Journal of Mathematical Biology

, Volume 66, Issue 4, pp 915-933

First online:

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

  • J. A. J. MetzAffiliated withEEP, IIASAMathematical Institute and Institute of Biology and NCB Naturalis, Leiden Email author 
  • , Viet Chi TranAffiliated withUniversité des Sciences et Technologies Lille 1, Laboratoire Paul Painlevé, UFR de Mathématiques, UMR CNRS 8524CMAP, Ecole Polytechnique

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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.).


Birth and death process Age and size-structured populations Stochastic interacting particle systems Piecewise deterministic motion Large population limits

Mathematics Subject Classification (2000)

92D40 60J80 60K35 60F99