To make a crude model of a forest we might introduce states 0 grass, 1 bushes, 2 small trees, 3 large trees, and write down a transition matrix such as the following The idea behind this matrix is that if left undisturbed a grassy area will see bushes grow, and then small trees, which of course gro...

To make a crude model of a forest we might introduce states 0 = grass, 1 =

Question:

To make a crude model of a forest we might introduce states 0 = grass, 1 = bushes, 2 = small trees, 3 = large trees, and write down a transition matrix such as the following:

0 01/2 1 1/2 1/24 7/8 1/36 1/8 1 2 3 2 0 1/12 0 8/9 0 0 3 0 0 1/12 7/8

The idea behind this matrix is that if left undisturbed a grassy area will see bushes grow, and then small trees, which of course grow into large trees. However, disturbances such as tree falls or fires can reset the system to state 0.

Find the limiting fraction of land in each of the states.

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