now let's turn to exploring the dynamics at the logistic model so you might remember that n of sub (t+1) is the population that time t + 1 and that was equal to the birthrate – the death rate times the population at time t minus the numbers individuals who died times the population that time T minus the numbers individuals who died due to overcrowding which is the population a time T-square divided by maximum population or the carrying capacity I'm going to start by reading this in the simplest format. First I'm going to let R represents the birth rate minus the death rate and I'm going to let K equal the maximum population. now I can write this top equation using these new symbols now I'm going to do a little bit algebra so if you don't like algebra then just try and follow along and if you don't understand it doesn't really matter it's only the endpoint that you need to know. So I'm gonna do is divide both sides of this equation by K the carrying capacity. And now I'm gonna define one more new symbol and that's going to be X sub t it equals to n sub t over K. So now I can rewrite this equation using my new symbol X This equation represents the fraction the current population is of the carrying capacity at a given time and that's equal to R times the fraction that the previous time step minus that same fraction square and this is known as the logistic map. And this turns out to be the most famous equation in the field of chaos theory. Let's rewrite the logistic map here for clarity Really simple huh as it happens though it's more interesting than it appears many people have studied this equation in depth since Verhulst proposed Two prominent examples have people who have studied it our Lord Robert May a theoretical biologist who had a very influential paper about sequestering in the 1970s and Michelle Feigenbaum a theoretical physicist who worked extensively on this equation in the nineteen eighties as probably the person most commonly associated with it in the scientific community note that X is the population at time sometime divided by the carrying capacity for the maximum population so X always is a real number between 0&1 this is why the question is called a map that is it takes in this on this side a current value of x between 0&1 and maps it into a new value of x which is also between 0&1 so let's look at an example let's let are equal 2 at our initial population over carrying capacity X 0 equal 0.2 that is our population is 20 percent at at the carrying capacity. Now i can iterate this map so get out a calculator let's calculate x1. That's gonna be equal to 2. that's our value times 0.2 0.2^2 and quite my calculator that's equal to 0.32 so we've gone from twenty percent are carrying capacity to 32 percent now what happens the next year. At the next year we have 2 times well now we gotta take this value for our previous generation 0.32 minus 0.32^2 and that's equal 0.4352 (writing) okay let's keep going but a little faster now (writing) (writing) (writing) (writing) and forever after will get .5 is our answer. This means that if your growth rate that is the birthrate months the death rate or R is equal to 2 and you start out at twenty percent of the carrying capacity under this model the population what always end up at fifty percent of the carrying capacity. There are two things I should note here first I am using the term model here to refer to a mathematical equation that is the logistic map. This is the model. It's called the model because its simplified representation of real phenomenon of population growth, also refer to the computer programs we write or use in NetLogo as models since they're also simplified representations of real phenomena. the word model is very general term in science for any simplified representation of nature whether it be an equation, computer program, a drawing or what have you. The second thing to note is that this value of .5 is called an attractor. It's an attractor for the system because the system is in some sense attracted to it. It turns out that even if we had started with different initial population say x0 equals 0.8 the system would still end up with a value 0.5 after some number of steps. When the system and sub a single value like 0.5 this value is called a fixed point. since the value with the point stays fixed Thus for this system 0.5 is called a fixed-point attractor and by this system i mean again this equation with R equals to 2. Often the terms model and system are used to synonymously I hope this doesn't get too confusing. In any case we'll see some other kinds of attractor in the next subunit. Finally let's look at a different way of visualizing the dynamics at the logistic map that is how it changes as you iterate it. I'm going to draw a plot logistic map equation for R equals 2 now I am plotting down here is X sub t and I am plotting over here X sub t+1. OK, so not a great drawing but this is roughly what it looks like it's a parabola and it goes between 0&1 here and here for E equals 2 right that goes between 0 and 0.5 just the maximum. Okay let's put 0.5 here on the x-axis and then we can follow the steps we took before in calculating the values taken. Okay so our first value if you remember x1 was point 0.32 thats x1 we find this on the .32 to its about right here and the y value for that on the parabola was point 0.4352, that was x2 okay so this was the point (X1,X2) then we take our value for X2 and we find it down here on that x-axis here because we're gonna calculate the next value of the function so 0.4352 you down here is around here and that corresponds up to this point on the problem that's X3 .49160192 okay this point here is (X2,X3) Okay and we take our X3 which was .49160192 We find that on the x-axis to make up here to find X4 which was .4 999999... et cetera okay I keep doing that and finally we get to you exactly .5 and .5 and once these are both .5 the system doesn't go anywhere you know it just stays at this point. So you can think of hopping from one point to another along this parabola as an example of the dynamics of the system with this r-value and are starting point and that is called a trajectory. Now it's time for our next quiz. You need a calculator for this quiz. Said R equals 2.5 and X0 equals to that 0.2. then you see equation for the logistic map filling in 2.5 for R and starting out with X0 of 0.2 To calculate X1 X2 X3 and so on until you reach a fixed-point. What is that fixed-point? and recall the fixed-point is the value of X such that X sub t is the same as X sub t+1