A numerical method for a nonlinear structured population model with an indefinite growth rate coupled with the environment

A numerical method is developed for a general structured population model coupled with the environment dynamics over a bounded domain where the individual growth rate changes sign. Sign changes notably exhibit nonlocal dependence on the population density and environmental factors (e.g., resource availability and other habitat variables). This leads to a highly nonlinear PDE describing the time‐evolution of the population density coupled with a nonlinear‐nonlocal system of ODEs describing the environmental time‐dynamics. Stability of the finite‐difference numerical scheme and its convergence to the unique weak solution are proved. Numerical experiments are provided to demonstrate the performance of the finite difference scheme and to illustrate a range of biologically relevant potential applications.     

A Second-Order High-Resolution Scheme for a Juvenile-Adult Model of Amphibians

In this article we consider a juvenile-adult population model of amphibians in which juveniles are structured by age and adults are structured by size. We develop a second-order explicit high-resolution scheme to approximate the solution of the model. Convergence of the finite difference approximation to the unique weak solution with bounded total variation is proved. Numerical examples demonstrate the high-resolution property and the achievement of the designed accuracy for the scheme. The scheme is then applied to understand the dynamics of an urban amphibian population.     

A finite difference approach to the equations of age-dependent population dynamics

We establish the convergence of the finite difference scheme for the nonlinear equations of population dynamics proposed by Guertin and MacCamy. The applicability of the discrete equations to establish qualitative properties of the solution to the continuous problem is also illustrated.     

A finite element method for a two-sex model of population dynamics

A finite element scheme is described to approximate the solution of a nonlinear and non-local system of integro-differential equations that models the dynamics of a two-sex population. Crank-Nicolson time discretization is used and error estimates are derived for the appoximation.     

Numerical Analysis for a Mean-Field Equation for the Ising Model with Glauber Dynamics

1.IntroductionWeconsiderthefollowingmean--fieldequationofmotionforthedynamicIsingmodelonaperiodiclatticeA:whereAdenotesthelatticeofZdwithNdsitesdefinedbyA:~{a:a=Zaie',i=1alEZ,15al5N}with{e'}beingthestandardunitvectorsofZd.WesaythatAisad-dimensionallattice.WedenotebyVAtheNddimensionalspaceoflatticevectorsv=(v.).6A*satisfyingv.+Nei=va'Hereu~(u.)..AandbadenotestheexpectationdrofthespinatsiteaofthelatticeandA*isdefinedby{a:a~Za.e',alEZ}.i=1TheNdxNdsymmetricmatrixAisdefinedby3forvEVAF'o…     

A finite difference scheme for nonlinear ultra-parabolic equations

In this paper, our aim is to study a numerical method for an ultraparabolic equation with nonlinear source function. Mathematically, the bibliography on initial–boundary value problems for ultraparabolic equations is not extensive although the problems have many applications related to option pricing, multi-parameter Brownian motion, population dynamics and so forth. In this work, we present the approximate solution by virtue of finite difference scheme and Fourier series. For the nonlinear case, we use an iterative scheme by linear approximation to get the approximate solution and obtain error estimates. A numerical example is given to justify the theoretical analysis.     

Global dynamics of a discretized SIRS epidemic model with time delay

We derive a discretized SIRS epidemic model with time delay by applying a nonstandard finite difference scheme. Sufficient conditions for the global dynamics of the solution are obtained by improvements in discretization and applying proofs for continuous epidemic models. These conditions for our discretized model are the same as for the original continuous model.     

A predator-prey model with Beddington-DeAngelis functional response: a non-standard finite-difference method

In this paper, we transform a continuous-time predator-prey model with Beddington–DeAngelis functional response into a discrete-time model by nonstandard finite difference scheme (NSFD). The NSFD model shows complete dynamic consistency with its continuous counterpart for any step size. However, the discrete model of same continuous system obtained by Euler forward method shows dynamic inconsistency for larger step size. Extensive numerical simulations have been done to compare the dynamics of NSFD system and Euler system. Our analysis reveals that dynamics of NSFD model is independent of the step-size, whereas the dynamics of the standard discrete model completely depends on the step-size and produces spurious dynamics like chaos.     

Stability preserving non-standard finite difference scheme for a harvesting Leslie–Gower predator–prey model

A non-standard finite difference scheme for a harvesting Leslie–Gower equations is constructed. It is shown that the obtained difference system has the same dynamics as the original continuous system, such as positivity of solutions, equilibria and their local stability properties, irrespective of the size of numerical time step. To illustrate the analytical results, we present some numerical simulations.     

Dynamic complexities in a single-species discrete population model with stage structure and birth pulses

Natural population, whose population numbers are small and generations are non-overlapping, can be modelled by difference equations that describe how the population evolve in discrete time-steps. This paper investigates a recent study on the dynamics complexities in a single-species discrete population model with stage structure and birth pulses. Using the stroboscopic map, we obtain an exact cycle of system, and obtain the threshold conditions for its stability. Above this, there is a characteristic sequence of bifurcations, leading to chaotic dynamics, which implies that this the dynamical behaviors of the single-species discrete model with birth pulses are very complex, including (a) non-unique dynamics, meaning that several attractors and chaos coexist; (b) small-amplitude annual oscillations; (c) large-amplitude multi-annual cycles; (d) chaos. Some interesting results are obtained and they showed that pulsing provides a natural period or cyclicity that allows for a period-doubling route to chaos.     

