I have the following discrete time dynamical system

$ y(t+1) = y(t) + \frac{1}{1+\exp(z+ u f y(t))} , 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 $y(t,f)$): 

$\lim_{n \to \infty} \frac{f}{n} y(nt,\frac{f}{n})$

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