Computational Project # 5
Due November 13
Consider a simple ecosystem of rabbits that has an infinite food supply and foxes that
prey upon rabbits for their food. A mathematical model for this system consists of the
following differential equations:
()
()
0
0
2, 0,
,0
dr rrf r r
dt
df ,
f
rf f f
dt
α
α
=− =
=− + =
where t is time, r is the number of rabbits, f is the number of foxes, and
α
is a positive
constant. When
α
is zero, the two populations do not interact, and the rabbits do what
rabbits do best, and the foxes die of starvation. This system has been studied for various
values of
α
and initial conditions. For the purpose of this problem, we fix the value of
α
to be 0.01. To answer the questions in parts a) and b) below, solve the system of
differential equations up to t=20.
a) Compute the solution with r0 = 300 and f0 = 150. You should observe that the
behavior of the system is periodic with period very close to five time units. Plot
the solution for both r and f as a function of time and in the (f, r) plane.
b) Compute the solution with r0 = 15 and f0= 22. What happens to the rabbit
population?
Interpret the results for both cases.
To solve this problem you do not need to program you own time integration scheme,
instead use Matlab function ode15s. In order to understand how the program works, use
the odeset to set up NormControl on, relative tolerance to 10-9, absolute tolerance to
10-6, and the maximum order of time integration scheme to 2.
P.S. Bring your solution to class. Email your code to [email protected]. The
program that you send should be a working program. All the codes will be checked
whether they run or not. If they are erroneous, but run, points will be taken for the errors.
If the code does not run (it has some syntax errors), an additional 25% will be taken off.
The goal of this class is for you to be comfortable solving engineering problems. Please
take your time and learn how to trust the computer.