Numerical Analysis of a Size-Structured Population Model with a Dynamical Resource

Oscar Angulo,M A Lopez-Marcos,J C Lopez-Marcos

doi:10.11145//j.biomath.2014.03.241

Abstract

In this paper, we analyze the convergence of a second order numerical method for the approximation of a size-structured population model whose dependency on the environment is managed by the evolution of a vital resource. Optimal rate of convergence is derived. Numerical experiments are also reported to demonstrate the predicted accuracy of the scheme. Also, it is applied for the solution of a problem that describes the dynamics of a Daphnia magna population, paying attention to the unstable case.

Highlights

Structured population models are based on the use of one or more attributes that structure the individuals in the population
We have proven the convergence of numerical methods which employ a selection criterion, whenever the positions, which are determined by the criterion we have chosen, lead us to subgrids which satisfy property (SR)
We have analyzed a second-order numerical method for a problem that describes a population with a possible shrinking size and with a dependency on the environment managed by the evolution of a vital resource

Summary

Introduction

Structured population models are based on the use of one or more attributes that structure the individuals in the population. We consider a size-structured population model nonlinearly coupled with an integroordinary differential equation accounting for substrate consumption and/or product formation. It was introduced first in [6] for modeling a Daphnia magna population. Note that all the vital functions (g, μ and α) depend on size x (the structuring internal variable), on time t and on the value of the resource at time t, which can reflect the influence of the environmental changes on the vital functions. I(t) (which represents the way of weighting the size distribution density in order to model the different influence of individuals of different sizes on such dynamics) and on time t.

Preliminary results

The numerical method

Convergence Analysis

Numerical results

Conclusions