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The Logistic map, otherwise known as the bifurcation diagram, is a mathematical graph used to describe the population of animal populations over time. The simple equation used to generate this graph is:
Xn+1 = r*Xn(1-Xn)
This equation historically speaking was created by biologist as a toy-model for modeling population growth in animals during the rise of computers around the 1970s. Xn+1 is the next generation, r is the rate of growth, Xn is the current population, and (1-Xn) is the death rate represented as a decimal from 0-1 which limits the population based on how big it is. This is a 'toy-model' because while the equation is too simple to acurately describe a single species, its general enough to apply to every species. Anything remotely alive which reproduces MUST be described by this model because every living thing dies and reproduces a next generation. What really matters is the numbers plugged into it. Thats where things get interesting.
What really matters in the logistic map is the rate of growth R. What happens to populations after many many generations given a constant rate of growth. whatever starting population you pick for Xn becomes completely irrelevant after many generations.
I encourage you to play with this equation on your own calculator and seeing what happens to the values as you continuously press the 'ans' key. Alternatively, here is a github of a python program which does this for you and generates visual graphs with the data (I highly recommend checking it out)
github of the Logistic map visualization program by jonnyhyman
Okay, lets discuss the values of R and their corrisponding results with a population. If we plug in a value of r less than 1, the population will inevitably go extinct no matter the starting population. This makes sense, any species which dies more than it reproduces consistently across generations is an evolutionary failure doomed to extinction.
Logistic map where R = 0 to 1 (extinction)
From R = 1 to 3, the population always stablizes on a single value after many generations. The higher r is, the higher the stablized population. Population stablization is also seen in nature from time to time, so this checks out.
Logistic map where R = 1 to 3 (single value population)
After 3 is where things get interesting. once we pass the threshold of R = 3, the stablized population value splits in two, cycling between two constant values over and over. This splitting is known as 'bifurcation' in the STEM community. The population will NEVER stablize on a single value, it will keep jumping between two values forever. One year the population might be higher and the next year it will be lower, cycling over and over. This population cycling is also observed in nature. The graph at this stage has forked in two representing the ocilating values.
Logistic map where R = 3 to 3.5 (Period 2 Bifurcation)
At roughly R=3.5, the graph bifurcates again. There are now 4 values the population cycles through. Instead of a two year cycle it becomes a 4 year cycle. Because the period of time has doubled, these are known as 'period doubling bifurcations'. As we keep raising R, these period doubling bifurcations continue coming faster and faster. periods of 8, 16, 32, 64, and finally we reach R = 3.57. This is where the magic happens.
Logistic map where R = 3.57 and up (Deterministic Chaos)
At r=3.57 this orderly bifurcation falls apart and complete chaos sets in. The population NEVER settles down, it bounces around values seemingly at random. If you look at the Logistic map, this chaos is represented with the messy cobweb part towards the end of the graph. Fun fact: This equation was actually one of the first methods computers used to generate psudo-random numbers. Its quite amazing that such a simple equation contains so much complexity and deterministic chaos.
You might expect that this chaos continues indefinitely, but that is not the case. Unexpectedly, As we keep increasing r there are small patches of values in the chaos in which periodic order returns. For example, at R=3.83 there is a stable cycle with a period of 3 and as r increases it splits into periods of 6, 12, 24 and so on before returning to chaos. In fact, this one equation contains stable periods of EVERY length. 21, 69, 420, 1 billion. Whatever you like as long as you have the right value of r. Magical right? These small islands of stability within the chaos are represented by the tiny gaps in the cobweb part.
Logistic map where R = 3.83 (stable period 3 bifurcation)
To quote John Gribbin's book 'Deep Simplicity' (which I highly recommend reading): "Feigenbaum (1945-) showed that the period doubling route to chaos is not some special feature of the logistic map equation, but that it is a product of the iterative process by which the system feeds back on itself - whether the system is an animal population, an oscillator in an electrical circut, an oscillating chemical reaction, or even (in principle) the business cycle of the economy. What mattered was that the systems had to be 'self-refferential'."
While ive stuck to the animal population explaination of the logistic map, it actually shows up in MANY MANY MANY different areas of math and science. from biology and chemistry, to electrical analysis and thermodynamics, to the economy. There is something connecting them all together in a deep, fundimental way. The universe seems to have a hard-on for self-refferential dynamical systems that feedback into themselves. It also seems to REALLY like fractal geometry visually describing both abstract and natural processes. Speaking of fractals....
Looking at the logistic map, you might notice it looks sort of self-similar and fractal-like with large scale features repeated at smaller and smaller scales. Yes indeed, sure enough it IS a fractal. Its crazy to think that ALL populations obey the properties of this graph to some degree. In reality R is never constant as birth rates fluxuate, but as long as that birthrate is a real number (and it has to be) it follows the properties of this graph.
There is something special about this particular fractal, though. As it turns out: The logistic map is actually the real number slice of the Mandelbrot set which exist in the complex plane. Please check out this gif and prepare to be blown away:
https://github.com/jonnyhyman/Chaos/blob/master/images/logistic-mandelbrot.gif?raw=true
(sorry about the https link, the gif is too big for me to host)
You might be wondering why the logistic map is part of the mandelbrot set and what that really means. To get to that, we need to first go over what the mandelbrot set even is and this article is getting long enough already. Ill need to write a whole nother' article to give the explaination it deserves (stay tuned!) What I will say is this: The Mandelbrot set is the poster-child of fractal geometry and a cornerstone to modern chaos theory. It beautiful to look at and rich with complexity, thats just the tip of the iceburg.
To learn more about the logistic map, check out these videos:
"The Mandelbrot set and its bifurcation tree Fractal zoom" by Logicedges
"Logistic map zoom to magnification x 1,000,000,000" by Logicedges
If you want to learn more about The Mandelbrot Set I recommend this video series by The Mathemagician Guild detailing everything about it. Please check it out if you have the time.
