Anyway, we have a matrix equation:
and an initial population distribution:
where is the set of eigenvectors of A.
Let be the set of eigenvalues corresponding to the (respective) eigenvector.
Iterating this process gives:
Now, if is the dominant eigenvalue (I think this said the first eigenvalue. It was hard to read.) then we may assume
for all . (Obviously only a positive eigenvalue can dominate the population.) This means that for large k there is effectively only one term in the that is significant: the first term. So
for large k.
This means the population is stable in the state represented by eigenvector and has a growth rate represented by .
If there is no dominant eigenvalue then the population will fluctuate in a series of eigenstates with no specific growth rate dominating. However, for large enough k the population should stabilize around a vector that is the sum of eigenvectors with the largest eigenvalues. For example, say that we know that for . Then, given a large enough k we would have:
So the population would settle into a growth state vector with (in this case) a growth of .