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!

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    $\begingroup$ I agree with you, the logistic map is not really a model of population dynamics at all. If it were, it would presumably satisfy $F(x)>x$ if $x \in (0,1)$, $F(x)<x$ if $x \in (1,\infty)$. $\endgroup$ – Ian Mar 20 '17 at 15:53

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.

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  • $\begingroup$ Thanks! After reading your answer, I realize there were some issues with my question. Specifically, I should have used $K=1$ also in the continuous model, i.e. normalized to carrying to capacity. I shall make some edits. $\endgroup$ – Étienne Bézout Mar 20 '17 at 18:16

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