Consider a linear birth–death process where the individual birth rate is =1, the

Question

Consider a linear birth–death process where the individual birth rate is =1, the individual death rate is = 3, and there is constant immigration into the population according to a Poisson process with rate . Please explain and show work!

(a) State the rate diagram and the generator.

(b) Suppose that there are 10 individuals in the population. What is the probability that the population size increases to 11 before it decreases to 9?

(c) Suppose that = 1 and that the population just became extinct. What is the expected time until it becomes extinct again? in Greek letter alpha

Explanation / Answer

Solution

Part (a)

Let the initial population be P. Then, during the next period unit, it will increase by 1 due to birth, but decrease by 3 due to death. Thus, if there is no other factor affecting the population, the population in the next period unit will be P + 1 – 3 = P – 2. But, due to immigration, the population will increase on an average by [Note: in a Poisson process with parameter , represents the average.]

Hence, the population will increase by ( – 2) if > 2 and decrease by (2 - ) if < 2. Clearly, population remains stagnant if = 2.

Part (b)

The population size increases to 11 before it decreases to 9 (from 10) => there is an immigration of 3 so that the net increase is 1 and current population is 11. Thus, the required probability = P(a Poisson process with parameter , assumes a value 3) = e- . 3/3! = (1/6)e- 3 ANSWER

Part (c)

Since population decreases at 2 per unit time without immigration, it will not be extinct if immigration per unit time > 2. Hence, P(population becomes extinct) = P(immigration is less than or equal to 2) = e- 0/0! + e- /1! + e- 2/2! = e- {1 + + (2/2)}

= 5/2e [substituting = 1]. ANSWER

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