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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 $$\aligned &\dot x_1= F_1(x_1,x_2),\quad \dot x_2= F_2(x_1,x_2),\\ &x_1(t^+_i)= \theta_1(x_1(t_i), x_2(t_i)),\quad x_2(t^+_i)= \theta_2(x_1(t_i), x_2(t_i)),\endaligned\tag 1$$ with $t_{i+1}- t_i= \tau> 0$, $i= 0,1,2,\dots$, $x_1,x_2\in \bbfR$, $\theta_1$, $\theta_2$ 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.

34C25Periodic solutions of ODE
92C37Cell biology
34A37Differential equations with impulses
34D30Structural stability of ODE and analogous concepts
92B05General biology and biomathematics