Lab 1 -- Population Growth ("Your Report" Section)

I'm getting a lot of good questions about the "Your Report" section of Lab 1. For your information, most labs will not require a "Report" section like this one.

(1): What differential equation was solved and how was the solution obtained? Exactly how were the coefficients computed?

It asks what equations were solved, and how. To save yourself a lot of writing, you could attach your assignment, and say, "I solved Equations (a), (b), and (c) for different sets of initial values." Then you can state the methods you used to solve each of them (integration, separation of variables, etc), and refer to 'the attached computations in Assignment (parts I and II) which show my work,' or something like that.

(2): What, according to each model, will the population in 2050 be? Which prediction do you have the most confidence in? Why? Be sure to discuss the fit to the historical data (both recent and past) using both your graphs and the values of epsilon.

The prediction that you have the most confidence in doesn't HAVE TO come from the most complex model [the logistic model]. But do explain why your work supports your conclusion, whatever it is -- that's what you'll get credit for. 

(3): At what population level will New York stop growing? How confident are you in this answer?

I think the lab should ask instead, "What does your solution to the logistic model of population growth predict to be the maximum population of New York?" Answer this and then explain whether or not you think the logistic model's prediction is likely to be accurate.

(4): What factors might cause your answers to deviate from the actual answers? Specifically, what variables do your various models take into account and what are some variables not accounted for?

 My favorite way to go about answering is this:

"Model _______ predicts that the population grows __________." (List some reasons why that prediction might be true. Then list some possible real world occurrences which would make that prediction false.)

For example, I'll give you a response for part (a):

Model (a), the linear model, predicts that the population grows at a linear rate, that population is proportionate to time.

This model takes time into account, because P(t) = kt+C. In other words, the number of people added to the population in one year will be the same as are added to the population every other year.

- BUT, suppose that when that population is bigger, a similar percentage of the people in the population are having sex and reproducing. This would suggest that population growth is proportionate to the current population (i.e., P' = kP) instead of simply being constant.

- Also, population growth is unlimited over time under the linear model, but the actual size of New York is not. Thus this model doesn't take crowding into account.

- Furthermore, if you use this model to try to figure out population in the distant past, you can get negative populations, which wouldn't make any sense.

- The linear model predicts that population grows continuously. But in some cases, sudden events can cause very discontinuous population changes (e.g., the Chernobyl accident).


I don't expect nearly that much detail for full credit, but I think you'd get the most out of this lab if you think about it in those kinds of terms.