Math 221

Homework #2

Instructions. This is due on September 16 at the beginning of class. It is worth 20 points. Late papers lose 3 points per day late. You must work by yourself, but feel free to consult with me. Explain what you did and why you did it: good explanations are essential! Your paper should have an introduction, a main body and a conclusion. Start early, ask questions and do well.

Introduction. Biology is increasingly using mathematics to describe phenomena. Some common biological phenomena are difficult to describe mathematically. One example is the human heartbeat, which has a stable cycle. The heart is essentially an electrical device that must coordinate the contractions of ventricles and atria. Polar coordinates can be used to describe the electrical potential, with r representing the strength of the signal and q representing the polarization of the wave. In this assignment,you will look at a simple model that illustrates the same dynamics as a stable heart beat.

Problems. As with many models, the starting place is differential equations describing how the variables change. In this case, we assume that (for some constant a)

r’(t) = a r (1 - r) with r(0) = r₀

q’(t) = 2p with q(0) = q₀

It can be shown (you do not need to do this; details can be found in section 6.5 of the book) that solutions of these equations are given by

r(t) = r₀ / [r₀ - (r₀ - 1)e^{-a t}]

q(t) = 2pt + q₀

You will start by graphing examples of this solution for different initial conditions. Explain why you can’t use the polar coordinates graphing mode on your calculator. Use parametric equations with x(t) = r(t) cos q(t) and y(t) = r(t) sin q(t) to graph the solution with a = 1, r₀ = 0.5 and q₀ = 0. Briefly explain what the solution does as t increases.

Next, graph the solution with a = 1, r₀ = 1.5 and q₀ = 0.

Then graph the solution with your own choice of a > 0, your own choice of r₀ > 1 and your choice of q₀ > 0.

Finally, graph the solution with your own choice of a > 0, your own choice of 0 < r₀ < 1 and your choice of q₀ > 0.

For each graph, briefly explain what the solution does as t increases. Mathematicians say that this solution has a limit cycle. Briefly explain why both words are appropriate.

The human heart has a limit cycle that is a nice steady beat. Heart fibrillation is a type of heart attack where the beating of the ventricles and atria has gotten “out of sync.” A defibrillator is used to shock the heart back to a steady beat. Explain why the presence of a limit cycle makes this more likely to work.