On the asymptotic behavior of a size-structured model arising in population dynamics

Abstract

We study the asymptotic behavior of a semilinear size-structured population model withdelay when the nonlinearity is small in some sense. The novelty in this work is that theoperator governing the linear part of the equation does not generate a compact semigroupunlike in the results present in literature. In such a case the spectrum does not consist whollyof eigenvalues but also has a non-trivial component called Browder’s essential spectrum. Toovercome the lack of compactness, we give a localization of Browder’s essential spectrum of theoperator governing the linear part and we use the Perron-Frobenius spectral analysis adaptedto semigroups of positive operators in Banach lattices to investigate the long time behavior ofthe system.Keywords: Perron-Frobenius, positive operators, structured population models, Browder’sessential spectrum, asymptotic behavior, semigroups of operatorsAMS Subject Classifications: 35B40, 35R10, 47D06.