By Sebastian Aniţa (auth.)

ISBN-10: 9048155908

ISBN-13: 9789048155903

ISBN-10: 9401594368

ISBN-13: 9789401594363

The fabric of the current publication is an extension of a graduate path given through the writer on the college "Al.I. Cuza" Iasi and is meant for stu dents and researchers attracted to the functions of optimum keep an eye on and in mathematical biology. Age is likely one of the most crucial 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's basic in epidemiology and ecology too. this can be the 1st booklet dedicated to the keep an eye on of continuing age based populationdynamics.It specializes in the elemental homes ofthe options and at the keep watch over of age dependent inhabitants dynamics without or with diffusion. the most target of this paintings is to familiarize the reader with an important difficulties, techniques and leads to the mathematical idea of age-dependent versions. distinctive realization is given to optimum harvesting and to particular controllability difficulties, that are extremely important from the econom ical or ecological issues of view. We use a few new recommendations and methods in smooth keep an eye on conception similar to Clarke's generalized gradient, Ekeland's variational precept, and Carleman estimates. The tools and strategies we use might be utilized to different keep an eye on problems.

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**Sample text**

4) elsewhere. ANALYSIS OF AGE-DEPENDENT POPULATION DYNAMICS 43 It is obvious that F E C(R+), K E LOO(R+) and as a consequence b E C(R+) . So, we have that p is a continuous function on {(a, t) E Qj a < t}. First we shall discuss the asymptotic behaviour of b. 4) we may infer via Bellman's lemma that and this implies that b is absolutely Laplace transformable. t t K (t - s) b(s) ds dt , where £(g)(,X) denotes the Laplace transform of 9 in ,X E C. tg(t)dt. Since F and K vanish for t > at, their transforms £(F) and £(K) are entire analytical functions of X E C .

In QT and PI (a, t) = =0 . e, in QT. 2. A general nonlinear model A more realistic description of the dynamics of a biological population is given by nonlinear age-dependent models. The birth and mortality rates are here nonlinear functions of the population size. ", s) and JL(',', s) belong to LOO(QT) and Lloc([O, at) x [0, TJ), respectively. e. (a, t) E QT. e. in QT x (0, +00). e. e, (a, t) E QT. e. (a, t) E QT. e. e. e. in (0, T). 1) for any T E (0,+00) . 1) will be proved via the Banach fixed point theorem.

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. Suppose now that the vital rates are time independent and that there is no inflow (f == 0). e. t(a)da = +00 . tPO E LI(O, at), (A"3) po(O) rat f3(a)po(a)da . 3. 1) belongs to C([O, at] x [0,TJ) and ap sol deri ap and at ezssist aalmost everywhere the portia eriuatives aa ere iti m OT · Sketch of proof.

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

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