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

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.


Introduction
Many species of animals care of their offsprings.This phenomenon is native for many species of mammals and birds and forms the main difference between the behavior of the population taking care of offsprings and that without maternal (or parental) duties.But child care for every species is different.Offsprings of mammals and birds spend some time with their mother or both parents, while young offsprings of fishes, reptilia, and amphibia are left to one's fate.Mammals and birds feed, warm, and defend their young offsprings from enemies.If one of these native duties is not realized, young offsprings die and the population vanishes.For many species of mammals [1], e.g.bear (Thalarctos maritimus and Ursus arctos horribilis), whale (Balaenoptera musculus), and panther (Pannthera onca), only a female takes care of her young offsprings.For some species of mammals and birds, e.g.red fox (Vulpes vulpes), gnawer (Dolichotis patagonium), penquin (Pygoscelis adeliae), heron (Ardea purpurea), falcon (Falco ciolumbarius), and tawny owl (Strix aluco), both parents take care of their young offsprings.
The Sharpe-Lotka-McKendrick-von Foerster (see, e.g., [2]) and Fredrickson-Hoppensteadt-Staroverov [3][4][5] models are well known in mathematical biology.In the case when information about sex ratio is not important the Sharpe-Lotka-McKendrick-von Foerster one sex model (or its Gurtin-MacCamy generalization [6]) is usually used to describe dynamics of age-structured population.The other one (or its Hadeler [7] modification involving a maturation period) describes the evolution of populations forming permanent pairs.All these models do not include a female gestation period.
Models involving a gestation period were first proposed and analyzed in papers [8][9][10][11].However, all these models do not treat the child care phenomenon.Therefore, all models mentioned above have to be applied for the population which does non care its young offsprings, e.g.some species of fishes, reptilia, and amphibia.In papers [12][13][14][15][16] we proposed and examined four population dynamics models with child care: two for one-sex and the other two for two-sex population.The main requirement in these papers is that all offsprings under maternal (or parental) care are killed if their mother (or any of their parents) dies.These models are based on the notion of the density of young (under maternal or parental care) offsprings which has to be a C 1 -function at least on the characteristic lines of the equation for this density.However, the differentiability assumption of this density is questionable for many species of mammals and birds.There exists the other essential requirement in the case of the population with the spatial diffusion.In this case, all young offsprings have to move together with their mother (or pair of parents).To describe the diffusion of young offsprings and their mothers the Ficke law for fluxes of young offsprings and their mothers with the same diffusion coefficient is used in models [14] and [15].In the case of the homogeneous Neumann problem, each of these fluxes have to be zeroth on the boundary of the living area.But such the model does not ensure that young offsprings and their mothers move together.If we assume that diffusion flux of young offsprings is proportional to that of their mothers, then in the case of homogeneous Neumann problem at the same time these both fluxes will be zeroth on the boundary of their living region.But the gradient of the young offsprings density on this boundary may not be equal to zero and we have a loss or gain of youngs through the boundary.This shows that this model is biologically incorrect, too.Therefore, there arises the problem of the construction of a biologically correct model in the case of a population with the spatial diffusion.
This problem can be solved by using a notion of the complex (family) which consists of mother (or both parents) and a discrete set of her (their) young offsprings.In [17], we proposed a model for two-sex population taking into account temporal pairs, a discrete set of offsprings, and child care and examined its separable solutions.In [9], a model of two-sex population is studied taking into account permanent pairs, child care, and a discrete set of offsprings.
In the present paper we present and examine a one-sex age-structured population dynamics deterministic model with child care and a discrete set of offsprings of the same generation produced by an individual.A preprint version of this paper has been used in [18] (see literature cited there) for numerical solving of the model discussed in the present paper.This model could be used to describe the evolution of the population for which only one mother takes care (see above) of her young offsprings.We consider the population dynamics both with and without spatial diffusion ant take into account an environmental influence (pressure) which depends on population overcrowding.All individuals have pre-reproductive, reproductive, and post-reproductive age intervals.All individuals of reproductive age are divided into fertile single (without offsprings under maternal care at the given time) and individuals taking child care groups.Individuals of pre-reproductive age are divided into young and juvenile (offsprings who can live without maternal care) classes.We assume that the ecological pressure does not influence the dynamics of the young offsprings directly, that youngs move together with their mother, and that after the death of mother all her young offsprings are killed.The model consists of a system of integro-partial differential equations subject to conditions of the integral type.The number of these equations depends on a biologically possible maximal number of newborns of the same generation produced by an individual.
The paper is organized as follows.In Section 3, we present and examine the model for a non-dispersing population.In Section 3.1, separable solutions are studied for the general type of stationary vital rates.In Section 3.2, the existence and uniqueness theorem is proved for the unlimited population.Section 3.3 is devoted to the analysis of the long time behavior of the solution to the model without spatial diffusion and with general type of the initial distributions.In Section 4, we consider the model with spatial dispersal.Separable and more general solutions and their long time behavior are studied in Sections 4.1 and 4.2, respectively.The structure of steady-state solutions is examined in Section 4.3.Remarks in Section 5 conclude the paper.

