MATH 462: Homework 5
Winter 2006
Set: 20 Feb. Due: 6 Mar.
1. Consider the following equation:
dN
dt =κ(C0−α N)N,
where N(t) is the bacterial density, C0is the available nutrient.
(a) Explain what the parameters represent, using the dimensions known.
(b) Show that the equation can be rewritten as
dN
dt =r(1 −
N
B)N,
do not nondimensionalize.
(c) Let N(0) = N0. Solve the new version of the equation.
2. Consider the following predator/prey model
dH
dt =rHH(1 −
H
K)−
a H P
1 + a H TH
,dP
dt =rPP(1 −
P
k H ).
Take the following parameter values: rH= 0.2, K = 500, TH= 0.5, rP, k = 0.2 and
a= 0.001,0.1,0.3.
(a) When we find equilibrium points we set the time-derivative and solve for the dependent
variables. Here, plot, in the P-H plane, the equations you arrive at when setting the
time derivative equal to zero. Use the ”hold on” command in matlab to plot them on
the same graph. One graph for each value of a.
(b) Write a matlab function to solve these equations. Plot your results in the P-H plane.
For each value of atest and state the effect of initial conditions (only try 3 or so).
(c) Why is the a= 0.3 case different to what we’ve seen before (this will only be 1 point).
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