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Discrete Dynamics in Nature and Society
Volume 2017 (2017), Article ID 8162865, 10 pages
https://doi.org/10.1155/2017/8162865
Research Article

Neimark-Sacker Bifurcation in Demand-Inventory Model with Stock-Level-Dependent Demand

1Institute of Logistics and Warehousing, Poznań, Poland
2Lodz University of Technology, Łódź, Poland
3University of Białystok, Białystok, Poland

Correspondence should be addressed to Ewa Schmeidel

Received 4 September 2016; Accepted 20 November 2016; Published 1 January 2017

Academic Editor: Zhan Zhou

Copyright © 2017 Piotr Hachuła et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

An analysis of dynamics of demand-inventory model with stock-level-dependent demand formulated as a three-dimensional system of difference equations with four parameters is considered. By reducing the model to the planar system with five parameters, an analysis of one-parameter bifurcation of equilibrium points is presented. By the analytical method, we prove that nondegeneracy conditions for the existence of Neimark-Sacker bifurcation for the planar system are fulfilled. To check the sign of the first Lyapunov coefficient of Neimark-Sacker bifurcation, we use numerical simulations. We give phase portraits of the planar system to confirm the previous analytical results and show new interesting complex dynamical behaviours emerging in it. Finally, the economical interpretation of the system is given.