I am trying to get an intuitive understanding of the meanings of right and left eigenvectors. I guess the best thing you can do is to provide examples of application. (Examples from the field of biology would be especially welcome).

Example:

A population have individuals that are classified in three classes, class A, B and C. Na, Nb, Nc represents the number of individuals in each class.

$$Na(t+1) = m*Na(t) + n*Nb(t) + o*Nc(t)$$ $$Nb(t+1) = p*Na(t) + q*Nb(t) + r*Nc(t)$$ $$Nc(t+1) = s*Na(t) + t*Nb(t) + u*Nc(t)$$

$$v(t+1) = A * v(t)$$

What is the long term equilibrium of this system of equations? Do we use left or right leading eigenvector of A. Why would be the biological meaning (if any) of the other (left/right) eigenvector?

The current intuitive sense I have is that an eigenvector of a matrix is a measure of how oriented is the distortion caused by the multiplication by this matrix. The eigenvalue is the strength of this distortion. Therefore the eigenvector linked with the biggest eigenvalue determines the long term behaviour of a system.

Here are two links on Stack exchange that did not help me answering my question. How to intuitively understand eigenvalue and eigenvector? question on left and right eigenvectors