A One-Parameter Class of Separable Solutions for An Age-Sex-Structured Population Model with an Infinite Range of Reproductive Ages, A Discrete Set of Offspring, and Maternal Care
Abstract
A mathematical model based on a discrete newborn set is proposed to describe the evolution of a sex-age-structured population, taking into account the temporary pair of sexes, infinite ranges of reproductive age of sexes, and maternal care of offspring. Pair formation is modeled by a weighted harmonic mean type function. The model is based on the concept of density of families composed of mothers with their newborns. All individuals are divided into the pre-reproductive and reproductive age groups. Individuals of the pre-reproductive class are divided into the newborn and teenager groups. Newborns are under maternal care while the teenagers can live without maternal care but cannot mate. Females of the reproductive age group are divided into singles and those who care for their offspring. The model is composed of a coupled system of integro-partial differential equations. Sufficient conditions for the existence of a one-parameter class of separable solutions of this model are found in the case of stationary vital rates.
Keywords
Age-sex-structured population models , Population models with parental care , Two sex population models
References
- [1] G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, New York: M. Dekker, 1985. ISBN 9780824772901, 0824772903.
- [2] A. G. Fredrickson, A mathematical theory of age structure in sexual population: random mating and monogamous marriage models, Math. Biosci., 10 (1971), 117-143.
- [3] F. C. Hoppensteadt, Mathematical theories of populations, genetics, and epidemics, In CBMS Appl. Math. SIAM, 20, Philadelphia, 1979, 1-71.
- [4] O. V. Staroverov, Reproduction of the structure of the population and marriages, Ekonomika i matematicheskije metody, (1977), 77-82 (in Russian).
- [5] K. P. Hadeler, Pair formation with maturation period, J. Math. Biol., 32 (1993), 1-15.
- [6] J. Pruss, W. Schappacher, Persitent age-distributions for a pair-formation model, Math. Biol., 33 (1994), 17–33.
- [7] R. Zacher, Persistent solutions for age-dependent pair-formation models, J. Math. Biol., 42 (2001), 507-531.
- [8] V. Skakauskas, A population dynamics model with parental care, Lith. Math. J., 42(1) (2002), 71-80.
- [9] V. Skakauskas, An age-structured population dynamics model with females pregnancy and child care, Lith. Math. J., 44(3) (2004), 251-271.
- [10] V. Skakauskas, A two sex population dynamics model with strong parental care, Nonlinear Analysis: Real World Applications, 6 (2005), 609-636.