Notation
The following notation is used for the analysis of the population dynamics.
R m : the Euclidean space of dimension m with x = (x 1 , . . ., x m ), κ: the diffusion modulus, (0, T ) and (T 1 , T 3 ) (T < T 1 < T 3 ): the child care and reproductive age intervals, respectively, u(t, τ 1 , x): the age-space-density of individuals aged τ 1 at time t at the position x who are of juvenile (τ 1 ∈ (T, T 1 )), fertile single (τ 1 ∈ (T 1 , T 3 )), or post-reproductive (τ 1 > T 3 ) age, u k (t, τ 1 , τ 2 , x): the age-space-density of individuals aged τ 1 at time t at the position x who take care of their k offsprings aged τ 2 at the same time, ν(t, τ 1 , x): the natural death rate of individuals aged τ 1 at time t at the position x who are of juvenile or adult age, ν k (t, τ 1 , τ 2 , x): the natural death rate of individuals aged τ 1 at time t at the position x who take care of their k offsprings aged τ 2 , ν ks (t, τ 1 , τ 2 , x): the natural death rate of k − s young offsprings aged τ 2 at time t at the position x whose mother is aged τ 1 at the same time, α k (t, τ 1 , x) dt: the probability to produce k offsprings in the time interval [t, t + dt] at the location x for an individual aged τ 1 , N : sum of spatial densities of juvenile and adult individuals, ρ(N ): the death rate conditioned by ecological causes (overcrowding of the population), ρ(0) = 0, u 0 (τ 1 , x), u k0 (τ 1 , τ 2 , x): the initial age distributions, [u| τ1=τ ]: the jump discontinuity of u at the point ν ks , T 2 = T 1 + T : the minimal age of an individual finishing care of offsprings of the first generation, T 4 = T 3 + T : the maximal age of an individual finishing care of offsprings of the last generation, In what follows κ, T, T 1 , and T 3 are assumed to be positive constants.In the case of non-dispersing populations all functions u, u k , ν, ν k , ν ks , α k , u 0 , and u k0 do not depend on the spatial position x.

The non-dispersing population dynamics model
In this section, we present a deterministic model for a non-dispersing age-structured population with discrete set of offsprings of the same generation produced by an individual and prove the existence and uniqueness theorem.In the case of stationary vital rates, we examine separable solutions and find the long time behavior of the solution to this model with initial distributions of the general type.We take into account the environmental pressure by letting the death rates of juvenile and adult individuals depend on the sum of their spatial densities, N, and assume that young offsprings are subject to natural mortality and are protected from density related increases of mortality dependent on N directly.Note that in more general case the environmental pressure depends on N, x, t, and age of the individuals.At age τ 1 = T all young offsprings go to the juvenile group and at age τ 1 = T 1 all juveniles become adult individuals.Let n be the biologically possible maximal number of newborns of the same generation produced by an individual.Using the balance law, we derive the density-dependent population dynamics model which consists of the equations subject to the conditions Here ∂ t and ∂ τ k signify partial derivatives.The first term on the right-hand side in equation ( 1) means the part of individuals who produces offsprings, the second and third terms describe the part of individuals whose all young offsprings die and who finish child care, respectively.The transition term k−1 s=0 ν ks u k on the lefth-hand side in equation ( 2) describes the part of individuals aged τ 1 at time t who take child care of k young offsprings 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 offsprings aged τ 2 whose number after the death of the other offsprings is equal to k.The condition [u| τ1=τ ] = 0, τ = T 1 , T 2 , T 3 , T 4 means that function u must be continuous at the point, τ 1 = τ, discontinuity of the righthand side of equation (1).
As follows from the foregoing, the given functions ν, ν k , ν ks , α k , u 0 , and u k0 and the unknown ones u and u k are to be positively valued, otherwise they have no biological significance.The positive constants T and T s are to be given, too.The assumption T < T 1 given in Section 2 is natural.
In order that conditions (4) would be consistent we formulate the following compatibility conditions: Inserting into ( 1)-( 4), we split this system into the problem for U and U k , subject to the conditions and the equations for f and N, Function f means the ratio of the total limited (under ecological pressure) population N and the total unlimited population β.

