Also, how would these commands be entered in MATLab? Population dynamics An isla

Question

Also, how would these commands be entered in MATLab?

Population dynamics An island is divided into three regions, A, B, and C. The yearly migration of a certain animal among these regions is described by the following table. From C From B 15% 80% 5% From A 70% 15% 15% 10% ToB ToC 30% 6 60% For example, the first column in the table tells us, in any given year, that 70% of the population in A remains in region A, 15% migrates to B, and 15% migrates to C. The total population of animals on the island is expected to remain stable for the foreseeable future and a census finds the current population consists of 300 in region A, 350in region B, and 200 inregion C. Corresponding to the migration table and the census we define a matrix A and a vector xo: .70 .15 .10 1 300 A .15 80 .30 Xo350 200 15 .05 .60 The matrix A is called the transition matrix and the vector xo is the initial state vector. In general, let xk = [X1, X2, X3)" denote the state vector for year k. (The state vector tells us using the transition matrix, we find in year k +1 that the population distribution is given by

Explanation / Answer

Part (a)

Population Distribution one year after the census = x1 = Ax0

So, x1T =

282.5

385

182.5

ANSWER

Part (b)

We have x1 = Ax0

So, x2 = Ax1 = A.Ax0 = A2x0

x3 = Ax2 = A.A2x0 = A3x0

x4 = Ax3 = A.A3x0 = A4x0

Or, in general, xn = Anx0 ANSWER

Part (c)

Following the above formula and method, the transposes of x1, x2, ……, x10 as

x1

282.5

385

182.5

x2

273.75

405.125

171.125

x3

269.506

416.5

163.994

x4

267.529

422.824

159.647

x5

266.658

426.283

157.059

x6

266.309

428.143

155.548

x7

266.193

429.125

154.682

x8

266.172

429.634

154.195

x9

266.185

429.891

153.924

x10

266.205

430.018

153.777

ANSWER 1

From the above table, it is apparent that the population reaches a steady state of 266, 430 and 153 for 3 regions respectively. ANSWER 2

Part (d)

By following the very same method and formula,

x20T =

266.265

430.1209

153.6144

ANSWER 1

By comparing the above with the steady state values as obtained under Part (c), we see these figures are the same as the steady state values. ANSWER 2

282.5

385

182.5

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