Analysis and Control of Age-Dependent Population Dynamics by Sebastian Aniţa (auth.)

By Sebastian Aniţa (auth.)

The fabric of the current e-book is an extension of a graduate direction given by means of the writer on the collage "Al.I. Cuza" Iasi and is meant for stu­ dents and researchers drawn to the functions 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. no matter if 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 can be the 1st publication dedicated to the keep watch over of continuing age dependent populationdynamics.It makes a speciality of the fundamental houses ofthe strategies and at the regulate of age dependent inhabitants dynamics without or with diffusion. the most target of this paintings is to familiarize the reader with an important difficulties, ways and ends up in the mathematical thought of age-dependent versions. particular consciousness is given to optimum harvesting and to specified controllability difficulties, that are vitally important from the econom­ ical or ecological issues of view. We use a few new suggestions and methods in smooth keep watch over idea comparable to Clarke's generalized gradient, Ekeland's variational precept, and Carleman estimates. The tools and methods we use may be utilized to different keep watch over problems.

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Example text

The solution b of (3. 7) O. ~ Proof. R'x>0 £(K)(>') = lim I,X I..... 8) (see [34]) we conclude that lim 1,X1..... R>' > 0 £(F)(>' )£(K)(>') = O. :. :iY:.. :. :. :)(:.. 9) ANALYSIS OF AGE-DEPENDENT POPULATION DYNAMICS where x E R 'is such that £(K)(x + i y) implies that m(x) = inf 11 yeR - =1= 45 1, Vy E R. 8) £(K)(x + iy) 1> 0. Define the functions e-xtF (t), Ix (t) = { 0, and t ~ 0, t < 0, e-xtK(t) , t~O , kx (t) = { t < 0. 0, Since Ix and k x vanish outside of [0, at], their Fourier transforms lx, k x belong to L 2(R) and we can also verify that lx(y) = £(F)(x Thus i: I and so + iy) , kx(Y) = £(K)(x + iy).

15) for any sp E ell. e. e. e. in R . 17) . ) sign w(a, t + a)da, tER .

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 . tK(t - s)b(s)dt ds = £(F)('x) + £(b)('x)£(K)('x) , £(F)('x) = £(F)('x) and in conclusion £(b)('x) = 1 - £(K)('x) + £(F)('x)£(K)('x) .

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