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11. There are three kinds of vegetation in an ecosystem: grass, shrubs, and trees. Every year, 25% of grassland plots are converted to shrubs, 20% of shrub plots are converted to trees, 8% of trees are converted to shrubs, and 1% of trees are converted to grass. The other transition probabilities between different states are zero. (a) Draw a Markov model for this system. (b) Find a transition matrix for this system. (c) Suppose that initially the ecosystem is evenly split: one-third each grass, shrubs, and trees. What fraction of the ecosystem is covered in shrubs after 2 years?

11. There are three kinds of vegetation in an ecosystem: grass, shrubs, and trees. Every year, 25% of grassland plots are converted to shrubs, 20% of shrub plots are converted to trees, 8% of trees are converted to shrubs, and 1% of trees are converted to grass. The other transition probabilities between different states are zero. (a) Draw a Markov model for this system. (b) Find a transition matrix for this system. (c) Suppose that initially the ecosystem is evenly split: one-third each grass, shrubs, and trees. What fraction of the ecosystem is covered in shrubs after 2 years?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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No 11 a, b and c 

11. There are three kinds of vegetation in an ecosystem: grass, shrubs, and trees. Every year, 25% of grassland plots are converted to shrubs, 20% of shrub plots are converted to trees, 8% of trees are converted to shrubs, and 1% of trees are converted to grass. The other transition probabilities between different states are zero. (a) Draw a Markov model for this system. (b) Find a transition matrix for this system. (c) Suppose that initially the ecosystem is evenly split: one-third each grass, shrubs, and trees. What fraction of the ecosystem is covered in shrubs after 2 years?
Transcribed Image Text:11. There are three kinds of vegetation in an ecosystem: grass, shrubs, and trees. Every year, 25% of grassland plots are converted to shrubs, 20% of shrub plots are converted to trees, 8% of trees are converted to shrubs, and 1% of trees are converted to grass. The other transition probabilities between different states are zero. (a) Draw a Markov model for this system. (b) Find a transition matrix for this system. (c) Suppose that initially the ecosystem is evenly split: one-third each grass, shrubs, and trees. What fraction of the ecosystem is covered in shrubs after 2 years?
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