Nonlinear semigroups and age-dependent population models

Abstract

This paper treats the nonlinear age-dependent population problem (1)ϱ(0,a)=ϕ(a), a e I; (2)ϱ(t, 0)=F(ϱ(t, ·)), t⩾0; (3)\(\mathop {\lim }\limits_{h \to 0} (\varrho (t + h,a + h) - \varrho (t,a))/h = G(\varrho (t, \cdot ))(a),a \in I\),t⩾0,where I is the age range of the population, ϱ(t, ·) is the unknown age density at time t, ϕ is the known initial age distribution, and the functionals F and G are nonlinear. The problems of existence, uniqueness, continuous dependence upon initial values, and the positivity of solutions are investigated using the method of nonlinear semigroups.