Complexity_ A Guided Tour - Melanie Mitchell [14]
I won’t give the actual equation for the logistic model here (it is given in the notes), but you can see its behavior in figure 2.4.
As a simple example, let’s set birth rate = 2 and death rate = 0.4, assume the carrying capacity is thirty-two, and start with a population of twenty rabbits in the first generation. Using the logistic model, I calculate that the number of surviving offspring in the second generation is twelve. I then plug this new population size into the model, and find that there are still exactly twelve surviving rabbits in the third generation. The population will stay at twelve for all subsequent years.
If I reduce the death rate to 0.1 (keeping everything else the same), things get a little more interesting. From the model I calculate that the second generation has 14.25 rabbits and the third generation has 15.01816.
FIGURE 2.4. A plot of how the population size next year depends on the population size this year under the logistic model, with birth rate equal to 2, death rate equal to 0.4, and carrying capacity equal to 32. The plot will also be a parabola for other values of these parameters.
Wait a minute! How can we have 0.25 of a rabbit, much less 0.01816 of a rabbit? Obviously in real life we cannot, but this is a mathematical model, and it allows for fractional rabbits. This makes it easier to do the math, and can still give reasonable predictions of the actual rabbit population. So let’s not worry about that for now.
This process of calculating the size of the next population again and again, starting each time with the immediately previous population, is called “iterating the model.”
What happens if the death rate is set back to 0.4 and carrying capacity is doubled to sixty-four? The model tells me that, starting with twenty rabbits, by year nine the population reaches a value close to twenty-four and stays there.
You probably noticed from these examples that the behavior is more complicated than when we simply doubled the population each year. That’s because the logistic model is nonlinear, due to its inclusion of death by overcrowding. Its plot is a parabola instead of a line (figure 2.4). The logistic population growth is not simply equal to the sum of its parts. To show this, let’s see what happens if we take a population of twenty rabbits and segregate it into populations of ten rabbits each, and iterate the model for each population (with birth rate = 2 and death rate = .4, as in the first example above). The result is illustrated in figure 2.5.
FIGURE 2.5. Rabbit population split on two islands, following the logistic model.
At year one, the original twenty-rabbit population has been cut down to twelve rabbits, but each of the original ten-rabbit populations now has eleven rabbits, for a total of twenty-two rabbits. The behavior of the whole is clearly not equal to the sum of the behavior of the parts.
The Logistic Map
Many scientists and mathematicians who study this sort of thing have used a simpler form of the logistic model called the logistic map, which is perhaps the most famous equation in the science of dynamical systems and chaos. The logistic model is simplified by combining the effects of birth rate and death rate into one number, called R. Population size is replaced by a related concept called “fraction of carrying capacity,” called x. Given this simplified model, scientists and mathematicians promptly forget all about population growth, carrying capacity, and anything else connected to the real world, and simply get lost in the astounding behavior of the equation itself. We will do the same.
Here is the equation, where xt is the current value of x and xt+1 is its value at the next time step:1
xt+1 = R xt (1 − xt).
I give the equation for the logistic map to show you how simple it is. In fact, it is one of the simplest systems to capture the essence of chaos: sensitive dependence on initial