By Sebastian Aniţa (auth.)

The fabric of the current booklet 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 regulate 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 really lengthy interval age constitution has been thought of basically in demography, these days it truly is basic in epidemiology and ecology too. this is often the 1st publication dedicated to the regulate of continuing age dependent populationdynamics.It makes a speciality of the fundamental homes ofthe strategies and at the keep an eye on of age based inhabitants dynamics without or with diffusion. the most target of this paintings is to familiarize the reader with crucial difficulties, techniques and ends up in the mathematical idea of age-dependent versions. unique consciousness is given to optimum harvesting and to specific controllability difficulties, that are extremely important from the econom­ ical or ecological issues of view. We use a few new techniques and methods in glossy regulate idea resembling Clarke's generalized gradient, Ekeland's variational precept, and Carleman estimates. The equipment and methods we use might be utilized to different keep an eye on problems.

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R(AJ') < 1. 1) has at most Proof. 15) for any sp E ell. e. e. e. in R . 17) . ) sign w(a, t + a)da, tER .

2) that t-+oo lim t-++oo where P (a) = 'UZo exp { -O'*a and _ p(a) ='Uzoexp { * -0' a- l lip (t) - pllu"'(o a -l J-L (s) ds } , a E (0, at). 1). In the same manner follows the conclusion for 0'* [] Va E 0, at , = O. 4. 1) (a,t) E Q, where Q = [0, at) x R. l are T-periodic with respect to the time t . Moreover, the population is subject to a T-periodic external flow f. The density of the population is supposed to be independent with respect to the spatial position (this is a model without diffusion).

3) we have that for almost any 0 r: + io < t < at , e- i:' . e, t E (0,T) (we have used the Lebesgue theorem and (AnS) ). 13) ). 14) we get the conclusion. L o(a, t) + M(P). Lo are the natural fertility and mortality rates, respectively; M(P) is an additional mortality rate due to the limited resources of the habitat. 15) p(a,O) = po(a) rat p(a, t)da t E (O,T). e. e. e. 15) belongs to the class of "separable" models considered in [27] . 1). e. t E (0, T). e. in (0, T). 16) x(O) = Xo E (0, +00).

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