We transform a population dynamics problem with an additional structure into an abstract Cauchy problem, and use functional analytic method to solve it. Instead of using the familiar theory of resolvent, we rather replace the condition of well-posedness by its equivalent form in order to prove the existence of a unique weak solution. An {\it a priori} estimate of the solution is also given as well as the perturbation about the equilibrium point, and by a bounded linear operator.