Gurtin-MacCamy Model Research Articles

Numerical Analysis of Linearly Implicit Methods for Discontinuous Nonlinear Gurtin-MacCamy Model.

In this study, we study a nonlinear age-structured population models with discontinues mortality and fertility rates, motivated by the fact that different maturation period may cause the significant difference in rates. We develop a novel numerical method with two-layer boundary conditions, the linearly implicit -methods on a special mesh. With a uniform boundedness analysis of numerical solutions, the finite time convergence is proved piecewisely according to the fundamental approach for the smooth rates. For juvenile-adult models, the existence of numerical endemic equilibrium is determined by a numerical basic reproduction function, which converges to the exact one with accuracy of order 1. Moreover, it is shown that for juvenile-adult models, the global stability of the disease-free equilibrium and the local stability of the endemic equilibrium are approximately exhibited by the numerical processes. Finally, some numerical experiments on the Logistic models and tadpoles-frogs models illustrate the verification and the efficiency of our results.

Numerical analysis of linearly implicit Euler-Riemann method for nonlinear Gurtin-MacCamy model

In this paper, we deal with the existence and stability of an equilibrium age distribution of the linearly implicit Euler-Riemann method for nonlinear age-structured population model with density dependence, i.e., the Gurtin-MacCamy models. It is shown that a dynamical invariance is replicated by numerical solutions for a long time. With the help of infinite-dimensional Leslie operators, the numerical processes are embedded into a nonlinear dynamical process in an infinite dimensional space, which provides a numerical basic reproduction function and numerical endemic equilibrium distributions. As an application to the Logistic model, a numerical reproduction number R0h ensures the global stability of disease-free equilibrium whenever R0h<1 and the existence of the numerical endemic equilibrium for R0h>1. Moreover, instead of the convergence of numerical solutions, it is much more interesting that the numerical solutions preserve the existence of endemic equilibrium for small stepsize, since the numerical reproduction numbers, numerical endemic equilibrium and distribution converge to the exact ones with accuracy of order 1. Finally, some numerical experiments illustrate the verification and the efficiency of our results.

Investigate future population projection of Bangladesh with the help of Malthusian model, Sharpe-lotka model and Gurtin Mac-Camy model

A nonlinear deterministic model of age-dependent population growth is formulated in terms of joint integral equations in the total population size function and the total birth rate function. Existence of a unique solution of the model equation is established using Banach fixed point theorem. Then we investigate the total population size of Malthusian model, Sharpe-Lotka linear model and Gurtin Mac-Camy model for some specific age group and compare our investigated solutions with the total population of Bangladesh by using numerical method “Newton divided difference formula”. In our present study, we also investigate future population projection of Bangladesh with the help of Malthusian model, Sharpe-Lotka model and Gurtin Mac-Camy model.

Periodic Solutions of Gurtin-MacCamy Model and Extended Gurtin-MacCamy Model

Periodic Solutions of Gurtin-MacCamy Model and Extended Gurtin-MacCamy Model

Global Hopf Bifurcation Analysis of a Nicholson’s Blowflies Equation of Neutral Type

We investigate Hopf bifurcations in a delayed Nicholson’s blowflies equation of neutral type, derived from the Gurtin–MacCamy model. A key parameter that determines the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived. Global extension of local Hopf branches is established by combining a global Hopf bifurcation theorem with a Bendixson criterion for higher dimensional ordinary differential equations. We show that a branch of slowly varying periodic solutions and a branch of fast oscillating periodic solutions coexist for all large delays.

On the Multispecies Delayed Gurtin-MacCamy Model

The paper deals with the description of multispecies model with delayed dependence on the size of population. It is based on the Gurtin and MacCamy model. The existence and uniqueness of the solution for the new problem ofnpopulations dynamics are proved, as well as the asymptotical stability of the equilibrium age distribution.

Computing the eigenvalues of Gurtin-MacCamy models with diffusion

We address numerically the question of the asymptotic stability of equilibria for a Gurtin–MacCamy model with age-dependent spatial diffusion. The problem reduces to the study of a finite number of simpler models without diffusion, which are parametrized by the eigenvalues of the Laplacian operator. Here the approach in Breda et al. (2007, Stability analysis of age-structured population equations by pseudospectral differencing methods. J. Math. Biol., 54, 701–720; 2008, Stability analysis of the Gurtin–MacCamy model. SIAM J. Numer. Anal., 46, 980–995), which is based on pseudospectral methods, is adapted to the reduced models and the error analysis is revisited in order to prove the preservation of convergence of infinite order.

