Mathematics of chaos and nonlinear dynamics are so important for biological modeling that we thought it would help our readers, if we explain them in simple language. The toy model presented below was already mentioned in an earlier commentary, but here we will add code and other details so that you can play with it.
The concept of toy model is very important in data analysis and in understanding ‘big data’ or large amount of likely useless data. For example, imagine you are studying how people get fat. You may collect extensive amount of data on their food habits, what languages they speak, how many sentences they utter in a day, how many hours they sleep, but ultimately may find that their weight gain is directly proportional to number of bottles of sugary syrup they drink. So, the toy model is a straight line correlating weight gain and amount of sugar input, while other factors contribute to noise.
Mathematicians studied many types of simple toy models and observed that complex behavior appeared in very simple models as long as they included some nonlinearity. Try this equation -
It is a discrete equation and here is how to run it. Let us say r=1. We will start with x_0 between 0 and 1 (let’s say 0.5), and compute x_1=r *x_0 * (1-x_0)=0.25. Then we will plug-in x_1 into the equation and compute x_2. Here is the code you can play with, where r is the input parameter.
for($i=0; $i<20000; $i++)
print “$i $x\n”;
With r between 0 and 1, the population will eventually die, independent of the initial population.
With r between 1 and 2, the population will quickly approach a steady state value, independent of the initial population.
With r between 2 and 3, the population will also eventually approach the same value, but first will fluctuate around that value for some time. The rate of convergence is linear, except for r=3, when it is dramatically slow, less than linear.
With r between 3 and (approximately 3.44949), from almost all initial conditions the population will approach permanent oscillations between two values. These two values are dependent on r.
With r between 3.44949 and 3.54409 (approximately), from almost all initial conditions the population will approach permanent oscillations among four values.
With r increasing beyond 3.54409, from almost all initial conditions the population will approach oscillations among 8 values, then 16, 32, etc.
At r approximately 3.56995 is the onset of chaos, at the end of the period- doubling cascade. From almost all initial conditions we can no longer see any oscillations of finite period.
Beyond r = 4, the values eventually leave the interval [0,1] and diverge for almost all initial values.
Play with the equation yourself and see how it behaves. You can think about x_0 as the part of land on earth covered by trees, and x_1 as the the part covered by trees next year. Imagine you have a model, where ‘climate change’ increases amount of rainfall, which changes the part of land on earth covered by trees in a nonlinear way. Or x_0 can also be proportion of bugs in a forest, part of test-tube filled with a bacteria, normalized count of a type of transcription factor in a cell and so on.