A small town has no residents. An individual produces n offspring with probabili

Question

A small town has no residents. An individual produces n offspring with probability Pn forn0,1,2, All children are independent of each other. The lifetime of an individual follows a distribution F. Let X(t) represent the number of individuals at time t. X(t) is an age dependent renewal process. We are interested in understanding the mean value ofx(t), ie., m (t) = E(X(t)]. (1) What is the analytical expression of m(t) for a large t value? (2) What is the condition for m(t) to go to infinity? What is the condition for m(t) to go to 0? (3) Provide a simulation study to verify your results.

Explanation / Answer

1. m(t)=c, where c is a constant depending on d1 and d2 of F which will be birth and death rate.

2. a. m(t) will go to infinity if pn>0.5 for each n and F is not a drastically decreasing function that is death rate is less.

2. b. m(t) will be 0 if pn<0.5 and F is not a drastically increasing function that is death rate is high comparitevly to birth rate.

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