Questions #3 and #4 %233In 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 onlyif preceded by the grasses. In the figure below, the transfer coefficient of 0.3 indicates that, bythe end of the summer, 30% of the prior bare space in the field becomes occupied by grasses.Bare Space0.3/year0.15/year(death)0.05/year(death)0.2/year3SmallGrassesShrubs(a) Find the transfer matrix T.(b) Suppose X=| 0 | and the area is measured in acres%3Dc) 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 andmatrix decomposition to justify your answer#4Expand Ax) = 4x² – 1, –1

Name: Questions #3 and #4 %233In 2019, A field in California has been comple
Uploaded: 2024-03-05T11:53:49.000Z
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Description: Questions #3 and #4 %233In 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 onlyif preceded by the grasses. In t

“Since you have asked multiple questions, we will solve the first question for you only for the 1st three subparts i.e. (a), (b) & (c). If you want any specific question to be solved then please specify the question number or post only that question.” #3 (a) Tij specifies the rate of transfer to compartment i from j, we haveT21=0.3T23=0T12=0.05T13=0.15T31=0T32=0.2T=-0.050.150.3-000.2-Also, since the column entries should sum to 1, we haveT=0.70.050.150.30.75000.20.85is the required transfer matrix.(b) Suppose X=1100 and each area is measured in acres.This vector represents the areas of each vegetation.The bare space is of 11 acres while the grasses & the small shrubs is of 0 acres.(c) To find the ground cover of each of the next 4 years. In 1 year, Y=TX=0.70.050.150.30.75000.20.851100=7.73.30In 1 year, Bare space area is 7.7 acres, grass area is 3.3 acres and small shrubs is of 0 acres. In 2 years,Y=T2X=0.5050.10250.23250.4350.57750.0450.060.320.72251100=5.5554.7850.66In 2 years, bare space area will be 5.555 acres, grass area will be 4.785 acres and the small shrubs will be 0.66 acres. In 3 years, Y=T3X=0.384250.1486850.2733750.477750.4638750.10350.1380.38750.6231251100=4.226755.255251.518In 3 years, bare space area will be 4.22675 acres, grass area will be 5.25525 acres and the small shrubs will be 1.518 acres. In 4 years, Y=T4X=0.31356250.185356250.290006250.47358750.392493750.15963750.212850.422150.550356251100=3.44918755.20946252.34135In 4 years, bare space will be 3.4491875 acres, grass area will be 5.2094625 acres and the small shrubs will be 2.34135 acres.