Download Logistic Map: Understanding Population Growth and Simple Dynamics and more Study notes Physics in PDF only on Docsity! NOTES ON THE LOGISTIC MAP MAPS In science, we use numbers to describe the state of a system, such as ‘10,132 zebra fish are in the lake’ or ‘the asteroid is moving at 10 kilometers per second and is 32,000 km away from this point.’ Given those numbers and a model or theory for how the system changes in time, we can predict the future. If we consider making predictions at regularly spaced intervals or whenever specific conditions are met, we might represent our knowledge as a map. The map is the function that predicts the observations at a specific time in the future, based upon the current observations. Examples: Clock. Based upon experience, we note that every 3 hours, the little hand on an analog clock advances in angle by a right angle, 90?. So, if we measure the angle in degrees (with respect to the up direction), the map can be stated in English as: “if the angle of the little hand is a, then 3 hours later, the measured angle is a+90 (in clockwise degrees.)” The shorthand for this might be: “a ? a + 90, every 3 hours”. Slowing down of a rolling ball. Again, based upon experience, we might find that a rolling ball on a linoleum floor slows down to ½ of its speed every time it crosses a tile. If we measure the speed s in meters per second (m/s), we might write this as: “s ? s/2, each tile.” Repeating maps: These maps can be repeated (also called iterated) to predict further into the future. You can check, in these examples, that a ? a + 180 every 6 hours and that s ? s/8, every time the rolling ball crosses three tiles. We will use maps stated in this format to simulate how systems evolve in time. What we will find is that amazingly intricate behavior can arise from very simple maps. LOGISTIC MAP We will spend a little time in this class studying the famous “logistic map” and its relatives. Much of this discussion was gone over quickly in class on Sept. 12 and it overlaps very much with the discussion in the book Chaos by Gleick. Let’s see how the logistic map arises from a simplified analysis of population growth in a limited environment. This analysis will assume a reproductive rate that depends on the available resources: when the population is high, the available resources are reduced, so that the reproductive success is reduced. Let us assume that the animals reproduce to give the next generation, then die off before the next generation reproduces. Let the number of animals at any given generation be N (which is really an integer, but we will allow to be any value.) If there are no restrictions due to reduced resources, the number of animals in the next generation is assumed to be proportional to N – there is a fixed number of children per adult. We can write this as N ? a N (when N is small.) [Equation 1] This just says that the number of births is proportional to the total population. Now, lets say that reproductive success declines from its maximum a with population increase, due to reduced resources. The higher N is, the fewer offspring reproduced. Let C be the maximum capacity of the environment. We might see a reproductive success curve that looks something like this: The important feature of the reproductive success curve is that the reproductive success goes to zero as N increases to the capacity C. If you know the function b(N), the average births per adult in an environment witn N adults, you can find the size of the next generation by using the map N ? b(N) N [Equation 2] It turns out the general properties of this map are not sensitive to the details of the function b(N). So we can assume a simple form for how the number of children per adult depends on the current population. The simplest form that we can assume is a straight line, so let’s assume that, as plotted on the next page: Current population N b = # children per adult C a
