Read e-book online Analysis and Control of Age-Dependent Population Dynamics PDF

By Sebastian Aniţa (auth.)

ISBN-10: 9048155908

ISBN-13: 9789048155903

ISBN-10: 9401594368

ISBN-13: 9789401594363

The fabric of the current ebook is an extension of a graduate path given by way of the writer on the college "Al.I. Cuza" Iasi and is meant for stu­ dents and researchers drawn to the functions of optimum regulate and in mathematical biology. Age is likely one of 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 simply in demography, these days it really is basic in epidemiology and ecology too. this can be the 1st ebook dedicated to the regulate of constant age based populationdynamics.It specializes in the fundamental homes ofthe recommendations and at the keep watch over of age dependent inhabitants dynamics without or with diffusion. the most objective of this paintings is to familiarize the reader with crucial difficulties, techniques and leads to the mathematical thought of age-dependent versions. specific recognition 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 techniques and strategies in sleek regulate thought similar to Clarke's generalized gradient, Ekeland's variational precept, and Carleman estimates. The tools and methods we use may be utilized to different regulate problems.

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T E (0, T) . 2. e. t E (0, T), for any PI , P2 E H. rat Proof. e. t E (0, T). e, t E (0, T). 4) on (0, T) (which is nonnegative) . e, t E (0, T) . 12) we get that p(. , Q) E H and so we conclude that T maps H into itself. e. t E (0, T), = 0 , which obviously belong to H. 4) we get that for almost any t E (0, at) i II(Tpd(·, t) - (Tp2)( ', t)II£l (O,at ) ~ + + it it it i = IIp(·, t ;Pd - p(' , t j P2) II£l (O ,at) Ib(t - a; Pd - b(t - a; P2)IIl(a , t , a; Pdda Ib(t - a;P2)IIl(a,t,a ;Pd - Il(a,t ,a;P2)lda a at f(a - s , t - s)IIl (a, t , s; Pd - Il(a, t , s;P2) lds da + [ t po(a at r t)IIl(a, t , t ;Pd - Il(a, t , t ;P2) lda + [ t Jo f(a - s, t - s)IIl(a, t, s; Pd - Il(a, t, s;P2)lds da.

2. 1). Using now this remark and the analogy with the asymptotic behaviour of the solution to the mal thusian model , we can call 0'* the intrinsic malthusian parameter. 3. If we denote by (which is called the net reproduction rate), then the following relationship between the net reproduction rate and the intrinsic malthusian parameter holds: if 0'* > 0, R> 1, if 0'* = 0, R= 1, { R < 1, if 0'* < 0. 11) 48 CHAPTER 2 where P (t) rat P (a, t) da, t E (0, +00) and =1 0 (A4) M : [0, +00) -+ [0, +00) is a continuously differentiable and strictly increasing function , M (0) = 0 and lim M (r) = +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 t K (t - s) b(s) ds dt , where £(g)(,X) denotes the Laplace transform of 9 in ,X E C.

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

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