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


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 $0<r\leq 1$ there's extinction of the population.
  2. For $1<r\leq 3$ the sequence tends to a stable equilibrium $x_\infty:=1-1/r$.
  3. For $3<r\leq 1+\sqrt{6}$ there's convergence to a period-$2$ cycle.
  4. For $1+\sqrt{6}<r\leq r^*$ (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


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?


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.

  • $\begingroup$ 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... $\endgroup$ – Qfwfq Dec 9 '18 at 20:22
  • $\begingroup$ 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. $\endgroup$ – Qfwfq Dec 9 '18 at 20:25
  • $\begingroup$ It is even worse... if You already have issue with linear problems! $\endgroup$ – 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 $x_\infty=1$.

  • $\begingroup$ 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)? $\endgroup$ – Qfwfq Dec 9 '18 at 21:32
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    $\begingroup$ 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. $\endgroup$ – LutzL Dec 9 '18 at 21:50
  • $\begingroup$ So it's true that chaotic features are not present in the continuous model but appear only once I discretize in that specific way? $\endgroup$ – Qfwfq Dec 9 '18 at 22:19

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