A One-Sex Population Dynamics Model with Discrete Set of Offsprings and Child Care

Abstract

We present a one-sex age-structured population dynamics deterministic model with a discrete set of offsprings, child care, environmental pressure, and spatial migration. All individuals have pre-reproductive, reproductive, and post-reproductive age intervals. Individuals of reproductive age are divided into fertile single and taking child care groups. All individuals of pre-reproductive age are divided into young (under maternal care) and juvenile (offspring who can live without maternal care) classes. It is assumed that all young offsprings move together with their mother and that after the death of mother all her young offsprings are killed. The model consists of integro-partial differential equations subject to the conditions of the integral type. Number of these equations depends on a biologically possible maximal newborns number of the same generation produced by an individual. The existence and uniqueness theorem is proved, separable solutions are studied, and the long time behavior is examined for the solution with general type of initial distributions in the case of non-dispersing population. Separable and more general (nonseparable) solutions, their large time behavior, and steady-state solutions are studied for the population with spatial dispersal, too.

Modelling of a One-Sex Age-Structured Population Dynamics with Child Care

“…In this section, we give the nondispersing age-structured population model [1,8] with stationary vital rates. This model involves the environmental pressure, ρ(N ), by the death rates of juvenile an adult individuals depending on the sum, N, of their spatial densities.…”

“…Many species of wild animals care for their offsprings. This phenomenon is natural for many species of mammals and birds and forms the main difference between the behavior of the population taking child care and that without maternal (or parental) duties [1,8]. But child care for every species is different.…”

“…The Sharpe-Lotka-McKendrick-von Foerster one-sex population model and Fredrickson-Hoppensteadt-Staroverov two-sex population models (see, e.g., [1,8] and references there) are well known in mathematical biology. However, all these models do not treat the child care phenomenon and cannot be used to describe the evolution of the population taking care of its offsprings.…”

“…To solve this problem, some models of a sex-age-structured population, based on the discrete set of newborns, were proposed. In [1,7,8] two-sex with temporal pairs and one-sex population models, based on a discrete set of offsprings, are proposed. In [9], we discussed numerical results of model [2] for two-sex population with temporal pairs and child care.…”

“…In [9], we discussed numerical results of model [2] for two-sex population with temporal pairs and child care. The present paper is devoted to the numerical investigation of the models studied in [1,8].…”