# Leslie matrix with no dominant eigenvalue

## Homework Statement

What is the consequence for a population model if the Leslie matrix used has no dominant eigenvalue?

## Homework Equations

x(k) = Ax(k-1), where A is the Leslie matrix, x is a vector representing the initial population distribution.

x(k) is the vector of the population distribution for time period k.

## The Attempt at a Solution

All I understand of the dominant eigenvalue is that over long time periods (mutliple iterations of x(k) = Ax(k-1) ), x(k) approaches lambda*x(k-1). In other words, the population distribution tends towards that given by the eigenvector associated with the dominant eigenvalue.

I understand the proof of this http://online.redwoods.cc.ca.us/instruct/darnold/LinAlg/leslie2/context-leslie2-p.pdf [Broken]

which uses the diagonalisation of A.

I've read that one consequence of A not having a dominant eigenvalue is that the population numbers will rise and fall in waves. Is this the only consequence? Can anyone direct me to resources on this? All I seem to find are pages about matrices with dominant eigenvalues, rather than ones with no dominant eigenvalue.

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