Bifurcation Analysis in a Delay Differential Equations, which Confers a Strong Allee Effect in Escherichia Coli

Abstract

The paper addresses the bifurcations for a delay differential model with parameters which confers a strong Allee effect in Escherichia coli. Stability and local Hopf bifurcations are analyzed when the delay τ or σ as parameter. It is also found that there is a non-resonant double Hopf bifurcation occur due to the vanishing of the real parts of two pairs of characteristic roots. We transform the original system into a finite dimensional system by the center manifold theory and simplify the system further by the normal form method. Then, we obtain a complete bifurcation diagram of the system. Finally, we provide numerical results to illustrate our conclusions. There are many interesting phenomena, such as attractive quasi-periodic solution and three-dimensional invariant torus.

Keyword : double Hopf, bifurcation, delay, center manifold