Question

Some ecological system has 6 stages of capacity and transferring randomly one step up with probability ½ and down with the same probability. At stage 1 and stage 6 some resistance mechanism is rising and not permits to system to extinct or to overload from stage 6.   So from stage 1 system is transferring to stage 2 with probability 1 and from stage 6 to stage 5 with probability 1 as well. Describe types of all stages. Find limiting frequencies of visits of each stage if they exists.

Answer 1

From the given problem the transition probability matrix P is

Next Stage 0.0 0.0 0.0 0.0 0.5 0.0 0.0 0.0 0.0 0.5 0.0 1.0 0.0 0.0 0.5 0.0 0.5 0.0 1.0 0.0 0.5 0.0 0.0 0.0 0.0 0.0 0.5 0.0 0.0 0.0 0.0 0.5 0.0 Current Stage 0.5 0.0 0.0

Let $\pi_{j}$ denote the limiting frequency of visit at jth stage.

Considering the property of limiting distribution

$\pi \ast P = \pi$

0.0 1.0 0.0 0.0 0.0 0.0 0.5 0.0 0.5 0.0 0.0 0.0 0.0 0.5 0.0 0.5 0.0 0.0 πι π2 T3 π4 TT5 π6)*1 0.0 0.0 0.5 0.0 0.5 0.0|-(m π2 π3 π4 π5 πο) 0.0 0.0 0.0 0.5 0.0 0.5 0.0 0.0 0.0 0.0 1.0 0.0

The system of equations are

$0.5\ast\pi_{2} = \pi_{1}$

$\pi_{1}+0.5\ast\pi_{3} = \pi_{2}$

$0.5\ast\pi_{2}+0.5\ast\pi_{4} = \pi_{3}$

$0.5\ast\pi_{3}+0.5\ast\pi_{5} = \pi_{4}$

$0.5\ast\pi_{4}+\pi_{6} = \pi_{5}$

$0.5\ast\pi_{5} = \pi_{6}$

Also

$\pi_{1} + \pi_{2}+\pi_{3} + \pi_{4} +\pi_{5} + \pi_{6} = 1$

Solving the above set of equations we obtain limiting frequencies of visits of each stage

$\pi_{1} = \frac{1}{10} = \pi_{6}$

$\pi_{2} = \pi_{3} = \pi_{4} = \pi_{5} = \frac{1}{5}$