Consider a one-dimensional sequence (temporal or spatial) of samples from a many-species community. The sequence may exhibit no steady change, or gradual, steady change, or one or more abrupt, ...
If $A = (a_{ij})$ is an $n \times n$ irreducible matrix, then there are positive numbers $d_1, d_2, \cdots, d_n$ so that $\sum_k d_ia_{ik}d^{-1}_k = \sum_k d_ka_{ki}d ...
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