Global dynamics of the periodic logistic system with periodic impulsive perturbations.
The authors consider the impulsive differential logistic equation \[ x'(t)= x(t)\bigl(r(t) -a(t)x(t)\bigr), \quad t\neq t_k,\;k\in \mathbb N,\tag{1} \] \[ \Delta x(t_k)= b_k x(t_k),\;k\in \mathbb N,\tag{2} \] where \(t_0=0<t_1< \cdots < t_k<t_{k+1}<\dots\), \(\Delta x(t_k)=x(t^+_k)-x(t_k)\), the functions \(r\) and \(a\) are piecewise continuous with a discontinuity of first kind \(t_k\), \(k=0,1,\dots\). (1) is \(\omega\)-periodic, i.e., \(r(t+\omega)= r(t)\), \(a(t+\omega)= a(t)\), \(t\in\mathbb R\) and (2) is \(T\)-periodic. Sufficient conditions for the existence of solutions of (1), (2), are derived taking into account whether \(\gamma=\frac{\omega}{T}\) is a rational or irrational number.
scientific article; zbMATH DE number 2032970
scientific article