Bio 292
Projection matrix problem set:
In constructing the projection matrix for the semipalmated sandpiper, we assumed a “prebreeding census” – birds are censused as they return to the breeding grounds in spring
just prior to establishing their nests. The picture we had was this (where dashed arrows
represent reproduction, and solid arrows represent survival):
s2
3+ year olds
3+ year olds
s2
b3 c
2 year olds
2 year olds
s1
1 year olds
1 year olds
b2 c
b1 c
Census t
s0
Census t+1
Juveniles
Now imagine that instead, we census the population just after the breeding season (a
“postbreeding census”), so that the youngest individuals we observe at the census are not
1 year olds, but juveniles, and we will lump all individuals 2 years old and greater to keep
the same number of age classes as before. The picture would then look like this:
2+ year olds
s2
2+ year olds
s1
s0
1 year olds
Juveniles
1 year olds
Juveniles
s0 b1 c
Census t
s1 b2 c
s2 b3 c
Census t+1
where the vital rates have exactly the same meanings as before.
a) Explain why the dashed arrows now include different survival rates.
b) Draw the life cycle diagrams for the two cases.
c) The projection matrices corresponding to the two cases are:
A pre
b1cs0
  s1
 0
b2cs0
0
s2
b3cs0 
 s0b1c s1b2c s2b3c 

0  and A post   s0
0
0 
 0
s2 
s1
s2 
Show that the eigenvalues of these two matrices are the same, as they must be (Hint: you
don’t need to compute the actual eigenvalues; simply show that the characteristic
equations are the same).