Exponential and Logistic Population Models In the text, we modeled the U.S. population...

Exponential and Logistic Population Models

In the text, we modeled the U.S. population over the last 210 years using both an exponential growth model and a logistic growth model. For this lab project, we ask that you model the population growth of a particular state. Population data for several states are given in Table 1.11. (Your instructor will assign the state(s) you should consider.)

We have also discussed three general approaches that can be employed to study a differential equation: numerical techniques yield graphs of approximate solutions, geometric/qualitative techniques provide predictions of the long-term behavior of the solution, and in special cases analytic techniques provide explicit formulas for the solution.

In your report, you should use as many of these techniques as is appropriate to help understand the models.

Your report should address the following items:

Your report: The body of your report should address all three items, one at a time, in the form of a short essay. For each model, you must choose specific values for certain parameters (the growth-rate parameter and the carrying capacity). Be sure to give a complete justification of why you made the choices that you did. You should include pictures and graphs of data and of solutions of your models as appropriate. (Remember that one carefully chosen picture can be worth a thousand words, but a thousand pictures aren’t worth anything.)

Produce a logistic growth model for the population of your state. What is the carrying capacity for your model? Using Euler’s method, predict the population in the years 2010 and 2050. Using analytic techniques, obtain a formula for the population function P(t) that satisfies your model. To what extent do solutions of your model agree with the historical data?