Hits:

Date:2019-03-09

Indexed by:Journal Article

Date of Publication:2009-09-01

Journal:INTERNATIONAL JOURNAL OF BIOMATHEMATICS

Included Journals:Scopus、SCIE

Volume:2

Issue:3

Page Number:321-338

ISSN:1793-5245

Key Words:Stability; time delay; Hopf bifurcation; periodic solution; phase plane

Abstract:Introducing discrete time delay into the model for producing 1,3-propanediol by microbial continuous fermentation, the stability and Hopf bifurcation of a delay differential system for microorganisms in continuous culture are considered in this paper, including the changing regularity of bifurcation value and oscillating period. Algebraic criteria for absolute stability, as well as the transversality condition for Hopf bifurcation of this kind system are obtained. Explicit algorithm for determining the direction of Hopf bifurcation and the stability of periodic solution is derived, using the theory of normal form and center manifold. Finally, numerical simulations show the effectiveness of our results. The pictures of periodic solutions and phase planes with specified parameters suggest that our results can qualitatively describe oscillatory phenomena occurring in experiments.