Abstract
A five-dimensional delay differential system is proposed to describe oscillatory behaviour in microbial continuous culture. Stability and Hopf bifurcation are discussed, including the changing regularity of the bifurcation value. An explicit algorithm for determining the direction of Hopf bifurcation and the stability of periodic solution is derived, using the theory of normal form and centre manifold. Finally, some numerical simulations are performed to illustrate the analytical results found.
Acknowledgements
This work was supported by HIT(WH)2008009, HIT(WH)ZBQD201006, Weihai Municipal Science and Technology Development Program, the National Natural Science Foundation of China (No. 10471014, No. 10671126), and National Basic Research Program of China (No. 2007CB714304).