Section 2.1

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# Section 2.1 - PowerPoint PPT Presentation

Section 2.1. MODELING VIA SYSTEMS. A tale of rabbits and foxes. Suppose you have two populations: rabbits and foxes. R(t) represents the population of rabbits at time t. F(t) represents the population of foxes at time t . What happens to the rabbits if there are no foxes?

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## Section 2.1

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### Section 2.1

MODELING VIA SYSTEMS

A tale of rabbits and foxes

Suppose you have two populations: rabbits and foxes.

R(t) represents the population of rabbits at time t.

F(t) represents the population of foxes at time t.

• What happens to the rabbits if there are no foxes?

Try to write a DE.

• What happens to the foxes if there are no rabbits?

Try to write a DE.

• What happens when a rabbit meets a fox?
• If R is the number of rabbits and F is the number of foxes, the number of “rabbit-fox interactions” should be proportional to what quantity?
The predator-prey system

A system of DEs that might describe the behavior of the populations of predators and prey is

• What happens if there are no predators? No prey?
• Explain the coefficients of the RF terms in both equations.
• What happens when both R = 0 and F = 0?
• Are there other situations in which both populations are constant?
• Modify the system so that the prey grows logistically if there are no predators.
Exercises

Page 164, 1-6. I will assign either system (i) or (ii).

P(0) = 0

R(0) = 0

predators

prey

predators

prey

Graphing solutions

Here are some solutions to

predators

prey

A startling picture!

Here’s what happens if we start with R(0) = 4 and F(0) = 1.

This is the graph of the

parametric equation

(x,y) = (R(t), F(t)) for the IVP.

R(0) = 4

F(0) = 1

The phase plane

Look at PredatorPrey demo.

Exercises
• p. 165 #7a, 8ab
• Look at GraphingSolutionsQuiz in the Differential Equations software (hard!)
Spring break!

Now for something completely different…

Suppose a mass is suspended on a spring.

• Assume the only force acting on the mass is the force of the spring.
• Suppose you stretch the spring and release it. How does the mass move?
Quantities:

y(t) = the position of the mass at time t.

• y(0) = resting
• y(t) > 0 when the spring is stretched
• y(t) < 0 when the spring is compressed

Newton’s Second Law: force = mass  acceleration

Hooke’s law of springs: the force exerted by a spring is proportional to the spring’s displacement from rest.

k is called the spring constant and depends on how powerful the spring is.

DE for a simple harmonic oscillator

Combine Newton and Hooke:

Sooo….

which is the equation for a simple (or undamped) harmonic oscillator. It is a second-order DE because it contains a second derivative (duh).

Comes from our assumption

Comes from the original DE

How to solve it!

Now we do something really clever. We don’t have any methods to solve second-order DEs.

Let v(t) = velocity of the mass at time t.

Then v(t) = dy/dt and dv/dt = d2y/dt2. Now our DE becomes a system:

Exercises

p. 167 #19

• Rewrite the DE as a system of first-order DEs.
• Do (a) and (b).
• Check (b) using the MassSpring tool.
• Do (c) and (d).
Homework (due 5pm Thursday)