Daphnias: from the individual based model to the large population equation

TL;DR

This paper derives an individual-based model for size-structured Daphnia populations interacting with algae and proves that, for large populations, it can be approximated by a delay differential equation, linking individual dynamics to population-level models.

Contribution

It formulates the underlying individual-based models for Daphnia populations and establishes their approximation by delay equations in large populations.

Findings

The IBM models size-structured Daphnia and resource interactions.

Large populations allow approximation of stochastic models by delay equations.

The delay equation accurately captures the population dynamics.

Abstract

The class of deterministic 'Daphnia' models treated by Diekmann et al. (J Math Biol 61: 277-318, 2010) has a long history going back to Nisbet and Gurney (Theor Pop Biol 23: 114-135, 1983) and Diekmann et al. (Nieuw Archief voor Wiskunde 4: 82-109, 1984). In this note, we formulate the individual based models (IBM) supposedly underlying those deterministic models. The models treat the interaction between a general size-structured consumer population ('Daphnia') and an unstructured resource ('algae'). The discrete, size and age-structured Daphnia population changes through births and deaths of its individuals and throught their aging and growth. The birth and death rates depend on the sizes of the individuals and on the concentration of the algae. The latter is supposed to be a continuous variable with a deterministic dynamics that depends on the Daphnia population. In this model setting we prove that when the Daphnia population is large, the stochastic differential equation describing the IBM can be approximated by the delay equation featured in (Diekmann et al., l.c.).