated and the existence and uniqueness of positive periodic solutions are investigated. The model designs in assuming, that two patches are isolated, and the dynamics of both classes are governed by \[ \begin{aligned} \frac{dN_1} {dt}&=-\gamma_1N_1- \lambda_1A_1N_1-\delta_1N_1+ \nu_1A_1+\beta_1,\\ \frac{dA_1} {dt}&=\gamma_1N_1+ \lambda_1A_1N_1-\delta_1A_1- \nu_1A_1,\end{aligned} \] where \(N_1\) is the number of non-users in the first patch, \(A_1\) is the number of users in the first patch, \(\beta_1\) is the birth rate of the population in the first patch, \(\delta_1\) is the death rate of the population in the first patch, \(\gamma_1\) represents the intensity of advertisement for promoting product-users in the first patch, and \(\lambda_1\) is the valid contact rate of users of the product with non-users in the first patch. Similarly, it is supposed that the dynamics of non-user-class and user-class in the second patch are governed by \[ \begin{aligned} \frac{dN_2 } {dt}&=-\gamma_2 N_2 - \lambda_2 A_2 N_2 -\delta_2 N_2 + \nu_2 A_2 +\beta_2 , \\\frac{dA_2 } {dt}&=\gamma_2 N_2 + \lambda_2 A_2 N_2 -\delta_2 A_2 - \nu_2 A_2 ,\end{aligned} \] where \(N_2 \) is the number of non-users in the second patch, \(A_2 \) is the number of users in th…
reviewed by: Fatima T. Adylova
Mathematics Subject Classification ID: 91D25
Mathematics Subject Classification ID: 34K60
zbMATH DE Number: 1861284
zbMATH Keywords: stability
zbMATH Keywords: periodics
zbMATH Keywords: delay
MaRDI profile type: Publication
Wikidata QID: Q110648322
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full work available at URL: s0096-3003(01)00268-5
OpenAlex ID: W2093005211
description: scientific article; zbMATH DE number 1861284