1. Welcome to Differential Equations! Dr. Rachel Hall rhall@sju.edu http://www.sju.edu/~rhall/DiffEq

2. Predicting the future From the introduction to your textbook: “This book is about how to predict the future. To do so, all we have is a knowledge of how things are and an understanding of the rules that govern the changes that will occur. From calculus we know that change is measured by the derivative. Using the derivative to describe how a quantity changes is what the subject of differential equations is all about.” --Blanchard, Devaney, and Hall, Differential Equations, 3rd edition, Thomson Brooks/Cole, 2006.

3. What is a model? Read p. 2-3.

4. Unlimited population growth “The rate of growth of the population is proportional to the size of the population.” Quantities: t = time P = population k = proportionality constant (growth-rate coefficient) Identify independent variable, dependent variable, parameter. The differential equation: dP/dt = kP Explain…

5. What does the model predict? A solution is a function P(t) for which the statement dP/dt = kP is true. Can you think of any possible solutions? P = e kt P = ce kt (in fact, c must equal P(0), a.k.a. P 0 a.k.a. the initial condition) P 0 = 0 is the equilibrium solution because dP/dt = 0 forever. (booooring!) If P 0 ≠ 0, how does the behavior of the model depend on P 0 and k? In particular, how does it depend on the signs of P 0 and k?

6. General and particular solutions The general solution to dP/dt = kP is P(t) = ce kt. If we have a little more info (like, the value of P(0)), we can find a particular solution (something like P(t) = 3.9e kt ). If we have even more info (like the value of P(10) or whatever) we can also find k. Look at p. 7-8 for an example. Is this a “good” model? What are its flaws? Exercise: Use the census data for 1850 and 1950 to predict the population in 2007.

7. Logistic population model OK, let’s make some more sophisticated assumptions… 1.If the population is small, growth is proportional to size. 2.If the population is too large for its environment to support, it will decrease. We now have quantities t = time P = population k = growth-rate coefficient for small populations N = “carrying capacity” Let’s restate 1. and 2. in terms of derivatives: 1.dP/dt is approximately kP when P is “small.” 2.dP/dt is negative when P > N.

8. Big fat differential equation Check: 1.dP/dt is approximately kP when P is “small.” 2.dP/dt is negative when P > N.

9. Qualitative stuff Soooo, tell me about the function What kind of function is it? What does its graph look like? Now use what you know about the relationship between P and dP/dt to sketch some solutions to the logistic equation. Is this a realistic model of population growth?

10. A mystery Here’s a new population model… Find the equilibrium solutions. For what values of P is the population increasing? decreasing? Sketch some solutions to the differential equation. Can you think of a situation in which this model makes sense? Be creative…