Is the logistic map a good model for population dynamics? If so why? The logistic map $$x_{n+1}=rx_n(1-x_n)$$
is a well known system in chaos theory. Supposedly it is a discrete time model for population dynamics, with $x_n$ denoting the population at time step $n$ normalised to carrying capacity (hence $x_n \in [0,1])$, inspired by the logistic growth model $$\frac{dx}{dt} = rx(1-x)$$
(with $x$ normalised to carrying capacity).
However, I don't quite understand the connection. The natural discrete time version of the logistic growth model is a discretized version of the same, i.e. $$x_{n+1}=x_n+rx_n(1-x_n),$$
if we set out time step to unity. 
Following the suggestion of Julián Aguirre, we realize that $$x_{n+1} = (1+r)x_n\left(1-\frac{r}{1+r}x_n\right),$$
and with $y_n=\frac{r}{1+r} x_n$ we could write $$y_{n+1}=(1+r)y_n(1-y_n). $$
Indeed, this looks like the original logistic map, but we note some things:
1) If $0 \leq x_n \leq 1$, then $0 \leq y_n < 1$, and so we cannot reach the strange scenario where the population suddenly drops to zero due to setting $y_n=1$.
2) The factor $1+r$ has replaced $r$. This looks better if we want $r$ to mean a growth rate.
In summary, it appears the model wasn't so crazy after all!

Write the discretized model as
$$
x_{n+1}=x_n+r\,x_n(1-x_n/K)=(1+r)\,x_n\Bigl(1-\frac{r\,x_n}{(1+r)K}\Bigr).
$$
Now a change of variable of the form $x_n=\lambda\,y_n$ with the appropriate value of $\lambda$ will transform it into the logistic model.
