This equation will change how you see the world (the logistic map)

what's the connection between a dripping faucet the Mandelbrot set a population of rabbits thermal convection in a fluid and the firing of neurons in your brain it's this one simple equation this video is sponsored by fast hosts who are offering UK viewers the chance to win a trip to South by Southwest if they can answer my question at the end of this video so stay tuned for that let's say you want to model a population of rabbits if you have X rabbits this year how many rabbits will you have next year well the simplest model I can imagine is where we just multiplied by some number the growth rate R which could be say 2 and this would mean the population would double every year and the problem with that is it means the number of rabbits would grow exponentially forever so I can add the term 1 minus X to represent the constraints of the environment and here I'm imagining the population X is a percentage of the theoretical maximum so it goes from 0 to 1 and as it approaches that maximum then this term goes to 0 and that constrains the population so this is the logistic map xn plus 1 is the population next year and xn is the population this year and if you graph the population next year versus the population this year you see it is just an inverted parabola it's the simplest equation you can make that has a negative feedback loop the bigger the population gets over here the smaller it'll be the following year so let's try an example let's say we're dealing with a particularly active group of rabbits so R equals two point six and then let's pick a starting population of 40% of the maximum so point four and then times 1 minus 0.4 and we get 0.62 four okay so the population increased in the first year but what we're really interested in is the long term behavior of this population so we can put this population back into the equation and to speed things up you can actually type two point six times answer times one - answer get point six one so the population dropped a little hit it again point six one nine point six one three point six one seven point six one five point six one ...