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.
Highlights
Allee was initially stimulated by an example only loosely linked to the current interpretation of the Allee effect: he showed that goldfish grew faster in water which had previously contained other goldfish, than in water that had not
We only focus on Hopf and Double Hopf bifurcation
In the rest of this section, we show the Hopf bifurcation of the system when the time delay is used as parameter
Summary
Allee was initially stimulated by an example only loosely linked to the current interpretation of the Allee effect: he showed that goldfish grew faster in water which had previously contained other goldfish, than in water that had not (see [1]). (see [29]) discussed an impulsive delay differential equation with Allee effect, and obtained a periodic solution and its stability. They used the LuxR/LuxI quorum-sensing (QS) system from Vibrio fischeri (see [14]) and the CcdA/CcdB toxinCantitoxin module to control population survival.
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