Modelling of an age-structured population dynamics taking into account a discrete set of oﬀspring

. A model for description of an age-structured population dynamics taking into account a discrete set of oﬀspring, their care, and spatial diﬀusion in two-dimensional space is studied numerically. The model consists of a coupled system of integro-partial diﬀerential equations. Some results are illustrated by ﬁgures.


Introduction
The child care phenomenon is natural for many species of mammals and birds. It forms the main difference between the behaviour of populations that care for their young offspring and those without maternal (or parental) duties. Mammals and birds produce a small number of offspring then feed, warm, protect from predators and starvation, and train them to find food for themselves. If one of these natural duties is not realized, young offspring die and the population vanishes. For many species of mammals and birds both parents care for their offspring. It is also known that for some species of mammals (e.g., beer, whale, and panther) only a female provides care of her young offspring.
The one-sex Sharpe-Lotka-McKendrick-von Foerster [6,4,11,2] model and twosex Fredrickson-Hoppensteadt-Staroverov [1,3,10] permanent pairs model are known in mathematical biology. However, all these models do not treat the child care phenomenon and cannot be used to describe the evolution of populations that care for their offspring. They have to be applied only for the dynamics description of populations that do not take care of their young offspring, e.g. some species of fishes, reptile, and amphibia.
In recent years, three models for an age-structured wild populations based on the discrete set of newborns were proposed and examined analytically [7,8]. Numerical results of solving of a one-sex age-child-structured population model without and with spatial diffusion in R 1 are given in [5] taking into account a discrete set of offspring and their care.
The aim of this paper is numerical solving of this model in R 2 by using the alternating directions method.
The paper is organized as follows. In Section 2, we describe the model. Some numerical results are demonstrated Section 3.

Notation
We use the notation of paper [9]. ν ks (t, τ 1 , τ 2 , x): the natural death rate of k-s young offspring aged τ 2 at time t at the position x whose mother is aged τ 1 at the same time, s=0 ν ks , T 2 = T 1 + T : the minimal age of an individual finishing care of offspring of the first generation, T 4 = T 3 + T : the maximal age of an individual finishing care of offspring of the last generation, In what follows, κ, T , T 1 , and T 3 are assumed to be positive constants.

Model
To describe the population dynamics including the spatial diffusion in the Ω ⊂ R 2 with the extremely inhospitable ∂Ω and constant diffusion modulus κ we use the subject to the compatibility conditions Here ∆ = ∂ x1x1 +∂ x2x2 , ∂ t , ∂ τ k and ∂ xsxs denote partial derivatives, while the number n is the biologically possible maximal number of the newborns of the same generation produced by an individual. The first term on the right-hand side in Eq. (1) means the part of individuals who produce offspring, the second and third terms describe the part of individuals whose all young offspring die and who finish child care, respectively. The transition term k−1 s=0 ν ks u k on the left-hand side in Eq. (2) describes the part of individuals aged τ 1 at time t who take care of k young offspring and whose at least one young offspring dies. Similarly, the term on the right-hand side in this equation describes a part of individuals aged τ 1 at time t who take care of more than k, 1 k n − 1, young offspring aged τ 2 whose number after the death of the other offspring becomes equal to k. The condition [u| τ1=τ ] = 0, τ = T 1 , T 2 , T 3 , and T 4 , means that the function u must be continuous at the point, τ 1 = τ , of the discontinuity of the right-hand side of Eq. (1).
All calculations are performed for: values of parameters values of parameters from (8).
We conclude the paper by summarising main results. One-sex age-structured population model [8] is examined numerically taking into account an environmental pressure, a discrete set of offspring, child care, and spatial diffusion in Ω ∈ R 2 . Numerical scheme based on the method of the characteristic lines together with the iteration procedure is proposed and tested for initial functions of type (6). Numerical results show the rapid convergence of the scheme.