Question

5. In class, we considered an eco-system composed of two tree populations, red oak and hickory. Let R and H be the fraction of red oak and hickory respectively. If the fraction vector (R, H) changes acording to P. (R, H where P is given by p- (0.3 0.7 P0.7 0.3 a) What do you expect pn to be if noo? Why? b) If the initial population is (R, H)- (0.1,0.9), guess the possible equilibrium state, that is, the fraction vector (R, H) after a long time period.

Answer 1

a) The $P^n$ of the matrix value when n tends to infinity will lead to all entries being zero, since as soon as the power increases with the increase in value of n. Hence the matrix P^n will be

0.5	0.5
0.5	0.5

Reason: The rows and column are same with the sum of 1.0, hence it will lead to boost the value of lower possible values to 0.5 for all the 2x2 matrix values.

For an example: P^2 matrix will be

0.58	0.42
0.42	0.58

b) Hence the population will become equal, which is [0.5 0.5] after a long period of time, since the value of P^n with all values being equal to 0.5, hence on multiply to get the R and H population.

Note - Post any doubts/queries in comments section for any further explanation.