I have the following discrete time dynamical system $$ y(t+1) = y(t) + \frac{1}{1+\exp(z+ u f y(t))} ,\quad y(0)=0,$$ where $z$ is a real number $f$ and $u$ are non-negative reals. I know I have little hope of obtaining a closed form solution for this process. But, actually, for my application, a "better" solution is to find (making the dependence of $y$ on $f$ explicit by writing $y(t)$ as $y(t,f)$): $$\lim_{n \to \infty} \frac{f}{n}\cdot y\left(nt,\frac{f}{n}\right).$$ Simulations suggest that this converges to $$y(t) = \frac{ft}{1+\exp(z)}.$$ How can I rigorously derive this? Also appreciated are references to texts that discuss similar problems.

Thanks