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Dynamical behavior of an innovation diffusion model with intra-specific competition between competing adopters

Abstract

In this paper, we proposed an innovation diffusion model with three compartments to investigate the diffusion of an innovation (product) in a particular region. The model exhibits two equilibria, namely, the adopter-free and an interior equilibrium. The existence and local stability of the adopter-free and interior equilibria are explored in terms of the effective Basic Influence Number (BIN) RA. It is investigated that the adopter free steady-state is stable if RA < 1. By considering τ (the adoption experience of the adopters) as the bifurcation parameter, we have been able to obtain the critical value of τ responsible for the periodic solutions due to Hopf bifurcation. The direction and stability analysis of bifurcating periodic solutions has been performed by using the arguments of normal form theory and the center manifold theorem. Exhaustive numerical simulations in the support of analytical results have been presented.

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Acknowledgements

The authors gratefully acknowledge the support provided by the faculty and staff for providing the facilities to complete this research work in Numerical Analysis Laboratory of Shaheed Bhagat Singh State University, Ferozepur, Punjab, India. The authors want to thank the anonymous referees for their careful reading of the manuscript, valuable comments and constructive suggestions for the improvement of this work.

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Correspondence to Rakesh Kumar.

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Kumar, R., Sharma, A.K. & Sahu, G.P. Dynamical behavior of an innovation diffusion model with intra-specific competition between competing adopters. Acta Math Sci 42, 364–386 (2022). https://doi.org/10.1007/s10473-022-0120-1

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  • DOI: https://doi.org/10.1007/s10473-022-0120-1

Key words

  • intra-specific competition
  • basic influence number
  • local stability
  • Hopf-bifurcation
  • normal form theory
  • center manifold theorem

2010 MR Subject Classification

  • 34C23
  • 34D05
  • 34K18
  • 92D25