Hopf bifurcation and multiple limit cycles in bio-chemical reaction of the morphogenesis process

Morphogenetic process is an interesting but very hard bio-chemical problem. In this paper, we consider a bio-chemical model in temporal morphogenesis which is a generalization of the model studied by Gierer–Meinhardt. By using the theory of ordinary differential equations, it is shown that the model undergoes a Hopf bifurcation if the parameters in the model satisfy the following relationship: λ = 2/(ρ ₂ + x*)−1. It is also proved that the close orbit created by the Hopf bifurcation is stable. The conditions that guarantee the system has three closely nested limit cycles are also obtained in the paper.