Nonlinear Age-structured Population Model Research Articles

Coding Optimization for an Ecological non-linear Age-structured Fish Population Model with a Shepherd Recruitment Function

The study of the optimally density-regulated recruitment plays a crucial role in the overall management and determining of fishing efforts of marine fish populations. This paper will develop a Matlab code can determine what an optimal recruitment density regulation strategy would be for the un-fished population and what it changes to when fishing efforts are applied. This code can determine threshold values used to avoid fisheries from collapsing. The code will be applied to the Gulf of California Pacific sardine (Sardinops caeruleus) population. An age structured design with a Shepherd density regulating stock-recruitment function is used in this study. In this work, the parameter which is varied is captured in the Shepherd recruitment function and is associated with behavioral interventions such as selecting different drift routes and clumping on resources that will increase egg-larvae survival to the recruitment stage. A detailed background study of other models is explored, then a derivation of the Shepherd recruitment model is developed from the traditional discrete model. A special MATLAB code is then developed to illustrate the pair-wise invasive plots for the model structures. The code is for theoretical analysis of fish population models.

Modeling and analysis of a multilayer solid tumour with cell physiological age and resource limitations

We study an avascular spherical solid tumour model with cell physiological age and resource constraints in vivo. We divide the tumour cells into three components: proliferating cells, quiescent cells and dead cells in necrotic core. We assume that the division rate of proliferating cells is nonlinear due to the nutritional and spatial constraints. The proportion of newborn tumour cells entering directly into quiescent state is considered, since this proportion can respond to the therapeutic effect of drug. We establish a nonlinear age-structured tumour cell population model. We investigate the existence and uniqueness of the model solution and explore the local and global stabilities of the tumour-free steady state. The existence and local stability of the tumour steady state are studied. Finally, some numerical simulations are performed to verify the theoretical results and to investigate the effects of different parameters on the model.

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.

Monitoring in a Discrete-Time Nonlinear Age-Structured Population Model with Changing Environment

This paper is a contribution to the modeling–methodological development of the application of mathematical systems theory in population biology. A discrete-time nonlinear Leslie-type model is considered, where both the reproduction and survival rates decrease as the total population size increases. In this context, the monitoring problem means that, from the observation of the size of certain age classes as a function of time, we want to recover (estimate) the whole state process (i.e., the time-dependent size of the rest of the classes). First, for the linearization approach, conditions for the existence and asymptotic stability of a positive equilibrium are obtained, then the discrete-time observer design method is applied to estimate an unknown state trajectory near the equilibrium, where we could observe a single age class. It is also shown how the observer design can be used to detect an unknown change in the environment that affects the population dynamics. The environmental change is supposed to be generated by additional dynamics (exosystem). Now, the Leslie-type model is extended with this exosystem, and the observer design is applied to this extended system. In this way, an estimation can be obtained for different (constant or periodic) environmental changes as well.

Nonlinear age-structured population models with nonlocal diffusion and nonlocal boundary conditions

In this paper, we develop some basic theory for age-structured population models with nonlocal diffusion and nonlocal boundary conditions. We first apply the theory of integrated semigroups and non-densely defined operators to a linear equation, study the spectrum, and analyze the asymptotic behavior via asynchronous exponential growth. Then we consider a semilinear equation with nonlocal diffusion and nonlocal boundary condition, use the method of characteristic lines to find the resolvent of the infinitesimal generator and the variation of constant formula, apply Krasnoselskii's fixed point theorem to obtain the existence of nontrivial steady states, and establish the stability of steady states. Finally we generalize these results to a nonlinear equation with nonlocal diffusion and nonlocal boundary condition.

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.

Discontinuous Galerkin method for a nonlinear age-structured tumor cell population model with proliferating and quiescent phases

We propose a nonlinear age-structured model of tumor cell population with proliferating and quiescent phases. We apply the discontinuous Galerkin (DG) method to study its dynamical behavior. The DG numerical approximation is used for the spatial discretization and then the strong-stability-preserving explicit Runge–Kutta (SSPERK) method is performed for the temporal discretization. This paper aims to establish more efficient results in the sense of computational approach and compare these with analogous estimates for Weighted Essentially and Non-Oscillatory (WENO) scheme. Finally, some test examples and numerical simulations are given to illustrate theoretical results and to examine the behavior of the solution.

