Bifurcation of nontrivial periodic solutions for a biochemical model with impulsive perturbations (Q1049282)

The authors consider a basic biochemical reaction of the form (1.1): \[ \begin{aligned} \dot x(t) & = a-(kx(t)- bx(t)y(t),\\ \dot y(t) & = bx(t)y(t)- {qy(t)\over c+q(t)},\end{aligned}\qquad t\neq nT \] \[ \begin{aligned} \Delta x(t) & = p,\\ \Delta y(t) & = 0,\end{aligned}\qquad t= nT \] there are \(x(t)\) and \(y(t)\) the concentrations of the two internal reactants \(\Delta x(t)= x(t^+)- x(t)\), \(\Delta y(t)= y(t^+)- y(t)\), \(n= \{1,2,3,\dots\}\), \(P> 0\), \(T\) is the impulse period, the function \({q\over c+y}\) is the saturation of the input. By using Floquet theorem the authors find a number \(T^*\) so that the boundary-periodic solution \((\widetilde x(t),0)\) of (1.1) is locally stable if \(T^*< T\) and is unstable if \(T< T^*\).