An introduction to analysis by Kirkwood J.R.

By Kirkwood J.R.

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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 .

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