7
7.5 Logarithmic growth
We will look at how a rumor (or other information) spreads through out a population. This model is a more sophisticated model than the exponential growth model. Population often increase exponentially in its early stages but levels off eventually and approaches a carrying capacity because of limited sources. A mathematical model of the rate of diffusion is found by assuming that this rate is proportional to the number of people who have heard the rumor and those who have not heard it.
Logistic growth occurs when the rate of growth is proportional to the amount present and to the difference between the amount present and the fixed limit. Applications that can be modeled by logistic growth are long-term population growth, epidemics, sales of new products, and company growth.
The resulting differential equation is
[pic] where N is the size of the population.
Before we begin, we need to
1. Pick a number (not a nice round one like 7500, but a messy one) from 5000 to 10000 and record it here . This number is your seed value.
2. On your home screen, type in the number you selected above, then the [STO>] key. Now press [MATH] [left-arrow] (to select the probability sub-menu, [PRB], you can also press [right-arrow] three times), select [1:rand], and finally press [ENTER]. This command tells your calculator to start generating random numbers starting at the location specified by your seed value chosen in part (a). It is similar to the process of using Table B, starting at a certain row. Your screen should look something like the one below (with your seed value replacing 5678).
[pic]
Ok, let us begin the experiment:
Assume that there are 30 students enrolled in Calculus II. On Monday (day 0), one student invented a rumor that the next test was cancelled. On Tuesday (day 1), she passes this rumor on to two other students. On Wednesday, each of these three students passes it on to two other (who may or may not have already heard the rumor). This process continues with only two contacts per student per day occurring.
We can simulate the spread of the rumor through the class using the TI-83 command randInt(1,30, ___). Record the results below. Stop when all 30 have heard the rumor.
|Day |Number heard rumor |
|0 | |
|1 | |
|2 | |
|3 | |
|4 | |
|5 | |
|6 | |
|7 | |
|1 |2 |3 |4 |5 |
|6 |7 |8 |9 |10 |
|11 |12 |13 |14 |15 |
|16 |17 |18 |19 |20 |
|21 |22 |23 |24 |25 |
|26 |27 |28 |29 |30 |
Create a scatter plot of Days, t, and the number that heard the rumor, y. Label the axes appropriately. The data should be in lists L1={0,1, …} and L2 ={1, …}.
Use the differential equation [pic] to model the spread the rumor. N=30.
In order to complete the solution we need a general solution to the differential equation given above. Find the general solution.
Use your result from above to solve the differential equation for the particular solution with y(0)=1 and y(___) = _____ as you initial conditions. Note: you cannot use the ordered pair with the max value y=30.
Graph your equation on the calculator.
When will half of the student have heard the rumor?
When is the rumor spreading the fastest?
Use the function nDerive([pic] to graph the function and state any relationship.
[pic]
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