Stability Analysis of the Gurtin–MacCamy Model

In this paper we propose a numerical scheme to investigate the stability of steady states of the nonlinear Gurtin-MacCamy system, which is a basic model in population dynamics. In fact the analysis of stability is usually performed by the study of transcendental characteristic equations that are too difficult to approach by analytical methods. The method is based on the discretization of the infinitesimal generator associated to the semigroup of the solution operator by using pseudospectral differencing techniques. The method computes the rightmost characteristic roots, and it is shown to converge with spectral accuracy behavior.

Limit cycles in a chemostat model for a single species with age structure

We model an age-structured population feeding on an abiotic resource by combining the Gurtin–MacCamy [Math. Biosci. 43 (1979) 199] approach with a standard chemostat model. Limit cycles arise by Hopf bifurcations at low values of the chemostat dilution rate, even for simple maternity functions for which the original Gurtin–MacCamy model has no oscillatory solutions. We find the exact location in parameter space of the Hopf bifurcations for special cases of our model. The onset of cycling is largely independent of both the form of the resource uptake function and the shape of the maternity function.

Optimality conditions for age-structured control systems

We consider a fairly general model (extension of the Gurtin–MacCamy model of population dynamics) of an age structured control system with nonlocal dynamics and nonlocal boundary conditions. A necessary optimality condition is obtained in the form of Pontryagin's maximum principle, which is applicable to a number of practically meaningful models where the previously known results fail. We discuss such models (an epidemic control, and a capital accumulation model) as illustrations.

A Nonlinear Model of Age-Dependent Population Dynamics

We consider a mathematical model of age-dependent population dynamics that is a generalization of the Gurtin–MacCamy model. We study the existence and uniqueness of solutions of an initial boundary-value problem and the existence and stability of stationary age distributions.

Global boundedness of the solutions to a Gurtin-MacCamy system

This paper is concerned with the analysis of a generalized Gurtin-MacCamy model describing the evolution of an age-structured population. The problem of global boundedness is studied. Namely we ask whether there are simple general assumptions that one can make on the vital rates in order to have boundedness of the solution. Next a fully implicit finite difference scheme along the characteristic is considered to approximate the solution of the system. Global boundedness of the numerical solutions is investigated. The optimal rate of convergence of the scheme is obtained in the maximum norm. Numerical examples are presented.

A simplified model for age-dependent population dynamics

The Gurtin-MacCamy model for age-dependent population dynamics is reduced to a single ordinary differential equation by assuming a certain form of the mortality function, which is justified by several biological examples. Some results about harvesting are obtained.

Age structured populations with history dependent mortality and natality

We consider a modification of the Gurtin-MacCamy model of population dynamics, where the birth and death moduli depend on a time integral of the solution rather than on an age integral. This models the influence of past history in the time dependence of the death and birth rates. We propose a numerical algorithm to approximate the solution based on the finite difference method of characteristics and prove that it is optimally convergent. The algorithm is devised to minimize the storage requirement in the time variable.

A model of age-dependent population dynamics providing simple criteria for growth or extinction

The Gurtin-MacCamy model for age-dependent population dynamics is reduced to a single ordinary differential equation. (This is done by assuming a specific form of the mortality function, and by assuming fertility to be age-dependent only.) The model is then applied to a population of greater horseshoe bats.

Nonlinear Age-Dependent Population Dynamics with Constant Size

The existence of populations of constant size is studied for the Gurtin–MacCamy model. Necessary and sufficient conditions are given and some examples of their implications are considered.MSC codes92A15Keywordspopulation dynamicsage-structuredconstant size population

Hopf bifurcation and the stability of non-linear age-dependent population models

The possibility of Hopf bifurcation into stable orbits is considered for the Gurtin-MacCamy model of age-dependent population dynamics, in which the mortality function depends only on the population while fertility is a fairly general function of age as well as population. In addition an algorithm is produced which provides a necessary condition for Hopf bifurcation to occur for arbitrary systems of ordinary differential equations.

Dynamics of populations with age-difference and diffusion: localization.

"In this paper we study the behavior of the solutions of the Gurtin-MacCamy model for the dynamics of populations with [spatial] diffusion and age-dependence. We give sufficient conditions on the birth and death modules for the population to remain localized in a fixed interval or to ultimately cover all the domain."

Existence of solutions in a population dynamics problem

In this paper we show the existence of a solution for the Gurtin—MacCamy model in population dynamics with age dependence and diffusion. We also discuss the behavior of this solution.