A transition matrix is said to be regular if and only if for some ', P' has no zero elements.
If P is a regular transition matrix, then, for any initial ecosystem distribution x0, P'x0 approaches so-called limiting ecosystem distribution x* which is the unique ecosystem distribution vector such that
The regular transition matrix corresponds to a succession system where a given piece of land can be covered by any of the ecosystems under concern after some number of time steps, no matter what the initial cover. In other words, any transition is possible. If some transition is possible only for special '-values, then no power of P is positive. Different powers will have zeros in different positions and these zeros are changing cyclically for the powers. Hence, P'x0 cannot converge to a limiting ecosystem distribution.
The simplest transition matrix of this sort is
The even powers of this matrix are equal to '1 0'
and odd powers to
This transition matrix has fixed ecosystem distribution:
However, it does not converge to it.
For example, if initial ecosystem distribution is (1/4, 3/4)t, then ecosystem distribution will be cyclically changing at each step from (1/4, 3/4)T to (3/4, 1/4)T and back.
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