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Complexity
Volume 2017, Article ID 1409865, 20 pages
https://doi.org/10.1155/2017/1409865
Research Article

Bifurcations and Dynamics of the Rb-E2F Pathway Involving miR449

1Department of Basic Courses, Guangdong Polytechnic College, Zhaoqing 526100, China
2Institute of Applied Mathematics, Xuchang University, Xuchang, Henan 461000, China

Correspondence should be addressed to Jianwei Shen; moc.liamg@nehswjcx

Received 23 February 2017; Revised 19 April 2017; Accepted 14 May 2017; Published 19 October 2017

Academic Editor: Shanmugam Lakshmanan

Copyright © 2017 Lingling Li and Jianwei Shen. 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

We focused on the gene regulative network involving Rb-E2F pathway and microRNAs (miR449) and studied the influence of time delay on the dynamical behaviors of Rb-E2F pathway by using Hopf bifurcation theory. It is shown that under certain assumptions the steady state of the delay model is asymptotically stable for all delay values; there is a critical value under another set of conditions; the steady state is stable when the time delay is less than the critical value, while the steady state is changed to be unstable when the time delay is greater than the critical value. Thus, Hopf bifurcation appears at the steady state when the delay passes through the critical value. Numerical simulations were presented to illustrate the theoretical results.