**Luis M. Abia <sup>1</sup> , Óscar Angulo 2,\*, Juan C. López-Marcos1 and Miguel A. López-Marcos <sup>1</sup>**


Received: 05 August 2020; 24 August 2020; Published: 27 August 2020

**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

**MSC:** 92D25; 92D40; 65M25; 65M12; 35B40
