Intuitive meaning of right and left eigenvector

Intuitive meaning of right and left eigenvector

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