Answer Happy

In a single-species population, and to first order in dt, d(t)t is the probability that an individual chosen at random from the population at time t will die in the time interval [t, t + St). There is no birth. Let på(t) denote the probability that, at time t, the population size X is k. (a) Show that for k=1,2,... dpk dt d(!)((k+1)P&#1 - kpk). (b) If N(t) denotes the expected population size E(X, t) at time t, show that N(t) = exp exp(-S* (5)ds) No. where No = N(O). (c) Let V(t) be the variance of X at time t. Derive a differential equation for V(t). (d) Suppose that V(O) = V, and that the death rate satisfies: d(t) Ooststi Idi t>ti, where d, is a positive constant. Find N(t) fort > t . Using this expression and your solution to part (e), find V(t) fort >ti