Separable solutions to problem (7)-(9)
In this section we restrict ourselves by the case where ν, ν k , ν ks , and α k do not depend on t and are positive supported functions.Moreover, we assume that ν is continuous, while α k , ν k , and ν ks are C 1 -functions.We seek solutions of the form where Ũ > 0 is an arbitrary constant while the constant λ and positive functions v λ and v λ k are to be determined.Note that separable solutions to the Gurtin-MacCamy model and their application to genetics were first studied in [19] and [20], respectively, (see also [21] and [2]).Inserting (13) into equations ( 7)-( 9) gives the equations for v λ and v λ k , with the conditions with the condition and the characteristic equation for λ, Here and in what follows the prime indicates differentiation.Equations ( 15) can be solved in the recurrent way starting with k = n and have a unique positive C 1 -solution.
Equations ( 14) can be solved explicitly for ), they can be reduced into Volterra type integral equations with the delay T and, therefore, have a unique positive solution.Obviously, this solution is a C 1 -function except the points τ 1 = T 1 , T 2 , T 3 , and T 4 .From ( 14) and ( 15) it is easy to see that where v 0 and v 0 k satisfies equations ( 14) and ( 15) for λ = 0. From equations ( 16) and ( 17) we get the characteristic equation for λ, The distribution of roots of this equation is well known.It has a unique real root λ 0 and a discrete set of complex conjugate roots.The real part of complex roots is less than λ 0 .As a result we formulate Theorem 1.Let ν, ν k , ν ks , and α k be positive functions and Then problem (7)-( 9) has a oneparameter class of separable solutions of type (13) with the properties From the biological point of view death rates increase with age increasing and need not stay bounded.

3.2
The existence and uniqueness theorem to system (7)- (9) In this section, we consider the case T 3 − T 1 > 2T (the opposite case can be examined similarly) and prove the existence and uniqueness theorem to system (7)-( 9) with vital rates independent of t.We assume that conditions of Theorem 1 are satisfied and u 0 and u k0 are positive C 1 -functions.Integrating of equation ( 8), for t < τ 2 , yields 19) can be solved in the recurrent way starting with k = n − 1.
If t > τ 2 , we have with v 0 k defined in Section 1.It remains to determine Integrating equation (7), we get and for τ 1 > T 4 .We write two last terms of the right hand side of equations (7) and sets [0, γ1(τ1) Then, by integrating, reduce equation (7) with conditions (9) 3,4 into the integral equations obtaining: in with with with By changing variables equations ( 23)-( 29) can be written in a form of Volterra type integral equations with kernels independent of t and therefore have a unique positive C 0 -solutions.Once equations ( 23)-( 29) are solved, formulas (30)-( 33), (21), and (22) determine function U which is continuous in {(t, τ 1 ) : It is evident that function U is not differentiable at the points τ 1 = T 1 , . . ., T 4 and it is a C 1 -function along the characteristic lines except the points τ 1 = T 1 , . . ., T 4 .
If u 0 is a positive C 1 -function, then under the conditions of Theorem 1 on differentiation equations ( 23)-( 29) with respect to t, we derive integral equations of Volterra type for ∂ t U written on the characteristic lines.These equations show that Similarly, by using (30)-( 33) and (22) we prove that U is not differentiable at the lines τ 1 = T 4 and t = τ 1 − T s , τ 1 ∈ (T 3 , T 4 ), s = 1, . . ., 4.
If t > τ 1 − T, we have with v 0 determined in Section 1.By definition (see ( 9) 1 ) we get the equation for U (t, T ), which has a retarded structure with the delay T 1 and has a unique positive solution.It is easy to see that U (t, T ) ∈ C 0 ([0, ∞))∩C 1 ((0, T )∪(T, ∞)).Thus, we have the following result: Theorem 2. Let u 0 and u k0 be positive, Then under the conditions of Theorem 1 problem (7)-( 9) has a unique positive solution with the properties The proof of the solvability of equation ( 10) is evident.
Note that the similar result can be obtained for the case As a result we formulate Theorem 3. Assume that ρ ∈ C 1 ([0, ∞)), ρ(0) = 0 and ρ ′ > 0. Then under the conditions of Theorem 2 equations (10) and ( 12) have a unique positive global solution such that f and N ∈ C 1 ((0, ∞)).

