Bifurcation in Markov Chains with Ecological Examples

The adjacency matrix of a weighted directed graph contains information on both connectivity and the strength of that connection. When the special case of Markov chains are considered, the additional constraints permit the characterization of the eigenvalues of its transition matrix, and the change of the nature of those eigenvalues as the probabilities (weights) change. A change in the nature of the eigenvalues, bifurcations, signals a change in the dynamic approach to a limiting probability of a chain as well as other aspects that can be of interest in applications. In this paper, we first characterize eigenvalues of any weighted directed cycles and any 3-state Markov chain. Then we define and characterize a special case, zero trace chains, which is useful in an ecology application discussed.