(Note: This essay could also be used as a lesson on modeling logistic function)
The logistic model was developed by Belgian mathematician Pierre Verhulst (1838) who suggested that the rate of population increase may be limited, i.e., it may depend on population density
R=Ro(1-n/k)
At low densities (N < 0), the population growth rate is maximal and equals to Ro. Parameter Ro can be interpreted as population growth rate in the absence of intra-specific competition.
Before we move on to illustrate Verhulst's model, we first must understand the dynamics of the population. Population growth rate declines with population numbers, N, and reaches 0 when N = K. Parameter K is the upper limit of population growth and it is called carrying capacity. It is usually interpreted as the amount of resources expressed in the number of organisms that can be supported by these resources. If population numbers exceed K, then population growth rate becomes negative and population numbers decline. The dynamics of the population is described by the differential equation:
To illustrate this, we could form a simulation to explore how population is effected and how we can model the logistic function by Verhulst.
Description of Simulation: Each student in the class is given an identification number, a blank data sheet, and a die. The instructor is given a bell. After the instructor records all of the identification numbers given, they will ring the bell. This will begin a two minute stage in which the students will begin to have "encounters". To be considered an encounter, two students will shake hands, exchange each others identification numbers, and roll each die. If the sum of the numbers rolled is less than or equal to 5, then the students have had a risky encounter. However, if the sum of the two die rolled is greater than 5, the students may move on to another encounter. The students are to have as many encounters in the given two minute stage as reasonable. After the two minutes, the instructor will end the stage. There is a short break between each stage, maybe 10-15 seconds. This procedure can be repeated for any number of stages, just as long as there is enough stages to collect suitable data. After the students have completed the number of stages, by rolling an arbitrary number, a person will be chosen to be initial infected person. Then, each person who had a risky encounter in stage 1 will be circled in the collection of identification numbers taken down at the beginning. Once stage one has been completed, move on to stage two, and whoever had a risky encounter with anyone who is now infected, they become infected at that given time. This can be repeated to the completion of the stages of the experiment.
How does this model the logistic function?
Using this simulation, we can develop the recursive formula as In+1=In+Nn+1. The number of infected people is the I n+1 value is determined by adding the people who became infected in the previous stage which is In, and add the number of people who are newly infected, thus In+1. We can also determine the number of newly infected by letting N n+1= a* I n(T-I n), in this case, a = p/T, where p is probability of having a risky identification number, and T is the total population, and I n(T-I n) is the possibility of a person becoming infected.
To illustrate this, we could as well input some data into a spreadsheet to calculate what is happening to the population. Verhulst Model
(For the spreadsheet, we used T=30 for our total population, and we also used random values for people becoming infected. 10/36 for the possibility of becoming infected)