An Age-Structured Population Dynamics Model with Females' Pregnancy and Child Care

We present a deterministic model for an age-structured population dynamics taking into account females' pregnancy, maternal care of offsprings, and environmental pressure with or without spatial migration. The model is based on the age-density notion for a group formed by a mother and her offsprings under maternal care. A harmonic-mean-type mating function of sexes without formation of permanent pairs is used. It is assumed that each sex has the pre-reproductive, reproductive, and post-reproductive age intervals. All adult individuals are divided into males, single females, fertilized females, and females taking child care. Individuals of post-reproductive age belong to the group of singles. All individuals of pre-reproductive age are divided into the young and juvenile groups. Only young offsprings are assumed to be under maternal care. Juvenile individuals can live without maternal care. The model consists of integro-PDEs subject to the conditions of integral type. The existence and uniqueness theorem is proved in the case of unlimited population. Separable solutions and their long-time behavior are studied for the limited nondispersing population. In the case of random migration two types of separable solutions and their long-time behavior for the homogeneous Dirichlet and Neumann boundary conditions are studied. In the case of directed migration in one-dimensional domain with special initial and Dirichlet boundary conditions, the unlimited invasive population dynamics is studied. In particular, an explicit formula for the migration front is given.     

Exact Null Controllability of a Stage and Age-Structured Population Dynamics System

This paper is concerned with the exact null controllability of an age-dependent life cycle dynamics with nonlocal transition processes arising as boundary conditions. We investigate the controllability for the pest by acting on eggs in a small age interval. The main method is based on the derivation of estimations for the adjoint variables related to an optimal control problem. A fixed point theorem is then used to draw conclusions.     

基于切比雪夫小波基与年龄相关种群模型的数值解

基于切比雪夫小波基给出与年龄相关种群模型的数值解.利用切比雪夫小波基的性质使得所求偏微分方程转化为矩阵方程,从而简化了数值解的求解过程.最后通过数值例子验证其理论结果.     

A Population Dynamics Model with Parental Care

A model for an agestructured unlimited population dynamics with parental care of offspring is presented (migration of individuals is not taken into account). The model consists of six partial integrodifferential equations for single males, single females, pairs with offspring under parental care, pairs without offspring under parental care, and offspring of the male and female sex. A class of separable solutions is constructed.     

Uniqueness of the Solution of the Inverse Problem for a Model of the Dynamics of an Age-Structured Population

Mathematical Notes - For the McKendrick model of the dynamics of an age-structured population, we consider the inverse problem of reconstructing two coefficients of the model: in the equation and...     

A Side-by-Side Single Sex Age-Structured Human Population Dynamic Model: Exact Solution and Model Validation

A side-by-side single sex age-structured population dynamic model is presented in this paper. The model consists of two coupled von Foerster-McKendrick-type quasi-linear partial differential equations, two initial conditions, and two boundary conditions. The state variables of the model are male and female population densities. The solutions of these partial differential equations provide explicit time and age dependence of the variables. The initial conditions define the male and female population densities at the initial time, while the boundary conditions compute the male and female births at zero-age by using fertility rates. The assumptions of the nontime-dependence of the death and fertility rates and a specific factorization of the migratory balances allow us to obtain exact solutions for male and female population densities. In addition, the hypotheses about the mathematical structure of the input variables are formulated, and the exact solution of the model is obtained. Next, the model is applied to the case study of Spain for the time period 1996–2004. Model validation demonstrates that this approach is a powerful prediction tool. Code and data are available upon request.     

总人口规模变化的年龄结构MSEIR流行病模型的再生数

在总人口规模变化和疾病影响死亡率的假设下,讨论了带二次感染和接种疫苗的年龄结构MSEIR流行病模型.首先给出再生数R(ψ,λ)(这里ψ(a)是接种疫苗率,λ是总人口的增长指数)的显式表达式.其次,证明了当R(ψ,λ)<1时,系统的无病平衡态是稳定的;当R(ψ,λ)>1时,无病平衡态是不稳定的.     

A Box Method for a Nonlinear Equation of Population Dynamics

We analyse a box method for a nonlinear integro-differentialequation with a nonlocal boundary condition. The problem considereddescribes the evolution in time of the age structure of a population.We study the consistency, stability, and convergence propertiesof the scheme and show that it is second-order accurate. Thisanalysis is carried out employing an abstract theory of discretizations.     

A Semigroup Approach to Nonlinear Equation of Age-dependent Population Dynamics

This paper deals with nonlinear model, of age-dependent population dynamics, with nonautonomous time dependence. Using a nonlinear semigroup approach, the authors prove the existence of the problem under general assumptions.     

