Math 640:336
Dynamic Models in Biology
Homework 2 Solutions
1. ( Chapter 2, Problem 3) a) Denote the steady state by N . Then
N
N = N er(1 K )
The trivial N steady state that holds for the above equation is N = 0. To
nd the nontrivial steady states we div
Math 640:336
Dynamic Models in Biology
Homework 5 (Solutions)
1. MMB Chapter 5, Problem 13.
We begin with the equations
dN
dt
dC
dt
C
N N
1+C
C
=
N C + 2
1+C
= 1
(a) Obvious.
(b) If we let x = N + 1 2 then the equation reduces to
dx
= 1 2 x
dt
Using separ
Math 640:336
Dynamic Models in Biology
Homework 1
1. Consider the one-parameter family of maps f : R R given by
f (x) = x.
1
(a) Indicate the dynamics by drawing a staircase diagrams for = 2, 1, 1 , 0, 2 , 1, 2.
2
(b) Describe all the xed points as a func
y = 2x
y=x
x0 x1
x2
Figure 1: Staircase diagram for f (x) = 2x.
Math 640:336
Dynamic Models in Biology
Homework 1 (Solutions)
1. Consider the one-parameter family of maps f : R R given by
f (x) = x.
1
1
(a) Indicate the dynamics by drawing a staircase dia
Math 640:336
Dynamic Models in Biology
Homework 2
1. Mathematical Models in Biology, Chapter 2, Problem 3.
2. Mathematical Models in Biology, Chapter 2, Problem 8(b).
3. Consider a C 1 map f : R2 R2 with a xed point x. The stability of x
is determined by
Math 640:336
Dynamic Models in Biology
Homework 3
1. Consider the dierential equation x = K (t)x where x(0) = x0 . Show that
the solution is of the form
t
x(t) = x0 exp
K (s) ds
0
Consider an algae population whose growth rate is dependent upon the
presen
Math 640:336
Dynamic Models in Biology
Homework 3 (Solutions)
1. Consider the dierential equation x = K (t)x where x(0) = x0 . Show that
the solution is of the form
t
x(t) = x0 exp
K (s) ds
0
Consider an algae population whose growth rate is dependent upo
Math 640:336
Dynamic Models in Biology
Homework 4 (Solutions)
1. Mathematical Models in Biology, Chp 6. 13.
The victim-exploiter Lotka-Volterra equations take the form
v = av bve
e = ce + dve
We rescale variables by v = pV , e = qE , and t = T . This give
Math 640:336
Dynamic Models in Biology
Homework 5
1. Mathematical Models in Biology, Chapter 5, Problem 13.
2. Mathematical Models in Biology, Chapter 5, Problem 21.
3. Lecture Notes on Mathematical Systems Biology 1.10.2 Problem 6
4. Lecture Notes on Mat