Steady States Analysis of a Nonlinear Age-Structured Tumor Cell Population Model with Quiescence and Bidirectional Transition

A nonlinear age-structured tumor cell population model with quiescence and bidirectional transition is presented and studied. The division rate of the proliferating cells is assumed to be nonlinear due to the limitation of the nutrient and space. The model includes a proportion of newborn cells that enter directly the quiescent phase with age zero. This proportion can reflect the effect of treatment by drugs such as erlotinib. The bidirectional transition between proliferating cells and quiescent cells is considered. The local and global stabilities of the trivial steady state are investigated. The existence and local stability of the positive steady state are also analyzed. Numerical simulations are performed to verify the results and to examine the impacts of parameters on the nonlinear dynamics of the model.

Analysis of a nonlinear age-structured tumor cell population model

A nonlinear age-structured tumor cell population model is presented and analyzed. The population is divided into proliferating and quiescent cell compartments. The nonlinearity of the birth rate is designed to halt population’s exponential growth and force the system to converge to stable equilibria or stable cycles. The existence and uniqueness of solutions are studied. The local and global stabilities of the trivial steady state are investigated. Moreover, the existence and local stability of the positive steady state are analyzed. Numerical examples are performed to verify the validity of the results and to discuss the impacts of parameters on the nonlinear dynamics of the model.

Density–Dependent Feedback in Age–Structured Populations

The population size has far-reaching effects on the fitness of the population, that, in its turn influences the population extinction or persistence. Understanding the density- and age-dependent factors will facilitate more accurate predictions about the population dynamics and its asymptotic behaviour. In this paper, we develop a rigourous mathematical analysis to study positive and negative effects of increased population density in the classical nonlinear age-structured population model introduced by Gurtin \& MacCamy in the late 1970s. One of our main results expresses the global stability of the system in terms of the newborn function only. We also derive the existence of a threshold population size implying the population extinction, which is well-known in population dynamics as an Allee effect.

An improvement of Laguerre computational scheme for solving nonlinear age-structured population models

In this paper, we simultaneously implement two kinds of orthogonal polynomials for solving a nonlinear age-structured population model. This non-classic type of partial differential equation is typically defined in large domains that makes finding an accurate solution by common techniques to be difficult. The presented method namely modified generalized Laguerre-Chebyshev (MGLC), which is based on the modified generalized Laguerre functions and Chebyshev orthogonal polynomials provides the spectral accuracy. The theoretical and experimental analysis of the scheme reliability verifies the validity of the proposed method in large domains.

Age Structure Can Account for Delayed Logistic Proliferation of Scratch Assays.

Scratch assays are in vitro methods for studying cell migration. In these experiments, a scratch is made on a cell monolayer and recolonisation of the scratched region is imaged to quantify cell migration rates. Typically, scratch assays are modelled by reaction diffusion equations depicting cell migration by Fickian diffusion and proliferation by a logistic term. In a recent paper (Jin et al. in Bull Math Biol 79(5):1028-1050, 2017), the authors observed experimentally that during the early stage of the recolonisation process, there is a disturbance phase where proliferation is not logistic, and this is followed by a growth phase where proliferation appears to be logistic. The authors did not identify the precise mechanism that causes the disturbance phase but showed that ignoring it can lead to incorrect parameter estimates. The aim of this work is to show that a nonlinear age-structured population model can account for the two phases of proliferation in scratch assays. The model consists of an age-structured cell cycle model of a cell population, coupled with an ordinary differential equation describing the resource concentration dynamics in the substrate. The model assumes a resource-dependent cell cycle threshold age, above which cells are able to proliferate. By studying the dynamics of the full system in terms of the subpopulations of cells that can proliferate and the ones that can not, we are able to find conditions under which the model captures the two-phase behaviour. Through numerical simulations, we are able to show that the interplay between the resource concentration in the substrate and the cell subpopulations dynamics can explain the biphasic dynamics.

Asymptotic stability of delayed consumer age-structured population models with an Allee effect

In this article we study a nonlinear age-structured consumer population model with density-dependent death and fertility rates, and time delays that model incubation/gestation period. Density dependence we consider combines both positive effects at low population numbers (i.e., the Allee effect) and negative effects at high population numbers due to intra-specific competition of consumers. The positive density-dependence is either due to an increase in the birth rate, or due to a decrease in the mortality rate at low population numbers. We prove that similarly to unstructured models, the Allee effect leads to model multi-stability where, besides the locally stable extinction equilibrium, there are up to two positive equilibria. Calculating derivatives of the basic reproduction number at the equilibria we prove that the upper of the two non-trivial equilibria (when it exists) is locally asymptotically stable independently of the time delay. The smaller of the two equilibria is always unstable. Using numerical simulations we analyze topologically nonequivalent phase portraits of the model.

