MODELING VIA SYSTEMS

1
Section 2.1

• MODELING VIA SYSTEMS

2
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?

3
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.

4
Exercises

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

5
Graphing solutions

• Here are some solutions to

6
A startling picture!

• Heres what happens if we start with R(0) 4 and
  F(0) 1.

7
The phase plane

• Look at PredatorPrey demo.

8
Exercises

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

9
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?

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• Quantities
• y(t) the position of the mass at time t.
• y(0) resting
• y(t) gt 0 when the spring is stretched
• y(t) lt 0 when the spring is compressed
• Newtons Second Law force mass ? acceleration
• Hookes law of springs the force exerted by a
  spring is proportional to the springs
  displacement from rest.
• k is called the spring constant and depends on
  how powerful the spring is.

11
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).

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How to solve it!

• Now we do something really clever. We dont 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

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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).

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Homework (due 5pm Thursday)

• Read 2.1
• Practice p. 164-7, 7, 9, 11, 15, 17, 19
• Core p. 164-7, 10, 16, 20, 21
• Some of the problems in this section are really
  wordy. You dont have to copy them into your HW.