Oscillation and Global Attractivity in a Nonlinear Delay Periodic Model of Population Dynamics

In this article we shall consider the following nonlinear delay differential equation $$x'(t) + p(t)x(t)-\frac {q(t)x(t)}{r + x^{n}(t-m\omega )} = 0\eqno (*)$$ where m and n are positive integers, p ( t ) and q ( t ) are positive periodic functions of period y . In the nondelay case we shall show that (*) has a unique positive periodic solution $ \overline {x}(t), $ and provide sufficient conditions for the global attractivity of $ \overline {x}(t) $ . In the delay case we shall present sufficient conditions for the oscillation of all positive solutions of (*) about $ \overline {x}(t), $ and establish sufficient conditions for the global attractivity of $ \overline {x}(t). $     

与年龄相关的模糊随机种群扩散系统的均方散逸性

讨论了一类与年龄相关的模糊随机种群扩散系统,该系统受随机和模糊两种不确定性因素的影响.在有界的条件(弱于线性增长条件)和Lipschitz条件下,利用It公式和Bellman-Gronwall-Type引理,建立了均方意义下与年龄相关的模糊随机种群扩散系统均方散逸性的判定准则.并通过数值例子对所给出的结论进行了验证.     

A Discrete-Continuous Model for a Bisexual Population Dynamics

We consider a parametric model for the dynamics of an isolated population with sex structure which is realized as a system of ordinary differential equations with impulses. The birth rate in the population in this model is assumed to be of a discrete character and the appearance of new generation specimens occurs at fixed moments, while the death rate is of a continuous character. We examine dynamical regimes of the model; in particular, we show that cyclic and chaotic regimes may occur for some values of parameters.     

A Stochastic Model for Population and Well-Being Dynamics

This article presents a stochastic dynamic model to study the demographic evolution per sexes and the corresponding well-being of a general human population. The main model variables are population per sexes and well-being. The considered well-being variable is the Gender-Related Development Index (GDI), a United Nations index. The model's objectives are to improve future well-being and to reach a stable population in a country. The application case consists of adapting, validating, and using the model for Spain in the 2000–2006 period. Some instance strategies have been tested in different scenarios for the 2006–2015 period to meet these objectives by calculating the reliability of the results. The optimal strategy is “government invests more in education and maintains the present health investment tendency.”     

基于Shifted Legendre多项式非线性年龄结构种群模型的数值解

基于Shifted Legendre多项式研究非线性年龄结构种群模型的数值解问题.定义了在区间[0,A]×[0,T]上函数的Shifted Legendre逼近多项式,通过Shifted Legendre算子矩阵结合Tau方法,把求解非线性年龄结构种群模型的数值解问题转化成非线性代数方程的求解问题.数值算例的结果显示该算法有效.     

Approximation of Linear Age-Structured Population Models Using Legendre Polynomials

We develop a numerical algorithm for approximation of solutions for linear age-structured population models. The construction is based on approximation of age distributions by modified Legendre polynomials and uses the Trotter-Kato theorem of semigroup theory for the corresponding abstract Cauchy problem. Unbounded resp. nonintegrable mortality rates are admissible.     

具脉冲效应的阶段结构单种群动力学模型

讨论了生物资源管理中的具脉冲出生与脉冲收获的单种群阶段结构动力学模型.利用离散动力系统频闪映射理论,得到了脉冲投放幼体对整个种群持续生存的重要意义.为现实的生物资源管理提供了可靠的策略依据,也丰富了脉冲微分方程理论.     

Global Aspects of Age-Structured Cigarette Smoking Model

Smoking impacts health and as a result creates several problems related to age which means smoking has a strong correlation with age. Keeping this problem in view, we consider the global asymptotic properties of age-structured smoking model. First, we formulate the model and present the existence and uniqueness of solution. Then we discuss the equilibrium points and construct the Lyapunov function to examine global stability of the free smoking and positive smoking equilibrium points. Finally, we fixed the age factor and use the non-standard finite difference (NSFD) scheme for numerical solutions and compare our results obtained with RK4 and ODE45 graphically with the help of MATLAB.     

同类相食对两阶段结构种群模型的动力学影响

研究同类相食对成熟阶段个体具有密度制约的两阶段结构种群的动力学影响,分别分析了不具有同类相食和具有同类相食时两类模型的动力学性态,得到了它们具有相似的动力学性态,即种群灭绝平衡点总存在但不稳定,而种群存在平衡点总存在且是全局渐近稳定的结点.这意味着两个阶段种群密度的最终变化趋势是单调的.同时还讨论了种群存在平衡点的大小对同类相食的依赖性,以及同类相食存在时对种群存在平衡点的大小随自食相关参数的变化.