High-order IMEX-WENO finite volume approximation for nonlinear age-structured population model

ABSTRACTIn this article, we design a high-order implicit–explicit weighted non-oscillatory (IMEX-WENO) scheme for the solution of the population age density model with the nonlinear mortality rate as well as fertility rate. The mortality rate is the combination of natural mortality that comes to be unbounded at maximum age and bounded external mortality rate that includes external resources as well as seasonality. The existence of global terms in the mortality function as well as in the boundary condition is the main technical complication which provides high nonlinearity for the model equation. We carefully construct a numerical scheme in such way, that the high-order accuracy is maintained in the global terms and boundary condition. The performance of the designed scheme is shown by comparing with the exact solution of the examples considered.

Haar wavelet method for solving nonlinear age-structured population models

In this study, Haar wavelet method is implemented for solving the nonlinear age-structured population model which is the nonclassic type of partial differential equation associated with boundary integral equation. This paper develops the flexibility of Haar wavelet method for reduction of the partial differential equation with nonlocal boundary conditions to an algebraic system. In fact, the simple structure of piecewise orthogonal Haar basis functions which leads to sparse matrices causes the convergence and computational efficiency. Some illustrative results show the reliability and accuracy of the presented method.

Nonstationary and Chaotic Dynamics in Age-Structured Population Models

The dynamics from nonlinear discrete age-structured population models is under consideration. Focus is on bifurcations, as well as nonstationary and chaotic dynamics. We also explore different mechanisms which may lead to periodic phenomena. Some new results are also presented, in particular from models where both fecundity and survival terms contain nonlinear elements.

Adomian Decomposition Method for Nonlinear Age-Structured Population Models with Using Trapezoidal and Simpson Rules

Adomian Decomposition Method for Nonlinear Age-Structured Population Models with Using Trapezoidal and Simpson Rules

Chebyshev Cardinal Functions for Solving Age-Structured Population Models

In this paper, we use Chebyshev cardinal functions for solving a nonlinear age-structured population models. This partial integro-differential equation includes an integral equation as a boundary condition. The proposed method which approximates the highest derivative based on Chebyshev cardinal basis functions is flexible with local and non-local conditions. Numerical results show the efficiency and accuracy of the presented method.

A study of nonlinear age-structured population models

This paper presents a novel analysis for the solution of nonlinear age-structured problem which is of extreme importance in biological sciences. The presented model is very useful but quite complicated. Modified variational iteration method (MVIM) coupled with auxiliary parameter is used to cope with the complexity of the model which subsequently shows better results as compared to some existing results available in literature. Furthermore, an appropriate way is used for the identification of auxiliary parameter by means of residual function. Numerical examples are presented for the analysis of the proposed algorithm. Graphical results along with the discussions re-confirm the efficiency of proposed algorithm. The work proposes a new algorithm where He’s polynomials and an auxiliary parameter are merged with correction functional. The suggested scheme is implemented on nonlinear age-structured population models. Graphs are plotted for the residual function that reflects the accuracy and convergence of the presented algorithm.

Asymptotically stable states of nonlinear age-structured monocyclic population model I. Travelling wave solution

This paper is devoted to the study of evolutionary dynamics of monocyclic age-structured population including effect of nonlinear mortality (population growth feedback) and proliferation. The total population is considered as partitioned by fixing age into two subpopulations. Individuals of first population are born, mature, die and can at the final fixed age give birth for some new individuals (with null age). Individuals of the second subpopulation are older than those of the first one. They can mature, die and do not have possibility to proliferate. This model was considered as a system of two initial–boundary value problems for nonlinear transport equations with non-local boundary conditions. We obtained explicit travelling wave solution provided the model parameters (coefficients of equations and initial values) satisfy the restrictions that guarantee continuity and smoothness of solution. Explicit form of solution allowed us to perform numerical experiments with high accuracy using the set of parameterized algebraic functions which do not depend from the time. In all performed experiments solutions are attracted to some stationary functions for a long time period (asymptotically stable states of system). We indicated and studied three different regimes of population dynamics. The first is quasi-equilibrium regime, when the maximum value of population density by age, as a function of time, is attracted to the point from the neighbourhood of initial value. This is a result of balance between effects of proliferation and nonlinear mortality (like behaviour of microorganism population in the cases of asymptomatic or healthy carriers). The second and third regimes are characterized by increasing (decreasing) maximum by age values of population density with following attracting to values higher (lower) than initial one. We studied also the impact of parameters of nonlinear death rate on the tremendous growth of population density followed by transition to asymptotically stable states (like infection generalization process in living organisms).