(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...

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 4 X(t)5 . Hint: Using (12.5), calculate E 4 X(t + h) | X(t) = m 5 for an infinitesimal h. Then find