Second-order characteristic schemes in time and age for a nonlinear age-structured population model

In this paper, we analyze two new second-order characteristic schemes in time and age for an age-structured population model with nonlinear diffusion and reaction. By using the characteristic difference to approximate the transport term and the average along the characteristics to treat the nonlinear spatial diffusion and reaction terms, an implicit second-order characteristic scheme is proposed. To compute the nonlinear approximation system, an explicit second-order characteristic scheme in time and age is further proposed by using the extrapolation technique. The global existence and uniqueness of the solution of the nonlinear approximation scheme are established by using the theory of variation methods, Schauder’s fixed point theorem, and the technique of prior estimates. The optimal error estimates of second order in time and age are strictly proved for both the implicit and the explicit characteristic schemes. Numerical examples are given to illustrate the performance of the methods.     

Numerical solution of the nonlinear age-structured population models by using the operational matrices of Bernstein polynomials

In this paper a numerical method for solving the nonlinear age-structured population models is presented which is based on Bernstein polynomials approximation. Operational matrices of integration, differentiation, dual and product are introduced and are utilized to reduce the age-structured population problem to the solution of algebraic equations. The method in general is easy to implement, and yields good results. Illustrative examples are included to demonstrate the validity and applicability of the new technique.     

An improved model of age-structured population dynamics

The classical model of age-dependent population dynamics is improved. Instead of the traditional renewal equation, a new approach is developed to describe the reproduction process of the population. The composition of a population is redefined to contain the pre-birth individuals, and the disadvantages of the classical model avoided. Moreover, the improved model turns out to be an initial value problem, which is mathematically more convenient to deal with. Existence and uniqueness results for the nonlinear nonautonomous system of model equations are obtained. It is shown that the classical model and its time delay generalization are two degenerate cases of the improved model.     

Global analysis of an age-structured SEIR model with immigration of population and nonlinear incidence rate

Epidemic models with infection age of infectious individuals have been extensively studied, however, most of the existing works ignore the combined effects of immigration and nonlinear incidence. In this paper, we incorporate both the effects of immigration and nonlinear incidence, based on which we formulate an SEIR epidemic model. We give a rigorous mathematical analysis on some necessary technical materials. Then, by constructing a Lyapunov functional, we show that the endemic equilibrium is globally asymptotically stable. Numerical simulations of an application are given to support our theoretical results.     

Asymptotic behavior of an age-structured population model with diffusion

A reaction-diffusion system with stage-structure is studied. We provide well-posedness of the model and prove that time-dependent solutions evolve either towards a positive equilibrium or to the trivial one. Under suitable conditions, a branch of positive equilibrium is shown to exist.     

On age-structured SIS epidemic model for time dependent population

1.IntroductionandPreliminaryAgestructureinepidemicmodelshasbeenconsideredbymanyauthors,becauseoftherecoghtionthattransmissiondynamicsofcertaindiseasescouldnotbecorrectlydescribedbythetraditionalepidemicmodelswithnoagedependence.Especially,Busenbergetal.II'2]giveacompleteanalysisofafairlygeneralSISmodelwithagestructureandasteady-statetotalpopulation,showingtheekistenceofathresholdforendemicstates.In[tis],theyassumethatthepopulationhasreacheditssteadystate.Althoughitisnotilladeqilatetoassumet…     

Mathematical analysis of autonomous and non-autonomous age-structured reaction-diffusion-advection population model

In this paper, we study an age-structured reaction-diffusion-advection population model. First, we use a non-densely defined operator to the linear age-structured reaction-diffusion-advection population model in a patchy environment. By spectral analysis, we obtain the asynchronous exponential growth of the population model. Then we consider nonlinear death rate and birth rate, which all depend on the function related to the generalized total population, and we prove the existence of a steady state of the system. Finally, we study the age-structured reaction-diffusion-advection population model in non-autonomous situations. We give the comparison principle and prove the eventual compactness of semiflow by using integrated semigroup. We also prove the existence of compact attractors under the periodic situation.     

A numerical method for nonlinear age-structured population models with finite maximum age

We propose a new numerical method for the approximation of solutions to a non-autonomous form of the classical Gurtin-MacCamy population model with a mortality rate that is the sum of an intrinsic age-dependent rate that becomes unbounded as the age approaches its maximum value, plus a non-local, non-autonomous, bounded rate that depends on some weighted population size. We prove that our new quadrature based method converges to second-order and we show the results of several numerical simulations.     

