By Sebastian Aniţa (auth.)

The fabric of the current booklet 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 attracted to the purposes of optimum keep an eye on and in mathematical biology. Age is among the most vital parameters within the evolution of a bi ological inhabitants. whether for a truly lengthy interval age constitution has been thought of simply in demography, these days it's primary in epidemiology and ecology too. this can be the 1st ebook dedicated to the regulate of constant age based populationdynamics.It makes a speciality of the fundamental homes ofthe ideas and at the regulate of age dependent inhabitants dynamics without or with diffusion. the most aim of this paintings is to familiarize the reader with crucial difficulties, ways and ends up in the mathematical thought of age-dependent types. distinct cognizance is given to optimum harvesting and to detailed 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 thought akin to Clarke's generalized gradient, Ekeland's variational precept, and Carleman estimates. The tools and methods we use will be utilized to different keep watch over problems.

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**Extra resources for Analysis and Control of Age-Dependent Population Dynamics**

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E. e. 1) . e. 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 .

A* - For any h > 0 we denote by z (h) th e unique solution of the equation a* - Let us take any 0 that M(hz) = O. < hI < ho < h2. Since M is strictly increasing we deduce z (hd > Zo > z (h2) , As lim h (t) = ho, we infer that there exists T E (0, +00) such that t-too hI < h (t) < ba , 'Vt E [T, +00) . We shall prove that lim dist (y (t), [z (h2) , Z (h l )]) = O. t-too Indeed, if for a certain t E [T, +00) we have Y (t) > z (hd, then y' (t) = (a* - M (h (t) Y (t))) Y (t) < (a* - M (hiz (hd)) Y (t) = 0, and so y is strictly decreasing on it, t + c:] (e If for a certain t E [T, +00) we have > 0).

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. 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 .