Answer Happy

2. Throughout its lifetime, itself a random variable having distribution function F(x), an organism produces offspring according to a nonhomogenous Poisson process with intensity function (u). Independently, each offspring follows the same probabilistic pattern, and thus a population evolves. Assuming 1<j au – Flu F(u)(u) du <0, show that the mean population size m(t) asymptotically grows exponentially at rate r > 0, where r uniquely solves 1= L = – Flu)(u) du. e 1 Hint: Develop a renewal equation for B(t), the mean number of individuals born up to time t, and from this infer that B(1) grows exponentially at rate r. Then express m(t) in terms of B(u) for u St.