Journal of Mathematical Biology

, Volume 66, Issue 4–5, pp 915–933

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

Article

## 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 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

### Mathematics Subject Classification (2000)

92D40 60J80 60K35 60F99

### References

1. Champagnat N, Ferriére R, Méléard S (2006) Unifying evolutionary dynamics: from individual stochastic processes to macroscopic models. Theor Pop Biol 69:297–321Google Scholar
2. Champagnat N, Ferriére R, Méléard S (2008) Individual-based probabilistic models of adaptive evolution and various scaling approximations. In: Dalang, R.C., Dozzi, M., Russo, F (eds) Seminar on Stochastic Analysis, Random Fields and Applications V, Centro Stefano Franscini, Ascona, May 2005. Progress in Probability vol. 59, Birkhauser, pp 75–114Google Scholar
3. Diekmann O, Gyllenberg M (2012) Equations with infinite delay: blending the abstract and the concrete. J Diff Equ 252(2):819–851Google Scholar
4. Diekmann O, Metz JAJ (2010) How to lift a model for individual behaviour to the population level? Phil Trans Roy Soc London B 365:3523–3530
5. Diekmann O, Metz JAJ, Kooijman SALM, Heymans HJAM (1984) Continuum population dynamics with an application to Daphnia magna. Nieuw Archief voor Wiskunde 4:82–109Google Scholar
6. Diekmann O, Gyllenberg M, Metz JAJ, Thieme HR (1998) On the formulation and analysis of general deterministic structured population models I Linear theory. J Math Biol 36:349–388
7. Diekmann O, Gyllenberg M, Huang H, Kirkilionis M, Metz JAJ, Thieme HR (2001) On the formulation and analysis of general deterministic structured population models. II. Nonlinear Theory. J Math Biol 43:157–189
8. Diekmann O, Gyllenberg M, Metz JAJ (2003) Steady state analysis of structured population models. Theor Pop Biol 63:309–338
9. Diekmann O, Getto P, Gyllenberg M (2007) Stability and bifurcation analysis of Volterra functional equations in the light of suns and stars. SIAM J Math Anal 39(4):1023–1069
10. Diekmann O, Gyllenberg M, Metz J, Nakaoka S, de Roos A (2010) Daphnia revisited: local stability and bifurcation theory for physiologically structured population models explained by way of an example. J Math Biol 61:277–318Google Scholar
11. Durinx M, Metz JAJ, Meszéna G (2008) Adaptive dynamics for physiologically structured models. J Math Biol 56:673–742
12. Evans L (1998) Partial Differential Equations, Graduate Studies in Mathematics, vol. 19, American Mathematical SocietyGoogle Scholar
13. Ferriére R, Tran VC (2009) Stochastic and deterministic models for age-structured populations with genetically variable traits. ESAIM: Proceedings 27. pp. 289–310, Proceedings of the CANUM 2008 conferenceGoogle Scholar
14. Fournier N, Méléard S (2004) A microscopic probabilistic description of a locally regulated population and macroscopic approximations. Ann Appl Probab 14(4):1880–1919
15. Gurney WSC, Nisbet RM (1985) Fluctuation periodicity, generation separation, and the expression of larval competition. Theor Pop Biol 28:150–180
16. Jacod J, Shiryaev A (1987) Limit Theorems for Stochastic Processes. Springer, Berlin
17. Jagers P, Klebaner F (2000) Population-size-dependent and age-dependent branching processes. Stoch Proc Appl 87:235–254
18. Jagers P, Klebaner F (2011) Population-size-dependent, age-structured branching processes linger around their carrying capacity. J Appl Prob 48A: 249–260, special volume: New Frontiers in Applied ProbabilityGoogle Scholar
19. Joffe A, Métivier M (1986) Weak convergence of sequences of semimartingales with applications to multitype branching processes. Adv Appl Prob 18:20–65
20. Kurtz TG (1970) Solutions of ordinary differential equations as limits of pure jump Markov processes. J Appl Prob 7:49–58Google Scholar
21. Kurtz TG (1981) Approximation of population Processes. SIAM, Philadelphia, PA
22. Méléard S, Metz JAJ, Tran VC (2011) Limiting Feller diffusions for logistic populations with age-structure, 58th World Statistics Congress of the International Statistical Institute (ISI 2011), Dublin Ireland (July 2011). hal-00595928Google Scholar
23. Méléard S, Roelly S (1993) Sur les convergences étroite ou vague de processus à valeurs mesures. CRAcadSciParis, Serie I 317:785–788Google Scholar
24. Méléard S, Tran VC (2009) Trait substitution sequence process and canonical equation for age-structured populations. J Math Biol 58(6):881–921
25. Méléard S, Tran VC (2012) Slow and fast scales for superprocess limits of age-structured populations. Stoch Proc Appl 122:250–276
26. Metz JAJ, Diekmann O (1986) The dynamics of physiologically structured populations. Lecture Notes in Biomathematics, vol. 68. Springer, BerlinGoogle Scholar
27. Metz JAJ, de Roos AM (1992) The role of physiologically structured population models within a general individual-based modeling perspective. In: DeAngelis DL, Gross LJ (eds) Individual-based models and approaches in ecology. Routledge, Chapman& Hall, London, pp 88–111Google Scholar
28. Nisbet RM, Gurney WSC (1983) The systematic formulation of populationmodels with dynamically varying instar duration. Theor Pop Biol 23:114–135
29. Oelschläger K (1990) Limit theorem for age-structured populations. Ann Prob 18(1):290–318
30. de Roos A, Metz JAJ, Evers E, Leipoldt A (1990) A size dependent predator-prey interaction: who pursues whom? J Math Biol 28(6):609–643
31. Tran VC (2006) Modéles particulaires stochastiques pour des problémes d’évolution adaptative et pour l’approximation de solutions statistiques. PhD thesis, Université Paris X—Nanterre. http://tel.archives-ouvertes.fr/tel-00125100
32. Tran VC (2008) Large population limit and time behaviour of a stochastic particle model describing an age-structured population. ESAIM: P&S 12:345–386