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

##### 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