15. (Birth and Death with Disaster) Consider a population of a certain colonizing species. Suppose that each individual produces offspring at a Poisson rate of λ as long as it lives. Furthermore, suppose that the natural lifetime of an individual in the population is exponential with mean 1/μ and, regardless of the population size, individuals die at a Poisson rate of γ because of disasters occurring independently of natural deaths and births. Let X(t) be the population size at time t. If X(0) = n (n > 0), find E

X(t)

.

Hint: Using (12.5), calculate E

X(t + h) | X(t) = m

for an infinitesimal h. Then find

Letting h → 0 in

show that E
X(t)
satisfies a first-order linear differential equation. Solve that equation.
16. Let
X(t) : t ≥ 0
be a birth and death process with birth rates

and death rates

Show that if

then, with probability 1, eventually extinction will occur.

Transcribed Image Text:

E[X(t + h)] = E [E[X(t+h) | X(t)]].