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

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 through 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., loc. cit.).
KEYWORDS: Birth and death process; Age and size-structured populations; Stochastic interacting particle systems; Piecewise deterministic motion; Large population limits

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