Generators of translation semigroups and asymptotic behavior of the Sharpe-Lotka model

Abstract

Our first aim in this paper is to give a characterization of generators for the class of translation semigroups which are associated with a symbol $\phi$ and can brought into the form $u(t) = \phi (u_t).$ As applications, we consider two linear models from biology, a cell-cycle model based on unequal division and the Sharpe-Lotka age-dependent model. Our second aim is to show the relation of this last model to this class of semigroups in order to give a result on its asymptotic behavior. We show that the semigroup associated with the Sharpe-Lotka model is equivalent to a translation semigroup and both semigroups are essentially compact. It is also shown that the generators of the two semigroups have the same spectrum. We give properties of this spectrum and of its spectral bound.