Differential Equations Day 11 -- Project 2 -- Logistic Growth
Directions
Follow the steps below and do your work in the provided GeoGebra windows.
NOTE to VSC students taking this class with me: be sure you are accessing this through a GeoGebra Classroom Link! The URL of this page should have the word “classroom” in it. If not, then go back to Canvas and be sure to access this page from the Project 2 page. Also: I strongly recommend you login to GeoGebra.org with a free account so your work is saved, and you can come back later to review or modify it.
For external readers: these "project" activities are meant to be taken as part of my course, so these sections of the GeoGebra book my not be as intelligible as others. My apologies.
Goals
The primary goal of this project is to learn about the Logistic Growth Model, a modification of the Exponential Growth Model. A secondary goal is that you will also continue to gain experience with algebraic and numerical methods.
Content Overview
The logistic growth model is a first order differential equation model of the growth of a population. It is an extension of the exponential growth model.
Exponential Growth Model: The growth rate of a population is proportional to the size of the populationOr, in a first order differential equation notation: The logistic growth model adds an additional proviso to this:
Logistic Growth Model: The growth rate of a population is proportional to (1.) the size of the population, and (2.) the difference between the population and the environment's carrying capacity.It's not immediately clear how to express this in a first order differential equation. The first bit is just like exponential growth, so let's start with that, and leave the second bit as a question mark to figure out later. With that out of the way, let's turn our focus on "the difference between the population and the environment's carrying capacity." If we call the "environment's carrying capacity" a constant K (note: capitalized), then this second bit is or . It turns out we need to select so that when P is less than K the growth rate of P will be positive, and with P is greater than K, the growth rate will be negative. The only other tweak is a rather technical adjustment, and that's that we "normalize" this term. This means that we divide through by K so this term is not as sensitive to different population sizes. In other words, the correct term is . Thus the logistic growth model is . Note: it is common to refer to k as the growth rate, and K as the carrying capacity of the model. For values of P that are not near K, solutions of the model will very nearly match exponential models with growth rate k.
Steps For Part 1
- Declare
k=0.013andK=21000 - Declare a function
fof two variables x and y that is equal to the right hand side of the logistic growth model. Be sure to use x and y as appropriate. - Use
fto create a slope field that illustrates the nature of the solutions of the logistic growth model. - Plot an initial condition,
InitialCondition=(0,17000) - Plot a second observation,
SecondObservation=(1,18000) - Adjust your y axis so
InitialConditionandSecondObservationare plainly visible - Use Locus to create a numerical solution of the differential equation through InitialCondition
- Plot an exponential function that passes through InitialCondition and which has growth rate k (Hint: exponential models are of the form ). Note the similarity initially between the exponential function and the numerical integral.
- Use a "guess, and check, improve" method to select k so that the numerical solution is within 10 of
SecondObservation - Use your numerical solution to predict when the population, P, will be within 10% of the carrying capacity, K. Plot your response as a point on the numerical model.
Part 1 Applet
Steps For Part 2
- Review the analytical solution of the logistic growth model in section 3.4.2 here: https://sites.math.northwestern.edu/~mlerma/courses/math214-2-03f/notes/c2-logist.pdf
- Tips for reading the solution: note that the solution makes use of the method of separation of variables, but encounters a fairly difficult integral that requires the method of partial fractions from Calculus 2. Also, note that whether or not you fully understand the integral, the general solution is shown at the bottom of page 1.
- Starting from the general solution found in the linked document, find the specific solution that matches the
InitialConditionandSecondObservationfrom Part 1. - Plot it in the Part 1 GeoGebra applet above.
- Estimate the difference between the numeric integral from Part 1 and the specific algebraic solution from this Part at t=10. Either plot the difference graphically, or make a note of your findings in a textbox (or both).
Steps for Part 3
Replicate the solution of the example in the reading in the Part 3 Applet below. You are given leeway in making decisions about how you think it would be best to "replicate" the example. At the very least however, you should plot the slope field, the analytic solution, the initial condition, and other objects that illustrate the solution of the example question.
The reading:
https://sites.math.northwestern.edu/~mlerma/courses/math214-2-03f/notes/c2-logist.pdf