Perennial Plants flower some years but not others. Suppose that a plant that is

Question

Perennial Plants flower some years but not others. Suppose that a plant that is flowering one year will also flower in the following year with a probability of 80%. A plant that is not flowering in one year will flower the followig year with a probability of 60%. Suppose that the total # of plants is kept constant over the years. Denote tby xn the number of plants on a certain meadow that flower in year n and by yn the number of plants that do not flower.

a) Write a transition matrix A so that the iteration:

xn +1 =   

yn +1

Explanation / Answer

consider the situation in 'n' th year

let xn be the no.of plants that flower and yn,the no. that does not.

out of xn plants,80% will flower the next year and 20% wont.so,80% of xn contribute to xn+1.similarly 60% of yn contribute to xn+1.

=> xn+1 = 0.8*xn + 0.6*yn

similarly yn+1 can be written as

yn+1 = 0.2*xn + 0.4*yn

in matrix form

[xn+1;yn+1] = [0.8 0.6;0.2 0.4]*[xn;yn]

b)

given x0 = 200;y0 = 100;

=> x1 = 0.8*200 + 0.6*100 = 220

y1 = 0.2*200 + 0.4*100 = 80

x2 = 0.8*220 + 0.6*80 = 224

y2 = 0.2*220 + 0.4*80 = 76

x3 = 0.8*224 + 0.6*76 = 224.8 = 225(aaprox)

y3 = 75.2 = 75(aaprox)

it can also be obtained by from matrix equation [x3;y3] = A^3*[x0;y0]

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