Differential and Integral Equations

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

Larbi Alaoui

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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.

Article information

Differential Integral Equations, Volume 9, Number 2 (1996), 343-362.

First available in Project Euclid: 3 May 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47D03: Groups and semigroups of linear operators {For nonlinear operators, see 47H20; see also 20M20}
Secondary: 35Q80: PDEs in connection with classical thermodynamics and heat transfer 47H20: Semigroups of nonlinear operators [See also 37L05, 47J35, 54H15, 58D07] 47N20: Applications to differential and integral equations 92D25: Population dynamics (general)


Alaoui, Larbi. Generators of translation semigroups and asymptotic behavior of the Sharpe-Lotka model. Differential Integral Equations 9 (1996), no. 2, 343--362. https://projecteuclid.org/euclid.die/1367603351

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