3.3
The long time behavior of the solution to system (7)- (11) In this section, we find the asymptotic behavior of the solution to system ( 7)- (11).We first find an upper bound for U (t, T ).It follows from equation ( 20) and (32) that then and and, by induction, Therefore, there exists the Laplace transform U (λ, T ) of U (t, T ), with and I(λ) defined by equation (18).Hence, Roots of I(λ) are discussed in Section 1. Function I 1 (λ) is analytic and, using the method of a rectangle contour integral [22], we evaluate the inverse Laplace transform obtaining where µ < λ 0 is the real part of the first pair of conjugate complex roots.Then by equations ( 20) and (32), for large time (t > τ 1 − T ), we get the following formulas: with U (t, T ) defined by equation (36).The asymptotic behavior of β defined by equation ( 11) will now be studied.We assume that conditions of Theorem 2 for u 0 hold, while ν satisfies conditions of Theorem 1 and does not decrease as τ 1 → ∞.Function β can be written in the form By equation (36) we get J 1 = J 11 + J 12 with Now we get estimates of J s for large time.Set ν ∞ = lim ν as τ 1 → ∞.We first consider the case ν ∞ < ∞.From equations ( 14) and (15) where c 0 is a positive constant.Fix sufficiently large t 1 > T 3 and let t > t 1 .Then for all t > t 1 and tends to 0 as t 1 → ∞.Similarly, tends to 0 as t → ∞ since, for negative µ, it can be written as and tends to 0 if −µt 1 /(λ 0 − µ) < t → ∞, while, for µ ≤ 0, this assertion is evident; Here c 1 , . . ., c 4 are some positive constants.From equations ( 21 Since these estimates for J 11 , J 12 , J 2 , and J 3 are valid for every small ǫ > 0, we conclude that, for large time and ν ∞ < ∞, It is evident that (J 12 + J 2 + J 3 ) exp{−tλ 0 } for ν ∞ = ∞ is less than that given above for ν ∞ < ∞.Therefore formula (38) remains valid for the case ν ∞ = ∞, too.
It remains to find the asymptotic behavior of N defined by equation (12).Put f (t) = F (t) exp{−tλ 0 } in equation (10) to get The asymptotic behavior of F and N can be described by the unique solutions of the equations This enables us to formulate Theorem 4. Let conditions of Theorem 3 be satisfied, ν is non-decreasing, and β(λ 0 ) < ∞ where λ 0 is a unique real root of equation (18).Then the solution of problem ( 6)- (11) for large time behaves as follows: where asymptotic behavior of Ñ is given by (40).

A population dynamics model with spatial diffusion
In this section we generalize the model in Section 3 by including the random spatial diffusion in an open bounded domain Ω ⊂ R m with the extremely inhospitable boundary ∂Ω and examine two special and steady state solutions in the case of constant diffusion modulus κ and time-space-independent vital rates.The model reads as follows: if λ 0 > κΛ 1 and t → ∞ where v 0 and v 0 k are defined in Section 3.1, Λ 1 is the first eigenvalue of the Dirichlet problem to the operator −∆ in Ω, N * is a unique positive in Ω solution of the problem and β(λ 0 ) is defined by equation (39).