By Sebastian Aniţa (auth.)

The fabric of the current publication is an extension of a graduate direction given by means of the writer on the college "Al.I. Cuza" Iasi and is meant for stu­ dents and researchers drawn to the purposes of optimum keep an eye on and in mathematical biology. Age is among 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 primary in epidemiology and ecology too. this can be the 1st ebook dedicated to the regulate of continuing age dependent populationdynamics.It specializes in the fundamental homes ofthe recommendations 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 an important difficulties, methods and ends up in the mathematical concept of age-dependent versions. specific consciousness is given to optimum harvesting and to distinctive controllability difficulties, that are extremely important from the econom­ ical or ecological issues of view. We use a few new techniques and methods in sleek regulate concept comparable to Clarke's generalized gradient, Ekeland's variational precept, and Carleman estimates. The equipment and methods we use could be utilized to different keep watch over problems.

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1 we conclude that this solution is nonnegative. 1. e . e. e. t E (0, T) . 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] .

Denote by y(t) = it t E [a, b] . e. e. in (a, b). Multiplying the last inequality by exp . e. t E (a, b) . By int egration we conclude th at y(t) s it 'IjJ(s)cp(s) [ex p (it 'IjJ(T)dT)] ds, for any t E [a, b] and consequently we get the conclusion of the lemma. 24 CHAPTER 2 If in addition, cp(t) = M for each t E [a , b], from the inequality above we deduce the following LEMMA (Bellman). e. t E (a, b), MER and it x(t) sM+ x(t) ~ M exp for each t E [a, b], then 'ljJ(s)x(s)ds (it 'ljJ(S)dS) for each t E [a, b].

E. t E R. e. in Q. e. e. e. in Q. e. e . e. t E R. 4) . 8) for f := fn and f, respectively). 4). L Cp E L1((O,at) x (O ,T)) . e. 2), integrating by parts. 14) ANALYSIS OF AGE-DEPENDENT POPULATION DYNAMICS Denote by ell the set of all test functions properties. 13) for any ep E ell. 5. Suppose that one weak solution. 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 .

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