M. A. López-Marcos

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 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.

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.

Numerical study on the proliferation cells fraction of a tumour cord model

A characteristic curves numerical method based on a regular grid is performed for the solution of a kind of boundary-value problem for a hyperbolic first-order integro-partial-differential equation describing the stationary state of a tumour cord. Numerical evidence on the order of convergence is presented. And a study on the function that gives the fraction of cells which reenters into proliferation is developed.

Analysis of an efficient integrator for a size-structured population model with a dynamical resource

In this paper, an efficient numerical method for the approximation of a nonlinear size-structured population model is presented. The nonlinearity of the model is given by dependency on the environment through the consumption of a dynamical resource. We analyse the properties of the numerical scheme and optimal second-order convergence is derived. We report experiments with academical tests to demonstrate numerically the predicted accuracy of the scheme. The model is applied to solve a biological problem: the dynamics of an ectothermic population (the water flea, Daphnia magna). We analyse its long time evolution and describe the asymptotically stable steady states, both equilibria and limit cycles.

Study on the efficiency in the numerical integration of size-structured population models: Error and computational cost

Abstract We describe a procedure which is useful to select an appropriate numerical method in a size-structured population model. We consider four different numerical methods based on finite difference schemes or characteristics curves integration. We compute an analytical approximation in terms of the discretization parameters for the theoretical error principal terms and the computational cost. Thus, we show the efficiency curve that allows to select the best relationship between the discretization parameters for each numerical method. Finally, we obtain the most efficient numerical method for each test.

Numerical integration of a hierarchically size-structured population model with contest competition

We formulate schemes for the numerical solution to a hierarchically size-structured population 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 and to show their behaviour to approaching stable steady states.

Asymptotic behaviour of a mathematical model of hematopoietic stem cell dynamics

We deeply researched into the asymptotic behaviour of a numerical method adapted for the solution of mathematical model of hematopoiesis which describes the dynamics of a stem cell population. We investigated the stationary solutions of the original model by their numerical approximation: we proved the existence of a numerical stationary solution that provides a good approximation to the nontrivial equilibrium solution of the problem. Also, we presented a numerical simulation which confirms this behaviour.