A Mathematical Study in the Theory of Dynamic Population

Abstract

This article deals with a mathematical model of an age structured proliferating w cell population originally proposed by Lebowitz and Rubinow J. Math. Biol. 1 Ž . x 1974 , 17᎐36 . Individual cells are distinguished by age and by cell cycle length. The cell cycle length is considered as an inherited property determined at birth. Here, general boundary conditions are considered by means of a linear and bounded operator K. After establishing the theorem of traces, we show that the model is well posed in the sense of the theory of semigroup without restriction on the boundary operator K. We study the positivity and the irreducibility of the generated semigroup and we calculate its essential type. The asymptotic behavior is obtained in the uniform topology. ᮊ

On the Theory of a Growing Cell Population with Zero Minimum Cycle Length

“…As mentioned in the Introduction, the case l 1 > 0 is solved by Boulanouar (see [3,Theorem 3.2]) for arbitrary bounded transition operators K. In the present paper we deal with the delicate case…”

“…Our generation results, inspired by transport theory [10], rely on a "smallness assumption" of K in the neighborhood of l = 0, regardless of the global size of the boundary operator K. We begin by recalling a well-known generation result in the case of contractive transition operators (see [3] or [7, Chap. XI]), which is a simple consequence of the theorem of Hille and Yosida.…”

“…When l 1 > 0, any bounded transition operator K satisfies condition (2.5). In that case, λ 0 is proportional to 1/l 1 [3]. On the other hand, if l 1 = 0 then condition (2.5) is satisfied by any operator K "small in the neighborhood" of l = 0 (see Section 3).…”

“…This very special generation result is peculiar to L 1 and relies on the theorem of Batty and Robinson [2]. More recently, Boulanouar [3] showed, under the basic assumption l 1 > 0 that A K generates a c 0 -semigroup in L p 1 ≤ p < ∞ for an arbitrary boundary operator K. He also proved that the semigroup is compact (for large time) if K is a compact operator. Thus, if l 1 > 0 and if the boundary operator (1.2) reduces to the integral part (i.e., c = 0) then the asymptotic analysis t → +∞ fails within the framework of the classical theory of eventually compact semigroups (see Boulanouar [3] for more details).…”

“…More recently, Boulanouar [3] showed, under the basic assumption l 1 > 0 that A K generates a c 0 -semigroup in L p 1 ≤ p < ∞ for an arbitrary boundary operator K. He also proved that the semigroup is compact (for large time) if K is a compact operator. Thus, if l 1 > 0 and if the boundary operator (1.2) reduces to the integral part (i.e., c = 0) then the asymptotic analysis t → +∞ fails within the framework of the classical theory of eventually compact semigroups (see Boulanouar [3] for more details). To our knowledge, apart from the very special generation result given in [9] (which holds in L 1 ), there is no generation result when l 1 = 0 and (of course!)…”