2. GA 6.2 Discrete Time Systems. Now that we have some basics of linear algebra

Question

2. GA 6.2 Discrete Time Systems. Now that we have some basics of linear algebra under our collective belts, we can start looking at some applica- tions of linear algebra. One such example of a (class) of application is that of states and transitions. In a system (physical, economic, chemical biological, etc.) there are often certain quantities you can measure about the system at a given time (population, velocity, height, dollar value, for example). Suppose a system has measurable quantities z1, z2,... ,zn at time t. We can put these quantities into a vector and call the vector the state of the system at time t: To However, the system rarely stays in the same state for all t. Given we know the current state Xt, how can we determine the future state of the system X (one time-unit in the future)? IF we're lucky, we know the rules for how the current state Xt transitions into a future state Xt+1. If we're even luckier, that transition takes the form of a matrix A such that t-t-1 for all t. To understand this line of thinking, we will examine a basic example of a population model.

Explanation / Answer

a) Given
Xt+1 = A Xt
A = [0 2 ]
[0.6 0.85]
Xt = [jt]
[at]

i) Xo = [0]
[2]
X1 = [0 2 ][0] = [4] ; ratio = 2
[0.6 0.85][2] [2]
X2 = [0 2 ][4] = [4] ; ratio = 1
[0.6 0.85][2] [4]
X3 = [0 2 ][4] = [8] ; ratio = 1.33
[0.6 0.85][4] [6]
X4 = [0 2 ][8] = [12] ; ratio = 1.2
[0.6 0.85][6] [10]
X5 = [0 2 ][12] = [20] ; ratio = 1.25
[0.6 0.85][10] [16]
X6 = [0 2 ][20] = [32] ; ratio = 1.230
[0.6 0.85][16] [26]
X7 = [0 2 ][32] = [52] ; ratio = 1.238
[0.6 0.85][26] [42]
X8 = [0 2 ][52] = [84]
[0.6 0.85][42] [67]
X8 = [0 2 ][84] = [134]
[0.6 0.85][67] [108]
X9 = [0 2 ][134] = [216]
[0.6 0.85][108] [173]
X10 = [0 2 ][216] = [346]
[0.6 0.85][173] [277]
X11 = [0 2 ][346] = [554]
[0.6 0.85][277] [443.05]

b) as we can see the pattern for 1st, 2nd, 3rd, 4th years, population of adults = 2,4,6,10,16.. , population of juvenile = 4,4,8,12,20...
so the population of adults at+1 = at + at-1 [ from the pattern ]
so to make at+1 = 500, we have to add till tth term
so ffrom previous question, it will take 12 years to have a population of above 500
c)
From the last part we can see that the ratio oof jt / at = constsnt
d) X = [ 0 2 ][0] = 0
[0.6 0.85][0]
  

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