Innovation diffusion model with interactions and delays in adoption for two competitive products in two different patches

  • Rishi TuliEmail author
  • Joydip Dhar
  • Harbax S. Bhatti


The aim of the present paper, how the people behave towards the offer of two products in two different patches. In this work, an innovation diffusion model with six-compartments for two different patches is proposed. There is a delay in the adoption of product-1 in patch-2 and delay of adoption of product-2 in patch-1. The entire population in both the patches is classified into three different groups (i) non-adopter (ii) adopter of product-1 (iii) adopter of product-2. Dynamical behavior of the proposed system is studied, and Basic influence numbers (BINs) of the model are calculated. Stability analysis is executed for all the possible equilibrium points with and without delays. Hopf bifurcation analysis is too carried out taking the delay of adoption to adopt the product-1 in patch-2, and product-2 in patch-1 are bifurcation parameter and obtained the threshold values. Moreover, sensitivity analysis is carried out for the system parameter used in the interior equilibrium. Finally, exhaustive numerical simulations have been carried out by utilizing MATLAB, to supports analytical results.


Boundedness Positivity Delay Hopf bifurcation Sensitivity analysis 

Mathematics Subject Classification

34C23 34D05 34D020 



We are thankful to the managing editor and reviewers for there valuable suggestions to improve the manuscript. Also, I express the warm thanks to I.K.G. Punjab Technical University, Punjab for providing me the facilities for the research being required.

Compliance with ethical standards

Conflict of interest

The author declare that there is no conflict of interest.

Ethical standard

It is ensure that principles of ethical and professional conduct have been followed.


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Copyright information

© Università degli Studi di Napoli "Federico II" 2019

Authors and Affiliations

  1. 1.IKG-Punjab Technical UniversityKapurthalaIndia
  2. 2.Beant College of Engineering and TechnologyGurdaspurIndia
  3. 3.ABV-Indian Institute of Information Technology and ManagementGwaliorIndia
  4. 4.Department of Applied SciencesB.B.S.B.E.CFatehgarh SahibIndia

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