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Pair formation models with maturation period. (English) Zbl 0808.92024
This paper deals with a model of demography, taking into account the formation of pairs, and a maturation period from the time a pair has been formed and the time when it gives birth to new individuals. The model is a set of three equations describing the evolution of single males, single females and couples. No crowding effect is considered, so that birth and death effects are counted linearly. Nonlinearity occurs from the pair formation term which is supposed to be homogeneous of degree one.
An age-dependent version of the equation is presented in section 2 of the paper which, under some simplifying assumptions on the coefficients, leads to a system of equations of the above type satisfied by the respective total single male or female populations and couples. Analysis of the directional stability of the nontrivial steady-state (the asynchronous exponential stability) is then performed in sections 3 to 5 of the paper. This extends previous work by the author on homogeneous ordinary differential equations [see notably J. Differ. Equations 95, No. 1, 183-202 (1992; Zbl 0747.34030)].
There are two major problems faced here: 1) in contrast to the O.D.E. case, normalizing the solutions (in the vector case) by dividing them by a kind of norm does not lead to the same equation. The equation obtained is not even a delay equation. But, the normalized equation allows the author to show that the original equation has a simple dominant eigenvalue. 2) As in the O.D.E. case, the linear equation to be analyzed does not yield a positive semigroup with respect to the usual order. But, the author shows that it may be turned into a positive semigroup with respect to a suitable order and concludes stability from this.
Reviewer: O.Arino (Pau)

92D25 Population dynamics (general)
34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
34K20 Stability theory of functional-differential equations
Zbl 0747.34030
Full Text: DOI
[1] Arbogast, T., Milner, F. A.: A finite difference method for a two-sex model of population dynamics. SIAM J. Num. Anal. 26, 1474-1486 (1989) · Zbl 0683.92016
[2] Busenberg, S. N., Castillo-Chavez, C.: A general solution of the problem of mixing of subpopulations and its application to risk - and age structured epidemic models for the spread of AIDS. IMA J. Math. Appl. Med. Biol. 8, 1-29 (1991) · Zbl 0764.92017
[3] Busenberg, S., Hadeler, K. P.: Demography and epidemics. Math. Biosci. 101, 63-74 (1990) · Zbl 0751.92012
[4] Busenberg, S., van den Driessche, P.: Analysis of a disease transmission model in a population with varying size. J. Math. Biol. 28, 257-270 (1990) · Zbl 0725.92021
[5] Castillo-Chavez, C. (ed.): Mathematical and Statistical Approaches to AIDS Epidemiology. (Lect. Notes Biomath., vol. 83) Berlin Heidelberg New York: Springer 1989 · Zbl 0682.00023
[6] Castillo-Chavez, C., Blythe, S. P.: Mixing framework for social/sexual behavior. In: Castillo-Chavez, C. (ed.) Mathematical and Statistical Approaches to AIDS Epidemiology (Lect. Notes Biomath., vol. 83, pp. 275-288) Berlin Heidelberg New York: Springer 1989 · Zbl 0705.92017
[7] Caswell, H., Weeks, D. E.: Two-sex models: Chaos, extinction, and other dynamic consequences of sex. Am. Nat. 128, 707-735 (1986)
[8] Dietz, K., Hadeler, K. P.: Epidemiological models for sexually transmitted diseases. J. Math. Biol. 26, 1-25 (1988) · Zbl 0643.92015
[9] Gantmacher, F. R.: The Theory of Matrices, Chap. 13. New York: Chelsea 1959 · Zbl 0085.01001
[10] Grabosch, A., Heijmans, H.: Production, development, and maturation of red blood cells. A mathematical model. In: Arino, O., Axelrod, D. E., Kimmel, M. (eds.) Mathematical Population Dynamics, pp. 189-210. New York: Dekker 1991 · Zbl 0759.92003
[11] Greiner, G.: Linearized stability for hyperbolic evolution equations with semilinear boundary conditions. Semigroup Forum 38, 203-214 (1987) · Zbl 0683.47020
[12] Hadeler, K. P.: Pair formation in age structured populations. Acta Appl. Math. 14, 91-102 (1989a) · Zbl 0667.92013
[13] Hadeler, K. P.: Modeling AIDS in Structured Populations. Bull. Int. Stat. Inst. 53, Book 1, 83-99 (1989b)
[14] Hadeler, K. P.: Periodic solutions of homogeneous equations. J. Differ. Equations 95, 183-202 (1992) · Zbl 0747.34030
[15] Hadeler, K. P., Ngoma, K.: Homogeneous models of sexually transmitted diseases. Rocky Mt. J. Math. 20, 967-986 (1990) · Zbl 0733.92020
[16] Hadeler, K. P., Waldstätter, R., Wörz, A.: Models for pair formation in bisexual populations. J. Math. Biol. 26, 635-649 (1988) · Zbl 0714.92018
[17] Hale, J.: Theory of Functional Differential Equations. Berlin Heidelberg New York: Springer 1977 · Zbl 0352.34001
[18] Hoppensteadt, F.: Mathematical Theories of Populations: Demographics, Genetics and epidemics. (Reg. Conf. Ser. Appl. Math., vol. 20) Philadelphia: SIAM 1975 · Zbl 0304.92012
[19] Jacquez, J. A., Simon, C. P., Koopman, J., Sattenspiel, L., Perry, T.: Modelling and analysing HIV transmission: The effect of contact patterns. Math. Biosci. 92, 119-199 (1988) · Zbl 0686.92016
[20] Kendall, D. G.: Stochastic processes and population growth. J. R. Stat. Soc., Ser. B 11, 230-264 (1949) · Zbl 0038.08803
[21] Keyfitz, N.: Applied Mathematical Demography, 2nd ed. Berlin Heidelberg New York: Springer 1985 · Zbl 0597.92018
[22] Parrot, M.: Positivity and a principle of linearized stability for delay-differential equations. Differ. Integral Equations 2, 170-182 (1989)
[23] Schoen, R.: Modeling Multigroup Populations, New York: Plenum Press 1988
[24] Waldstätter, R.: Pair formation in sexually transmitted diseases. In: Castillo-Chavez, C. (ed.) Mathematical and Statistical Approaches to AIDS Epidemiology. (Lect. Notes Biomath., vol. 83) Berlin Heidelberg New York: Springer 1989
[25] Waldstätter, R.: Models for Pair formation with Applications to Demography and Epidemiology. Dissertation Tübingen (1990) · Zbl 0729.92021
[26] Webb, G. F.: Theory of Nonlinear Age-dependent Population Dynamics. New York: Dekker 1985 · Zbl 0555.92014
[27] Yellin, J., Samuelson, P. A.: A dynamical system for human population. Proc. Natl. Acad. Sci. USA 71 (no. 7), 2813-2817 (1974) · Zbl 0288.92016
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