ME17, Summer, 2008
Problem Set #4
Due Thursday, July 24
th
at class time
Reading:
Sections 13.1, 13.2, 13.3,
14.1, 14.3, 14.5, 19.1, 19.2, 19.3, 17.1, 17.2, 17.3, 17.4
Problem 1.
Here is one of the problems from a previous midterm:
Consider the following modification of the prey-predator scheme in which
there is now a third competing species, say eagles, denoted by z(t).
Eagles are
endangered species and have a very small birth rate:
they also die off due to
the use of pesticides in the first part of the 20th century.
They prey on both
wolves and bunnies, but with different efficiencies (since wolves are larger
and more difficult to kill than bunnies).
The resulting model is
dx/dt =
x
- xy - xz
dy/dt = -y + xy – yz
dz/dt = 0.02 z
2
– P + xz + 0.01 yz
Here terms have been added to the first and second equations to represent the
fact that eagles can kill both bunnies and wolves.
The third equation governs
the eagle population.
The small coefficients (0.02 and 0.01) reflect the low
birth rate of eagles and the relative difficulty of killing a large animal like a
wolf respectively.
P is the death rate due to pesticides.
Assume that P = 1. Write a program to solve for the steady states of this model by Newton’s
method.
Try to find at least one steady state solution to see if the eagles, bunnies and wolves can
coexist.
To apply Newton’s method, you will need (i) an initial guess, (ii) an evaluation of both the
equations and the Jacobian at the current iterate, and (iii) a way to solve the equations for the
next iterate.
The first of these will require some experimentation on your part, since there are
multiple solutions and Newton’s method might not converge if the initial guess is poor.
As an
initial guess, try something near the solution when there are no eagles:
x=1, y=1, z=0.1 (say).
Use the backslash operator to solve the sets of linear equations at each iteration.