Assuming the interaction matrix is diagonizable, the system state can be represented as a linear combination of the eigenvectors. The eigenvector with the largest positive eigenvalue grows the fastest under the system dynamics. Therefore, the respective compontent of the system state will become the dominating component, much larger than the others. (The growth of the components is exponential.) Ultimately, the normalized system state will be approximately equal to the fastest growing eigenvector, unless there are equally strongly growing other eigenvectors.
If we assume the eigenvalues are non-degenerate and thus sortable by size, one can identify the strongest growing eigenvector, the second strongest growing eigenvector, etc. I think this is what JenniferRM means with 'first' and 'second' eigenvector.
I wondered the same thing. The explanation I've come up with is the following:
See https://en.wikipedia.org/wiki/Linear_dynamical_system for the relevant math.
Assuming the interaction matrix is diagonizable, the system state can be represented as a linear combination of the eigenvectors. The eigenvector with the largest positive eigenvalue grows the fastest under the system dynamics. Therefore, the respective compontent of the system state will become the dominating component, much larger than the others. (The growth of the components is exponential.) Ultimately, the normalized system state will be approximately equal to the fastest growing eigenvector, unless there are equally strongly growing other eigenvectors.
If we assume the eigenvalues are non-degenerate and thus sortable by size, one can identify the strongest growing eigenvector, the second strongest growing eigenvector, etc. I think this is what JenniferRM means with 'first' and 'second' eigenvector.