Stability and Bifurcations in a Model of Bacteria Immunity with Quorum Sensing

Quorum sensing, a widespread phenomenon in bacteria that is used to coordinate gene expression among local populations, intervenes in the competition between bacteria and the immune system. The domain of attraction of the bacteria-free equilibrium results from a linear matrix inequality optimization with a multivariate polynomial objective under constraints. The Bogdanov-Takens singularity and bifurcation, including a saddle-node bifurcation, a Hopf bifurcation, and a homoclinic bifurcation, are obtained from normal form theory. The normal form of a bifurcation is a simple dynamical system which is equivalent to all systems exhibiting this bifurcation.