Math 2250 Lab 15
Due Date: 04/24/2014
Name/Unid:
Class ID:
Section:
References: Edwards-Penney: Sections 9.1-9.4, especially Section 9.3.
For additional references on modeling the biological systems described below see EdelsteinKeshet (2005) Mathematical Models in Biology and Strogatz (1994) Nonlinear Dynamics and
Chaos.
1. (40 points) Epidemic Model In Lab 14, we introduced the following set of differential
equations to describe an epidemic:
dS
= cI − aSI
dt
dI
= aSI − cI
dt
where S represents the number of susceptible people, and I represents the number of
infected people. Individuals become infected (move from the S class into the I class) at
a rate proportional to the product of the number of infected individuals with the number
of susceptible people. Individuals recover (move from the I class into the S class) at a
rate proportional to the number of infected individuals.
In part (f), we asked you to modify the above model by assuming that individuals
who leave the infected class through recovery become permanently immune rather than
becoming susceptible again. That is, we introduced a new state for permanently “recovered” individuals. Kermack and McKendrick (1927) gave the following set of differential
equations to incorporate this assumption:
dS
= −aSI
dt
dI
= aSI − cI
dt
dR
= −cI
dt
This is known as an S-I-R (susceptible-infected-recovered) model without vital dynamics
(no births or deaths in the population) where parameter a represents the rate of transmission of the disease and c, the rate of removal.
(a) Show that the total population N = S + I + R does not change over time.
(b) Consider the dynamics governing just the susceptible and infected populations.
That is,
dS
= −aSI
dt
dI
= aSI − cI
dt
1. Find and classify all the fixed points of the system.
2. Sketch the nullclines and the vector field.
3. Find a conserved quantity for the system. (Hint: Form a differential equation
for dy/dx. Separate the variables and integrate both sides.)
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2. (40 points) Predator Prey System Let F (t) denote the population of foxes and R(t),
the population of rabbits at a given time t. If there were no foxes, the population
of rabbits would grow at a natural rate. Similarly, if there were no rabbits, the fox
population would decay (since they no longer have a food source).
When both foxes and rabbits are present, we expect the fox-rabbit interactions to inhibit
the rabbit population and increase the fox population. This gives the equations
R0 (t) = aR − pF R
F (t) = −bF + qF R
where a, b, p and q are positive constants.
After observing and measuring the fox/rabbit population for a small period of time,
researchers are able to estimate the following values: a = 60, p = 15, b = 2.25, q = 51.75.
(a) Explain what each of the constants a, b, p and q represent.
(b) Compare the above predator-prey equations with the initial epidemic model given
above:
dS
= cI − aSI
dt
dI
= aSI − cI
dt
What is different in these equations?
(c) Find the critical points for this system of equations.
(d) Use the Jacobian to find the linearization of the system at these critical points.
Classify each point you found as stable or unstable, and determine whether each is
a node, spiral, or saddle.
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3. (40 points) Competing Species Recall the Rabbit-Sheep System from Lab 4 where
the two species, rabbits and sheep, competed for the same resource (i.e. grass). Suppose
the population dynamics of these two species are given as follows:
dR
= R(6 − 1.5R − 4S)
dt
dS
= S(6 − 2R − 3S)
dt
where R(t) represents the population of rabbits at time t, S(t) represents the population
of sheep and R(t), S(t) ≥ 0.
(a) Find and draw the nullclines of this system.
(b) Find the critical points for this system.
(c) Draw a few vectors of the vector field, indicating the general direction in each region
bounded by the nullclines.
(d) Use the Jacobian to find the linearization of the system at these critical points.
Classify each point you found (stability and type).
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