Homework March 24th 2003 due March 31
Bisc 471
Name:
1.
a) For each of the differential equations make the traveling wave substitution u(x,t)=f(x-vt), v is a
constant, and so derive a differential equation in one variable for the function f.
u
i) u
= 5(u-sin(u)) + 2
t
 

u
u
u
ii) t = u3 +  + 

b) For the equations you got in a) call the variable s, so that f is a function of s only, then write p=df/ds
and rewrite the differential equations as a pair of equations for f and p that only involve the first
derivative. Sketch the phase plane diagram for these 2 sets of equations. Label nullclines and equilibrium
points.
2. Do problem 17 p 486 from chapter 10 in E-K handout. If you don’t have the relevant pages 456-459, I
have made extra copies of them. They are outside my door as well as the printed version of this exam.
Continued on next page.
3. Assume we have a model which could be describing Predator Prey interactions.
dN
N
NP
= rN(1- ) - 
dt
K
+N
dP
NP
=
- P
dt
+N
a) We will use a simplified system given below.
dN
NP
= rN - 
= f(N,P)
dt
+N
dP
NP
=
- P =g(N,P)
dt
+N
r, , ,  are positive constants.
Under what type of conditions is this a reasonable approximation of the first set of differential equations.
b) Find the steady states. When does a non-trivial steady state exist?
c) Write up the stability Matrix (the Jacobian) for this system at the non-trivial steady state and use it to
determine for what conditions of r, , , the steady state is
i) a stable node (not spiral)
ii) a stable spiral
iii) a unstable node
iv) a saddle point
For i) draw the nullclines and do a phase plane analysis.
Which of these four could have a diffusive instability?
d) Now we will look at spatial variations in this system:
dN
NP
= rN - 
+DN
dt
+N
dP
NP
=
- P +DP
dt
+N
The spatial variation is 1-dimensional so becomes d2/dx2
Derive the conditions for diffusive instability for this system and show your work. Hint : linearize the
equations and assume perturbations of the type et cos(qx) away from steady state and find criteria for q.
Assume we are on an interval x[0,] and that there is no flux at the boundaries (x=0 and x=).
e) Let r=1, =2, =1
mm2/sec and DN =0.4 mm2/sec and if so describe the pattern
that will emerge, do the same for DP=1 mm2/sec and DN =0.05 mm2/sec
P=1
f*) Answer question e) in two dimensions, where x[0,] and y[0,] and no flux on the boundary.
2.Show that traveling wave solutions for yeast cells on glucose medium
n
= kng
t
g
g
= -ckng+D 2
t
x
would have to satisfy
-v
= kNG
-v
= -ckNG +D
G
where
z2
N(z)=n(x,t), G(z)=g(x,t) and z=x-vt.
Show that this leads to
kNG
=- v
-v
= -ckNG +D
G
z2
where