Problem 1:  In a 1982 paper (Ecology 63:1218-1222), Hal Caswell presented the following hypothetical life cycle which included both seed-based and vegetative reproduction:

a) Write down the projection matrix A for this life-cycle graph in the equation x_n+1= Ax_n, where x_n is a vector representing the abundance of individuals in each life stage.
 
 

b) Without computing the eigenvalues, etc. of this matrix, what can you say about the long term behaviour of this system?
 

c) Assume that all the parameters are set to 1.  Compute x₂ for the initial condition x₀=(0, 1, 3, 0, 2).  Also compute x₂for the (non-physical) initial condition x₀=(0, 1, 0, 0, -1).  What does this tell you about the eigenvalues of the matrix A?
Can you say anything more now about the long term behaviour of this system?
 

Problem 2: Edelstein-Keshet, p.62, Problem 3.

Problem 3: Edelstein-Keshet, p.62, Problem 4.