Bifurcation analysis of a non-standard finite difference scheme for a time-delayed model of asset prices

In this paper, we apply a non-standard finite difference scheme to a time-delayed model of speculative asset markets and discuss the effect of time delay on the dynamics of asset prices. Firstly, the stability of the positive equilibrium of the system is investigated by analysing the characteristic equation. By choosing the time delay as a bifurcation parameter, we prove that Hopf bifurcations occur when the delay passes a sequence of critical values. Then, the explicit algorithm for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are derived. Finally, some numerical simulations are given to verify the theoretical analysis.     

High order positivity-preserving finite volume WENO schemes for a hierarchical size-structured population model

In this paper we develop high order positivity-preserving finite volume weighted essentially non-oscillatory (WENO) schemes for solving a hierarchical size-structured population model with nonlinear growth, mortality and reproduction rates. We carefully treat the technical complications in boundary conditions and global integration terms to ensure high order accuracy and the positivity-preserving property. Comparing with the previous high order difference WENO scheme for this model, the positivity-preserving finite volume WENO scheme has a comparable computational cost and accuracy, with the added advantages of being positivity-preserving and having L¹ stability. Numerical examples, including that of the evolution of the population of Gambusia affinis, are presented to illustrate the good performance of the scheme.     

Global stability and a non-standard finite difference scheme for a diffusion driven HBV model with capsids

A diffusion driven model for hepatitis B virus (HBV) infection, taking into account the spatial mobility of both the HBV and the HBV DNA-containing capsids is presented. The global stability for the continuous model is discussed in terms of the basic reproduction number. The analysis is further carried out on a discretized version of the model. Since the standard finite difference (SFD) approximation could potentially lead to numerical instability, it has to be restricted or eliminated through dynamic consistency. The latter is accomplished by using a non-standard finite difference (NSFD) scheme and the global stability properties of the discretized model are studied. The results are numerically illustrated for the dynamics and stability of the various populations in addition to demonstrating the advantages of the usage of NSFD method over the SFD scheme.     

一种多介质流体计算的三维非守恒型差分格式

Motivated by the need for three-dimensional methods for interface calculations we describe a 3D non-conservative difference scheme in Lagrange coordinates for calculating multi-component fluid dynamics, according to the thoughts of 2D non-conservative difference scheme and making use of difference method of characteristic and Upwind form. Furthermore, we give a three-dimensional numerical simulation of explosion in concrete, and the results of simulation are consistent with examination.     

Almost Periodicity for a Nonautonomous Discrete Dispersive Population Model

In this article, study the almost periodic profile of solutions for nonautonomous difference equations in Banach spaces. We apply our results in population dynamics.     

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 STRONG ERGODIC THEOREM FOR SOME NONLINEAR MATRIX MODELS FOR THE DYNAMICS OF STRUCTURED POPULATIONS

A general class of matrix difference equation models for the dynamics of discrete class structured populations in discrete time which possess a certain general type of nonlinearity introduced by Leslie for age-structured populations is considered. Arbitrary structuring is allowed in that transitions between any two classes are permitted. It is shown that normalized class distributions for such nonlinear models globally approach a “stable class distribution” and thus possess a strong ergodic property exactly like that of the classical linear theory of demography. However, unlike in the linear theory according to which the total population size grows or dies exponentially, the dynamics of total population size in these nonlinear models are shown to be governed by a nonlinear, nonautonomous scalar difference equation. This difference equation is asymptotically autonomous, and theorems which relate the dynamics of total population size to those of this limiting equation are proved. Examples in which the results are applied to some nonlinear age-structure models found in the literature are given.     

Dynamics and family of equilibria in a population kinetics model with cosymmetry

We study dynamics in the population kinetics model which is given by the system of nonlinear parabolic equations with cosymmetry property. The cosymmetry implies the emergence of continuous families of steady states with variable spectrum of stability. Different scenarios of evolution of families of equilibria and nonstationary regimes are analyzed numerically by a finite-difference scheme which respects the cosymmetry property. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Basic Reproduction Number in a Growing Population

The basic reproduction number of a fast disease epidemic on a slowly growing network may increase to a maximum then decrease to its equi- librium value while the population increases, which is not displayed by classical homogeneous mixing disease models. In this paper, we show that, by properly keeping track of the dynamics of the per capita contact rate in the population due to population dynamics, classical homogeneous mixing models show simi- lar non-monotonic dynamics in the basic reproduction number. This suggests that modeling the dynamics of the contact rate in classical disease models with population dynamics may be important to study disease dynamics in growing populations.     

Numerical Solution of the Inverse Nonequilibrium Sorption Problem with a Time-Dependent Boundary Condition

We consider the quasi-linear problem of nonequilibrium sorption dynamics with external-diffusion kinetics and a boundary condition that contains the time derivative of a solution component. A numerical method is proposed for describing the inverse problem to recover the nonlinear parameter of the system of differential equations—the inverse of the sorption isotherm. Convergence of the difference scheme for the direct problem is proved. Numerical solutions of both the direct and the inverse problem are obtained for various parameter values.