The Convergence Analysis of a Numerical Method for a Structured Consumer-Resource Model with Delay in the Resource Evolution Rate

Luis M. Abia, Óscar Angulo, Juan C. López-Marcos and Miguel A. López-Marcos
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Luis M. Abia: Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad de Valladolid e IMUVa, 47011 Valladolid, Spain
Óscar Angulo: Departamento de Matemática Aplicada, ETS de Ingenieros de Telecomunicación, Universidad de Valladolid e IMUVa, 47011 Valladolid, Spain
Juan C. López-Marcos: Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad de Valladolid e IMUVa, 47011 Valladolid, Spain
Miguel A. López-Marcos: Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad de Valladolid e IMUVa, 47011 Valladolid, Spain

Mathematics, 2020, vol. 8, issue 9, 1-18

Abstract: In this paper, we go through the development of a new numerical method to obtain the solution to a size-structured population model that describes the evolution of a consumer feeding on a dynamical resource that reacts to the environment with a lag-time response. The problem involves the coupling of the partial differential equation that represents the population evolution and an ordinary differential equation with a constant delay that describes the evolution of the resource. The numerical treatment of this problem has not been considered before when a delay is included in the resource evolution rate. We analyzed the numerical scheme and proved a second-order rate of convergence by assuming enough regularity of the solution. We numerically confirmed the theoretical results with an academic test problem.

Keywords: size-structured population; consumer-resource model; delay differential equation; numerical methods; characteristics method; convergence analysis (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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