### Determine the limiting matrix

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Considering the ecological succession of barren dunes to an oak community using a Markov Chain. In this simple model, it is assumed that there are three states for a specic ecological region. These states are bare dunes (BD); a grass community (G); and an oak community (O). The transition probabilities are shown in Table 1.
(1) Using this information, construct the state diagram for the process
(2) Suppose initially 10% of the regions are bare dunes, 10% are grass communities, and the rest is oak communities. Determine the distribution of states after 2 time steps and after 10 time steps. Interpret the meaning of these results.
(3) Determine the limiting matrix L and the steady-state distribution for this ecological succession. Interpret the meaning of these results.

BD G O
BD 0.3 0.2 0.1
G 0.6 0.5 0.1
O 0.1 0.3 0.8
Table 1: Transition Probabilities for Ecological Succession

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