Question

A Markov chain model for the growth and replacement of trees assumes that there are three stages of growth based on the size of the tree; young tree, mature tree, and old tree. The transition period considered is 8 years. In each period of 8 years:

(1) 30% of the young trees remain young and the rest of the young trees become mature;

(2) Among the mature trees, 40% remain mature and the rest become old;

(3) Old trees die with probability p, 0 < p < 1, independently of their past history and of all other trees. Once a tree becomes old, it is considered to remain in that state until it dies, and any trees that die are immediately replaced by young trees.

(a) Considering the transition probability matrix of the MC, find the proportion of trees, which are mature, that will be replaced by a young tree within the next 16 years.

(b) What proportion of trees do you expect to be in old stage in the long run?

(c) Following a bushfire, a new growth forest starts with all trees being young. What proportion of trees in the forest will be in mature state in 24 years’ time?

(d) Find the mean time a young tree lives.

(e) The environmental benefit of a tree is measured by an index denoted by EBI, where higher EBI indicates greater environmental benefit. For a particular forest, the EBIs for each tree are given by 1, 4 and 10 for young, mature and old stages respectively. Calculate the expected total EBI from a young tree over the next 24 years.

Answer 1

States	Y	M	O
Y	0.3	0.7	0
M	0	0.4	0.6
O	p	0	(1-p)