In 2019, A field in California has been completely devastated by fire. Two types of vegetation, grasses and small shrubs, will first begin to grow, but the small shrubs can take over an area only if preceded by the grasses. In the figure below, the transfer coefficient of 0.3 indicates that, by the end of the summer, 30% of the prior bare space in the field becomes occupied by grasses. Bare Space 0.3/year 0.15/year (death) 0.05/year (death) 0.2/year 3 Small Grasses Shrubs (a) Find the transfer matrix T. (11) (b) Suppose X= 0 and the area is measured in acres 0. c) Determine the ground cover in each of the next 4years. (2023?) d) what happens after a long time? Does the system reach any equilibrium? Use eigenvalues and matrix decomposition to justify your answer

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A: See the detailed solution below.

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Permutations and Combinations

If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.

Counting Theory

The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.