P (t) t. dp (t) dt (rate of increase) P (t) = P (0)e λt (Malthusian population) (Maltusian parameter) (1) ) (Leonardo Pisano/Fibonacci)

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1 12 I 1. t P (t) β µ dp (t) = λp (t) (1) dt λ = β µ (rate of increase) P (t) = P ()e λt (Malthusian population) λ (Maltusian parameter) (1) P (t + 1) = (1 + r)p (t) (2) r t P (t) = (1 + r) t P () (T. R. Malthus) ( 1798) 17 [21] ) (Leonardo Pisano/Fibonacci) (Liber Abaci, 122) (L. Euler) (J. P. Süssmilch) [5] [28] 2. (A. Quetelet) (P. F. Verhulst) [31] (logistic model) dp (t) dt = λ ( 1 P (t) ) P (t) (3) K P (t) t K > K (carrying capacity) 1

2 192 (R. Pearl) (L. J. Reed)[22] 197 (Allee effect) 3. 2 (L. V. Bortkiewicz) (A. J. Lotka) [27] B(t) = G(t) + t β(a)l(a)b(t a)da (4) B(t) l(a) a β(a) a G(t) t = (stable population model) [6] [11] [16] [23] (A. G. McKendrick) [19] p(t, a) t + p(t, a) a = µ(t, a)p(t, a) (5) p(t, a) t a µ(t, a) t a (force of mortality) (5) (McKendrick equation) 195 (H. Von Foerster) t β(t, a) p(t, ) = β(t, a)p(t, a)da (6) (5)-(6) p(t, ) = B(t) (4) 198 [32] 197 (A. Rogers) [24] (R. Schoen) [26] : multistate population) 197 (Song Jian) 2

3 [1] [29] 4. (H. Bernardelli), (E. G. Lewis), (P. H. Leslie) [18] t =, 1, 2,... j p j (t), j =, 1, 2,... p(t) := (p (t),..., p ω (t)) T ω (Leslie matrix) A m 1 m m ω s A :=. s s ω 1 m j t t + 1 j t + 1 s j > j 1 j (Leslie matrix model) p(t + 1) = Ap(t) (8) p() t p(t) = A t p() m j λ λ λ < λ lim t λ t At p() = < v, p() > < v, u > u, (9) v, u A λ <, > u λ 1 u (stable age distribution) v (reproductive value) (generalized Leslie matrix) (9) [2] [3] [12] (V. Volterra) [25] P 1 (t), P 2 (t) t (Lotka-Volterra competition equation) { P 1 (t) = (ɛ 1 λ 1 P 1 (t) µ 12 P 2 (t))p 1 (t) P 2(t) (1) = (ɛ 2 λ 2 P 2 (t) µ 21 P 1 (t))p 2 (t) ɛ j λ j µ ij j i λ 1 = λ 2 =, ɛ 1 >, ɛ 2 <, µ 12 >, µ 21 < P 1 P 2 (7) 3

4 (prey-predator model) { P 1 (t) = (a bp 2 (t))p 1 (t) P 2(t) = ( c + dp 1 (t))p 2 (t) (11) a, b, c, d (P 1, P 2 ) = (c/d, a/b) [3] (Gurtin and MacCamy) [9] ( t + a) p(t, a) + µ(a, P (t))p(t, a) = p(t, ) = β(σ, P (t))p(t, σ)dσ (12) p(, a) = p (a), P (t) = p(t, σ)dσ P (t) β µ P (t) P (G. F. Webb) [32] (structured population dynamics) [2] (Eastalin model) B(t) = β(a, B(t a))l(a)b(t a)da. (13) β(a, B(t a)) t a B(t a) β(a, x) x (Easterlin hypothesis) [4] [7] (D. G. Kendall)[14] P m (t) P f (t) P c (t) P m(t) = (β m + σ)p c (t) µ m P m (t) Ψ(P m, P f ) P f (t) = (β f + σ)p c (t) µ f P f (t) Ψ(P m, P f ) (14) P c(t) = (µ m + µ f + σ) + Ψ(P m, P f ) 4

