# Analysis and Control of Age-Dependent Population Dynamics by Sebastian Aniţa (auth.)

By Sebastian Aniţa (auth.)

The fabric of the current publication is an extension of a graduate direction given through the writer on the collage "Al.I. Cuza" Iasi and is meant for stu dents and researchers drawn to the functions of optimum keep watch over and in mathematical biology. Age is likely one of the most vital parameters within the evolution of a bi ological inhabitants. no matter if for a truly lengthy interval age constitution has been thought of in simple terms in demography, these days it really is primary in epidemiology and ecology too. this can be the 1st ebook dedicated to the keep an eye on of continuing age based populationdynamics.It specializes in the fundamental houses ofthe ideas and at the keep an eye on of age established inhabitants dynamics without or with diffusion. the most aim of this paintings is to familiarize the reader with crucial difficulties, techniques and leads to the mathematical thought of age-dependent versions. specified cognizance is given to optimum harvesting and to distinct controllability difficulties, that are vitally important from the econom ical or ecological issues of view. We use a few new thoughts and strategies in sleek keep an eye on idea comparable to Clarke's generalized gradient, Ekeland's variational precept, and Carleman estimates. The tools and strategies we use should be utilized to different regulate problems.

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E. e. min{T, at} l at (3(a, t) < t < T. I(3l1 u xl(QT ) . lIn - IIILI(QT) ' which implies that Fn -+ F in LOO(O ,T) as n -+ +00. 7) we may infer via Bellman's lemma that bIn -+ b in Loo(O , T) . 7) correspo nding to F := Fn (and consequently to I := In) . 4) it follows that Pn -+ pin Loo(O,TjL1(O,at» and thus (iii) is proved. 3. e. e. 1) . These show that the biological meaning of at is the maximal age of the population. 13) is satisfied then (A4) holds. So, (A4) is the necessary and sufficient condition to have at as the maximal age of the population species.

F)()")£(K)()") 9 1 - £(K)()") (this follows by a classical result from Laplace transform theory; see [34]) . ,) = F(t) + g(t) , t E R+. 46 CHAPTER 2 So, the asymptotic behaviour of b is given by the asymptotic behaviour of g. RA < Xl . 10), from X to X l. tdAI l XI-i oo e- (o" - xJlt m (xd ° 1 - £(K)(A) IlfxIIL(R) · l k t 2(R). ~ ~ x I It is obvious that u = if and only if F (t) = 0, "It ~ 0. 7) is satisfied. 7). 1. e. aE (O,at), ° we where ,6 is extended by 0 outside [0, at]· This happens if the support of,6 lies to the left of the support of Po (po is called in this case a trivial datum) .

E, in (0, T) . e. in QT. 4 - continued. 23) cp(at> t) = 0 t E (O,T). 23) belongs to