The main aim of this paper is to study the oscillatory behaviors of gene expression networks in quorum-sensing system with time delay. The stability of the unique positive equilibrium and the existence of Hopf bifurcation are investigated by choosing the time delay as the bifurcation parameter and by applying the bifurcation theory. The explicit criteria determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are developed based on the normal form theory and the center manifold theorem. Numerical simulations demonstrate good agreements with the theoretical results. Results of this paper indicate that the time delay plays a crucial role in the regulation of the dynamic behaviors of quorum-sensing system.

中文翻译：

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具有时间延迟的群体感应系统中的周期性振荡

本文的主要目的是研究具有时延的群体感应系统中基因表达网络的振荡行为。以时滞为分岔参数，应用分岔理论，研究了唯一正平衡的稳定性和Hopf分岔的存在。基于范式理论和中心流形定理，提出了确定Hopf分岔方向和分岔周期解稳定性的明确判据。数值模拟表明与理论结果有很好的一致性。本文的研究结果表明，时间延迟在调节群体感应系统的动态行为中起着至关重要的作用。