Oscar Angulo

Numerical schemes for size-structured population equations.

We formulate schemes for the numerical solution of size-dependent population models. Such schemes discretize size by means of a natural grid, which introduces a discrete dynamics. The schemes are analysed and optimal rates of convergence are derived. Some numerical experiments are also reported to demonstrate the predicted accuracy of the schemes.

Numerical integration of autonomous and non-autonomous non-linear size-structured population models

We present an efficiency study for autonomous and non-autonomous non-linear size-structured population models. The study considers three numerical methods: a characteristics scheme, the Lax-Wendroff method and the box method, which are completely described in the paper. Five test problems are considered with diverse degree of complexity: non-trivial equilibrium, periodic solutions and diverse growth functions. The study of the efficiency takes into account the properties of the numerical schemes (such as the stability) and uses a multiple regression analysis to determine the constants of the leading terms of the corresponding global errors. We show how the tables of errors and cpu-times can be used to explain the meaning of the efficiency results. In addition, we present the convergence analysis of the box method.

Numerical integration of a mathematical model of hematopoietic stem cell dynamics

A mathematical model of hematopoiesis, describing the dynamics of stem cell population, is investigated. This model is represented by a system of two nonlinear age-structured partial differential equations, describing the dynamics of resting and proliferating hematopoietic stem cells. It differs from previous attempts to model the hematopoietic system dynamics by taking into account cell age-dependence of coefficients, that prevents a usual reduction of this system to an unstructured delay differential system. We prove the existence and uniqueness of a solution to our problem, and we investigate the existence of stationary solutions. A numerical scheme adapted to the problem is presented. We show the effectiveness of this numerical technique in the simulation of the dynamics of the solution. Numerical simulations show that long-period oscillations can be obtained in this model, corresponding to a destabilization of the system. These oscillations can be related to observations of some periodical hematological diseases (such as chronic myelogenous leukemia).

Numerical schemes for a size-structured cell population model with equal fission

We study numerically the evolution of a size-structured cell population model, with finite maximum individual size and minimum size for mitosis. We formulate two schemes for the numerical solution of such a model. The schemes are analysed and optimal rates of convergence are derived. Some numerical experiments are also reported to demonstrate the predicted accuracy of the schemes. We also consider the behaviour of the methods with respect to the different discontinuities that appear in the solution to the problem and the stable size distribution. In addition, the numerical schemes are used to study asynchronous exponential growth.

Numerical integration of nonlinear size-structured population equations

We formulate a second-order scheme for the numerical integration of nonlinear size-dependent population models. The scheme is completely analysed and some numerical experiments are also reported in order to demonstrate the predicted accuracy of the scheme.

Numerical approximation of singular asymptotic states for a size-structured population model with a dynamical resource

We compare two selection procedures in a numerical method proposed for the numerical simulation of the long time behaviour of solutions for a size-structured population model whose dependency on the environment is managed by the evolution of a vital resource and with a free sign growth rate. We obtain the approximation of singular asymptotic states such as stable steady ones and attractive limit cycles.

Numerical investigation of the recruitment process in open marine population models

The changes in the dynamics, produced by the recruitment process in an open marine population model, are investigated from a numerical point of view. The numerical method considered, based on the representation of the solution along the characteristic lines, approximates properly the steady states of the model, and is used to analyze the asymptotic behavior of the solutions of the model.

Numerical analysis of an open marine population model with spaced-limited recruitment

A numerical method for an open marine population model with spaced-limited recruitment is presented. The method takes into account the singularity of the mortality rate at the finite maximum age, and it is based on the representation of the solution along the characteristics lines. We analyze the numerical scheme and use it to approximate the asymptotic behavior of the solutions of the model.

DYNAMICS OF A STRUCTURED SLUG POPULATION MODEL IN THE ABSENCE OF SEASONAL VARIATION

We develop a novel, nonlinear structured population model for the slug Deroceras reticulatum, a highly significant agricultural pest of great economic impact, in both organic and non-organic settings. In the absence of seasonal variations, we numerically explore the effect of life history traits that are dependent on an individuals size and measures of population biomass. We conduct a systematic exploration of parameter space and highlight the main mechanisms and implications of model design. A major conclusion of this work is that strong size dependent predation significantly adjusts the competitive balance, leading to non-monotonic steady state solutions and slowly decaying transients consisting of distinct generational cycles. Furthermore, we demonstrate how a simple ratio of adult to juvenile biomass can act as a useful diagnostic to distinguish between predated and non-predated environments, and may be useful in agricultural settings.

A semi-Lagrangian method for a cell population model in a dynamical environment

Abstract We formulate a numerical method on a regular grid, which is based on integration along characteristics curves, to solve a size-structured cell population model in an environment with an evolutionary resource concentration. Numerical simulations are also reported to show the convergence of the approximations and to study the dynamics of the considered problem thanks to the fine behavior of the numerical scheme.