I am dealing with the system $$\begin{cases}y'(x)=y(x)(1-z(x))\\ z'(x)=y(x)-z(x)\end{cases}$$
Maybe there is an analytical solution?
I write $e^x(z'+z)=e^xy$, so $$z=e^{-x}\int_0^x e^t y(t) dt,$$ what leads to $$y'(x)=y(x)\left(1-e^{-x}\int_0^x e^t y(t) dt\right),$$ or $$y'(x)=y(x)\left(1-\int_0^x e^{-(x-t)} y(t) dt\right),$$
Some help?
Some clue about the maximum and minimum points?
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EDIT $$y=z′+z \\ y′=z′′+z′ \\ z′′+z′=(z′+z)(1−z)=z′−zz′+z−z^2 \\ z′′=−zz′−z^2+z.$$