Bifurcation of non trivial periodic solutions of impulsive differential equations arising chemotherapeutic treatment. (English) Zbl 1011.34031

Lakmeche, Abdelkader; Arino, Ovide

Dyn. Contin. Discrete Impulsive Syst. 7, No. 2, 265-287 (2000).

The authors study the nonlinear impulse differential equations in the plane \[ \begin{aligned} &\dot x_1= F_1(x_1,x_2),\quad \dot x_2= F_2(x_1,x_2),\\ &x_1(t^+_i)= \theta_1(x_1(t_i), x_2(t_i)),\quad x_2(t^+_i)= \theta_2(x_1(t_i), x_2(t_i)),\end{aligned}\tag{1} \] with \(t_{i+1}- t_i= \tau> 0\), \(i= 0,1,2,\dots\), \(x_1,x_2\in \mathbb{R}\), \(\theta_1\), \(\theta_2\) two positive, suitably smooth functions of \(x_1\): normal cell biomass and \(x_2\): tumor cell biomass. They describe the competition between normal and tumor cells. Sufficient conditions for the existence of nontrivial, periodic solutions to (1) are given.

Reviewer: Stepan Kostadinov (Plovdiv)

MSC:

34C25	Periodic solutions to ordinary differential equations
92C37	Cell biology
34A37	Ordinary differential equations with impulses
34D30	Structural stability and analogous concepts of solutions to ordinary differential equations
92B05	General biology and biomathematics

Keywords:

impulsive differential equations; bifurcation of nontrivial periodic solutions