3. We go back to the logistic model for population dynamics (without harvesting), but we now allow the growth rate and carrying capacity to vary in time: dP dt = k(t)P 1 ( t ) In this case the equation is not autonomous, so we can't use phase line analysis. We will instead find explicit analytical solutions. (a) Show that the substitution z = 1/P transforms the equation into the linear equation dz k(t) + k (t)= = M(t) (b) Using your result in (a), show that if k is constant but M varies, the general solution is pht PO)= C+ Mod (c) Similarly, show that if M is constant but k varies, the general solution is M P(t) = 1+CMe-Jk(t)dt (d) Consider the special case where M is constant but & decreases in time as k = e t. Suppose that the initial population is less than M. What happens to the population in the long run? Does it make sense