5 µ m, µ f β m, β f σ Ψ(x, y) (marriage function) [15] [a] (u, v) Ψ(u, v), [b] Ψ(u, ) = Ψ(, v) =, [c] (u, v) (u, v ) Ψ(u, v) Ψ(u, v ), [d] k > Ψ(ku, kv) = kψ(u, v), [d] [1] (monogamous marriage) (A. G. Fredrickson) [8] [13] [1] S. Anita (2), Analysis and Control of Age-Dependent Population Dynamics, Kluwer, Dordrecht. [2] H. Caswell (21), Matrix Population Models, 2nd Edition, Sinauer, Sunderland. [3] J. M. Cushing (1998), An Introduction to Structured Population Dynamics, SIAM, Philadelphia. [4] C. Y. Cyrus Chu (1998), Population Dynamics: A New Economic Approach, Oxford UP, Oxford. [5] L. Euler (176), Recherches générales sur la mortalité et la multiplication du genre humaine, Histoire de l Academie Royale des Sciences et Belles Lettres 16, [A general investigation into the mortality and multiplication of the human species, Translated by N. and B. Keyfitz, Theoretical Population Biology 1: ] [6] W. Feller (1941), On the integral equation of renewal theory, Ann. Math. Stat. 12: [7] J. C. Frauenthal and K. E. Swick (1983), Limit cycle oscillations of the human population, Demography 2(3): [8] A. G. Fredrickson (1971), A mathematical theory of age structure in sexual populations: Random mating and monogamous marriage models, Math. Biosci. 1: [9] M. E. Gurtin and R. C. MacCamy (1974), Non-linear age-dependent population dynamics, Archive for Rational Mechanics and Analysis 54: [1] K. P. Hadeler, R. Waldstätter and A. Wörz-Busekros (1988), Models for pair formation in bisexual populations, J. Math. Biol. 26, [11] M. Iannelli (1995), Mathematical Theory of Age-Structured Population Dynamics, Giardini Editori e Stampatori in Pisa. 5

6 [12] (1986),, 179: [13] H. Inaba (2), Persistent age distributions for an age-structured two-sex population model, Math. Pop. Studies 7(4): [14] D. G. Kendall (1949), Stochastic processes and population growth, J. Roy. Stat. Soc. B 11: [15] N. Keyfitz (1972), The mathematics of sex and marriage, In 6th Berkeley Symp. Math. Statist. Prob., Biology-Health Section, Part II, Berkeley, CA: [16] N. Keyfitz (1977), Introduction to the Mathematics of Populations with revision, Addison-Wesley: Reading. [17] N. Keyfitz (1985), Applied Mathematical Demography, Second Edition, Springer-Verlag: Berlin. [18] P. H. Leslie (1945), On the use of matrices in certain population mathematics, Biometrika 33: [19] A. G. McKendrick (1926), Application of mathematics to medical problems, Proc. Edinburgh. Math. Soc. 44: [2] J. A. J. Metz and O. Diekmann (1986), The Dynamics of Physiologically Structured Populations, Lecture Notes in Biomathematics 68, Springer-Verlag: Berlin. [21] (1944),,,. [22] R. Pearl and L. J. Reed (192), The rate of growth of the population of the United States since 179 and its mathematical representation, Proc. Nat. Acad. Sci. 6: [23] J. H. Pollard (1973), Mathematical Models for the Growth of Human Populations, Cambridge University Press: Cambridge. [24] A. Rogers (1975), Introduction to Multiregional Mathematical Demography, John Wiley: New York. [25] F. M. Scudo and J. R. Ziegler (eds.) (1978), The Golden Age of Theoretical Ecology: , Lecture Note in Biomathematics 22, Springer, Berlin. [26] R. Schoen (1988), Modeling Multigroup Populations, Plenum Press, New York and London. [27] F. R. Sharpe and A. J. Lotka (1911), A problem in age-distribution, Philosophical Magazine, Series 6, Vol. 21: [28] D. Smith and N. Keyfitz (1977), Mathematical Demography: Selected Papers, Springer-Verlag, Berlin. [29] J. Song and J. Yu (1988), Population System Control, Springer-Verlag, Berlin. [3] (1997),,, [31] P. F. Verhulst (1838), Notice sur la loi que la population suit dans son accroissement. Correspondance Mathématique et Physique Publiée par A. Quételet 1: [32] G. F. Webb (1985), Theory of Nonlinear Age-Dependent Population Dynamics, Marcel Dekker: New York and Basel. 6