Logistic map (discrete dynamical system) vs logistic differential equation

I have to roughly illustrate the logistic discrete dynamical system (as a model for population growth) to some non mathematics students. I'm not an analyst or an expert of dynamical systems.

Looking things up in the internet, I find the logistic map

$$x_{n+1}=rx_n(1-x_n)$$

with initial condition $$x_0\in [0,1]$$, and where $$r\in[0,4]$$ is a parameter (the condition $$0\leq r \leq 4$$ guarantees that $$x_n$$ doesn't escape the unit interval $$[0,1]$$ throughout the evolution of the system). Here $$x_n$$ represents the ratio between the population at time $$n$$ and the total population the environment is able to support.

I find also different behaviors according to the value of the parameter $$r\in[0,4]$$. For example,

1. For $$0 there's extinction of the population.
2. For $$1 the sequence tends to a stable equilibrium $$x_\infty:=1-1/r$$.
3. For $$3 there's convergence to a period-$$2$$ cycle.
4. For $$1+\sqrt{6} (where $$r^*$$ is a certain constant) several bifurcations occur with limit a cycle of period that doubles as $$r$$ traverses that range.
5. For $$r>r^*$$ there's chaotic behavior.

I would expect that a similar span of different behaviors also happens for the logistic differential equation

$$\dot{x}(t)=rx(t)(1-x(t))$$

upon varying the parameter $$r$$. But on the internet I found no reference to anything like this. On the contrary, many pages care to solve the differential equation explicitely and illustrate the solution, which is the famous logistic function: the S-shaped increasing curve (depending on $$r$$) with a horizontal asymptote and an inflection point. It seems this solution is obtainable no matter what $$r$$ is. This looks only like case number 2. of the discrete dynamical system above.

So where are the analogous to cases 1.,3.,4. and 5. where the time is continuous??

Or am I misunderstanding some aspects of how a continuous-time dynamical system gets discretized?

Also,

Which correspondence is there between the $$r$$ of the discrete version and the $$r$$ of the continuous version?

Only some elements

A differential equation and it associated discretized problem may have different stability behaviors depending on the way you perform the discretization. See for example Wikipedia - discretization, paragraph Approximations.

Nevertheless, approximating $$\dot{x}(t) = rx(t)(1-x(t))$$ by $$x_{n+1}=r x_n(1-x_n)$$ is rather strange. It should be better discretized by $$x_{n+1} -x_n =r x_n(1-x_n)$$. Because usually $$\dot{x}(t) = \frac{x_{n+1}-x_n}{t_{n+1}-t_n} =x_{n+1}-x_n$$ if you consider equal discretization of the time.

• I exactly thought the same thing: the ODE should be discretized by $x_{n+1}-x_n=rx_n(1-x_n)$ because the finite difference ratio $(x_{n+1}-x_n)/1$ plays the role of $dx/dt$. But I don't know... – Qfwfq Dec 9 '18 at 20:22
• In the paragraph you cite (of the wikipedia link) it seems they're considering linear ODEs, while the logistic is nonlinear - not sure how this affects the continuous/discrete relationship. – Qfwfq Dec 9 '18 at 20:25
• It is even worse... if You already have issue with linear problems! – mathcounterexamples.net Dec 9 '18 at 20:26

In the logistic equation you get transforming it as Bernoulli equation $$\frac{d}{dt}x(t)^{-1}=r(1-x(t)^{-1})\implies x(t)^{-1}=1+ce^{-rt}.$$ Now compare the values for $$t$$ and $$t+h$$, $$x(t+h)^{-1}-1 = ce^{-rt-rh}=e^{-rh}(x(t)^{-1}-1).$$ For the sequence $$x_k=x(kh)$$ you get thus the recursion formula $$x_{k+1}=\frac{x_k}{1-e^{-rh}+e^{-rh}x_k}$$ which looks rather different than the logistic map.

Any sane discretization using the Euler method or similar would use a sufficiently small $$h$$, so that in the Euler forward discretization $$x_{k+1}=x_k(1+hr)-hrx_k^2.$$ To get to the normal form one would have to rescale $$y_k=ax_k$$, so that then $$y_{k+1}=y_k(1+hr)-\frac{hr}ay_k^2=(1+hr)y_k\left(1-\frac{hr}{a(1+hr)}y_k\right)$$ giving $$a=\frac{hr}{1+hr}$$. This is indeed case $$2$$ in the list, with equilibrium at $$y_\infty=1-\frac1{1+hr}=\frac{hr}{1+hr}=a$$ or translated back at $$x_\infty=1$$.

• Ok, this is to say the recursion you get depends heavily on how you choose to discretize, I assume? But is it relevant to the question? Which is the discretization method in the OP case, and does it lead to different phenomena (differences between the continuum and the discrete)? – Qfwfq Dec 9 '18 at 21:32
• Too long for a comment, added remark on discretization to the answer. One can conclude that the interesting phenomena happen when $h$ is so large that the discretization error can no longer be classified as $O(h^2)$ as the higher order terms are equally large or even dominate. – LutzL Dec 9 '18 at 21:50
• So it's true that chaotic features are not present in the continuous model but appear only once I discretize in that specific way? – Qfwfq Dec 9 '18 at 22:19