A model of an age-structured population in a multipatch environment

We propose a model of an age-structured population divided into N geographical patches. We distinguish two time scales, at the fast time scale we have the migration dynamics and at the slow time scale the demographic dynamics. The demographic process is described using the classical McKendrick-von Foerster model for each patch, and a simple matrix model including the transfer rates between patches depicts the migration process.Assuming that 0 is a simple strictly dominant eigenvalue for the migration matrix, we transform the model (an e.d.p. problem with N state variables) into a classical McKendrick-von Foerster model (scalar e.d.p. problem) for the global variable: total population density. We prove, under certain assumptions, that the semigroup associated to our problem has the property of positive asynchronous exponential growth and so we compare its asymptotic behaviour to that of the transformed scalar model. This type of study can be included in the so-called aggregation methods, where a large scale dynamical system is approximately described by a reduced system. Aggregation methods have been already developed for systems of ordinary differential equations and for discrete time models.An application of the results to the study of the dynamics of the Sole larvae is also provided.     

A model for an age-structured population with two time scales

In the modelisation of the dynamics of a sole population, an interesting issue is the influence of daily vertical migrations of the larvae on the whole dynamical process. As a first step towards getting some insight on that issue, we propose a model that describes the dynamics of an age-structured population living in an environment divided into N different spatial patches. We distinguish two time scales: at the fast time scale, we have migration dynamics and at the slow time scale, the demographic dynamics. The demographic process is described using the classical McKendrick model for each patch, and a simple matrix model including the transfer rates between patches depicts the migration process. Assuming that the migration process is conservative with respect to the total population and some additional technical assumptions, we proved in a previous work that the semigroup associated to our problem has the property of positive asynchronous exponential growth and that the characteristic elements of that asymptotic behaviour can be approximated by those of a scalar classical McKendrick model. In the present work, we develop the study of the nature of the convergence of the solutions of our problem to the solutions of the associated scalar one when the ratio between the time scales is ε (0 < ε ⪡ 1). The main result decomposes the action of the semigroup associated to our problem into three parts:

• 1.(1) the semigroup associated to a demographic scalar problem times the vector of the equilibrium distribution of the migration process;
• 2.(2) the semigroup associated to the transitory process which leads to the first part; and
• 3.(3) an operator, bounded in norm, of order ε.

     

The comparison between Homotopy Analysis Method and Optimal Homotopy Asymptotic Method for nonlinear age-structured population models

This paper presents comparison between Homotopy Analysis Method (HAM) and Optimal Homotopy Asymptotic Method (OHAM) for the solution of nonlinear age-structured population models. Three examples have been presented to illustrate and compare these methods. In OHAM the convergence region can be easily adjusted and controlled. Comparison between our solution and the exact solution shows that the both methods are effective and accurate in solving nonlinear age-structured population models with HAM being the more accurate for the same number of terms. It was also found that OHAM require more CPU time.     

An exact solution for a cavity model

Résumé On considère l'écoulement plan, permanent et irrotationnel d'un jet gazeux aux grandes vitesses subsoniques. En appliquant le procédé deFalkovich, concernant la méthode hodographique deChaplygin, on obtient la solution exacte pour le modèle de Roshko situé dans un canal aux parois parallèles (Figure). On détermine l'expression exacte du coefficient de résistance et on donne quelques relations entre les différents paramètres de la configuration.     

On simple age-structured population models

This work addresses several aspects and extensions of the deterministic Leslie model, as a matrix-driven demographic evolution of an age-structured population. We first point out its duality with another matrix model, related to backward/forward in time ways of counting individuals. Then, in some special cases, we design explicitly both the eigenvalues and the offspring vector of the Leslie matrix in a consistent way. Finally, we show how embedding the dynamics in a space of larger dimension allows one to get various new results about the population. This includes access to the total lifetime asymptotic distribution and while including sterile and/or immortal individuals in the classical Leslie model, some insight into the trade-off between the different population species.     

Painlevé expansion and exact solution for nonlinear evolution equations

It is shown that the truncated Painlevé expansion provides a systematic procedure for obtaining exact as well as special solutions for nonlinear evolution equations. Several examples of nonintegrable equations of both infinite and finite dimensions are illustrated.Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chepauk, Madras-600 005, Tamilnadu, India. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 99, No. 3, pp. 528–536, June, 1994.