In probabilistic terms, the dynamics of a population with N individuals is no longer de- scribed by a deterministic function n(t), but by a probability pN(i) = Pr{N(t) = r1}. Note in the above formula, the N is an integer-valued random variable, which can take any values from 0,1,2, ---,rt,-- -, with the probability 31,16). In other words, we no longer talk about precisely the number of individuals at time t, instead we talk about the probability of having to number of individuals in the population at time i. If all the individuals in the population are statistically independent and identical (i.i.d.), with constant per capita birth rate b and death rate (1!, then pn(t) satises the Equation (6.26) (a) Explain why is the coefcient of the pn_1 term is (n 1)b, and coefcient of the p+l term is (11+ 1)d. Also explain what the two positive terms, b(n 1)p_1 and d{n+ 1) p+ 1 represent, and what about the two negative terms hop\" and dnpn? (b) The Equation (6.26) is really a system of ODEs for p06), p1(t),p2(t), - - - ,pnt), - - -. Write out the rst equation of the system: dn) = dt ..., and explain your logic. (c) Assuming that the order of summation and derivative can be exchanged, i.e., am) = s; ((111313)). n= n= show that {1 $090\") +P1(t)+'\") =0- Explain this mathematical equation. \f