By Sebastian Aniţa (auth.)

The fabric of the current ebook is an extension of a graduate direction given via 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 without doubt 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 basic terms in demography, these days it really is primary in epidemiology and ecology too. this can be the 1st booklet dedicated to the regulate of constant age based populationdynamics.It specializes in the elemental houses ofthe ideas and at the regulate of age based inhabitants dynamics without or with diffusion. the most aim of this paintings is to familiarize the reader with crucial difficulties, methods and ends up in the mathematical concept of age-dependent types. unique awareness is given to optimum harvesting and to specific controllability difficulties, that are vitally important from the econom ical or ecological issues of view. We use a few new ideas and methods in sleek keep watch over concept corresponding to Clarke's generalized gradient, Ekeland's variational precept, and Carleman estimates. The equipment and methods we use will be utilized to different keep an eye on problems.

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**Sample text**

Since K(t , s ; J-L) ~ 0, then for any J-L satisfying (A2) we have . ·K (t - SI - S2 - . ,sk- l, Ski J-L)ds 1 .. dSk , for k = 1, 2, ... 11) follows. 4. Suppose that r(AIl) < 1. (Q). Moreover we have (1) if f(a , t) ~ 0 c. e. e. e. e. e. in Q, then (4) if l« -+ f in L';(Q) , then in L';(Q) , where p~, pll are the solutions of (4·1) corresponding to f := fn and i , respectively. Proof. 1. e. e. in R. e . 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).

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.

4) imply now that p E C([O, at] x [0,TJ) and the partial deri . ap an d ap . e. e. a E (0, at) . Denote by * the set of all sp with the above mentioned properties. e . e, a E (O,at). 4. 1) has a unique weak solution. Proof. 1). 16) io io for any sp E . 5. Let 9 E Loo(QT). e. e. in QT. e. e, in (0, at) . 5. 10) . e, t E (0,T), n E N* . 18) . 15) is verified. e. e, in (0, T) . e. in QT. 4 - continued. 23) cp(at> t) = 0 t E (O,T). 23) belongs to *