By Sebastian Aniţa (auth.)

The fabric of the current booklet is an extension of a graduate direction 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 without doubt one of the most vital parameters within the evolution of a bi ological inhabitants. whether for a really lengthy interval age constitution has been thought of merely in demography, these days it's primary in epidemiology and ecology too. this is often the 1st publication dedicated to the regulate of continuing age based populationdynamics.It specializes in the fundamental homes ofthe recommendations and at the keep an eye on of age based inhabitants dynamics without or with diffusion. the most objective of this paintings is to familiarize the reader with an important difficulties, ways and ends up in the mathematical concept of age-dependent types. targeted cognizance is given to optimum harvesting and to distinct controllability difficulties, that are extremely important from the econom ical or ecological issues of view. We use a few new suggestions and methods in sleek keep an eye on concept comparable to Clarke's generalized gradient, Ekeland's variational precept, and Carleman estimates. The equipment and strategies we use might be utilized to different regulate problems.

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**Sample text**

16), corresponding to Xo := 1. 15). 3. 1) a E (0, at}, where Q = (0, at) x (0, +00) . This is the model with time-independent vital ra tes and without inflow. 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 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]. 2. 1). e. e. e. e. e. e. 1) corresponding to f := l«, respectively. Proof. 7), is nonnegative. e. e. in QT . e. 10) (they both depend on (3, /-L , Po , J) . e. e. e. e. e . in (O,T). 4) we may conclude that (ii) holds. 9), with I := In ). 9) we get Fn(t) - F(t) = I t (3(a , t) l a a e- I. ' (r,t-a+r)dr Un - J)(s , t - a rat (3(a ,t) i r e- J. e. e. min{T, at} l at (3(a, t) < t < T.

If r(Ai') Lr(R) . a i'o (u)du da . 3) being less than 1, the population would go extinct if the external supply f is null. 3. e. in Q. 11) Proof. 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.