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

## Authors

## 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 processAge and size-structured populationsStochastic interacting particle systemsPiecewise deterministic motion Large population limits### Mathematics Subject Classification (2000)

92D4060J8060K3560F99## 1 Introduction

The theory of physiologically structured populations as developed by Metz and Diekmann (1986); Diekmann et al. (1998); Diekmann et al. (2001); Diekmann et al. (2003); Diekmann et al. (2010) derives its motivation from its supposed ability to link population level phenomena to specific mechanisms in and around individual organisms. Yet, those individuals do not figure as such in the models, which treat the spatial concentrations of those individuals as continua. The supposition has always been that the proposed deterministic frameworks would in principle be derivable from individual-based stochastic models (c.f. Metz and de Roos 1986), but full derivations of this sort so far have only been done for the special cases of finite i-state models (i from individual; leading to ODEs in the large number limit, e.g. Kurtz 1970, 1981) and age-based models (e.g. Oelschläger 1990; Tran 2006; Tran 2008; Ferrière and Tran 2009; Jagers and Klebaner 2000, 2011). Moreover, it is possible to reinterpret the results by Champagnat et al. (2006) as pertaining to general structured models with only jump transitions. Although some age-structured models are phrased in terms of i-state variables that change with time in a fixed manner, uninfluenced by the environment, these can truly represent but a small fraction of the rich variety of physiological mechanisms seen in nature (e.g. Diekmann and Metz 2010; Metz and Diekmann 1986, Chapters I and III). For example, for the majority of species, size with a growth rate that depends on the environment is a far more important determinant of an individual’s population dynamical behaviour than is age. Except in the physiologically well buffered homeotherms (on which we anthropocentrically are inclined to focus) reproduction tends to be under a strong influence of past food availabilities, with reproductive size being reached far earlier when an individual has encountered good than when it has encountered bad feeding conditions. In, for example, arthropods or fishes, maturing to the reproductive states often largely depends on how much food they have encountered but little on their age. However, the toolbox for proving the appropriate law of large number results that was developed by Fournier and Méléard (2004), Tran (2006, 2008), Ferrière and Tran (2009) needs to be extended to deal with size-based models, referred to as ’Daphnia’ models by Diekmann et al. (2010), let alone to the even more general models considered by Metz and Diekmann (1986), Metz and de Roos (1986), Diekmann et al. (1998, 2001, 2003). In this paper we, as a birthday present to Odo Diekmann, remedy this lack, although so far only for the ’Daphnia’ models that appear as central example in his work (c.f. Diekmann et al. 2010).

Diekmann et al. (2010) lay out a general framework for investigating the interaction between a general age or size-structured consumer population (referred to as ’Daphnia’) and an unstructured resource (referred to as ‘algae’), a class of problems special forms of which were first considered by Nisbet and Gurney (1983); Gurney and Nisbet (1985) and by Diekmann et al. (1984); de Roos et al. (1990). Diekmann et al. (loc. cit.) show that stability properties and bifurcation phenomena can be understood in terms of solutions of a system of two delay equations that are analysed using results by Diekmann et al. (2007); Diekmann and Gyllenberg (2012). In this note, we derive their model from a microscopic description: starting from a stochastic, age- and size-structured, individual-based model (IBM) for the Daphnia population we recover the equations of Diekmann et al. (2010).

In our study, the Daphnia population is discrete and stochastic while the algal population is continuous. We treat the Daphnia population as a point measure on a space spanned by size and age and from this platform generalize the microscopic construction given in Tran (2008) to arrive at a stochastic differential equation (SDE) driven by Poisson point processes that gives a pathwise description of the population evolution. The main novelty is that the individual growth rate also depends on the population as a whole, instead of only on an individuals’ characteristics, since the algal concentration and thus each individuals’ resource access is affected by the entire Daphnia population.

We then, in the wake of Oelschläger (1990); Tran (2006, 2008); Ferrière and Tran (2009); Jagers and Klebaner (2000), (2011), provide a law of large numbers that allows approximating the dynamics of the individual-based size- and age-structured model by means of a partial differential equation (PDE) when the volume containing the Daphnia and algae as well as the population sizes are large. Méléard and Tran (2009) use similar limit theorems as a basis for obtaining adaptive dynamics approximations for the evolution of hereditary traits in age-structured populations, while Durinx et al. (2008) give informed conjectures about the latter approximations for the more encompassing model classes treated by Diekmann et al. (2003). Finally we study the limiting equations to recover the macroscopic (deterministic) system considered in Diekmann et al. (2010).

**Notation** We will denote the set of finite measures on \(\mathbb R _+^2\) endowed with the weak convergence topology as \(\mathcal{M }_F(\mathbb R _+^2)\). For \(\mu \in \mathcal{M }_F(\mathbb R _+^2)\) and a real measurable bounded function \(f\) on this space, we set \(\langle \mu , f\rangle =\int _\mathbb{R _+^2} f(\xi ,a)\mu (d\xi ,da)\). The set of bounded real functions of class \(\mathcal C ^1\) on \(E\) with bounded derivatives is denoted as \(\mathcal C ^1_\mathrm{{b}}(E,\mathbb R )\).

When \(X\) is a process indexed by time, we will interchangeably denote the value of \(X\) at time \(t\) as \(X(t)\) or \(X_t\) in order to avoid formulas becoming cluttered with too many brackets.

## 2 Individual-based Daphnia model

### 2.1 Model specification

**Assumptions 1**

- (i)
the birth rate \(\beta (\xi ,a,S)\) is piecewise continuous and bounded by \(\bar{\beta }\).

- (ii)
the death rate \(\mu (\xi ,a,S)\) is continuous and there exists a function \(\underline{\mu }(a)\) and a constant \(A\in (0,+\infty ]\) such that \(\forall (\xi ,a,S)\in \mathbb R _+^3,\)\(\mu (\xi ,a,S)\ge \underline{\mu }(a)\) and \(\int _0^A \underline{\mu }(a)da=+\infty \).

- (iii)
\(g\) is Lipschitz continuous with respect to \(\xi \), uniformly in \(S\) on compact intervals of \(\mathbb R _+\), and bounded by \(\bar{g}\).

- (iv)
\(f\) and \(\gamma \) are uniformly Lipschitz continuous with respect to \(S\) uniformly in \(\xi \) on compact intervals of \(\mathbb R _+\).

The assumption (i) on the birth rate ensures that in a short time interval a single individual can not beget too many young: intervals between births are stochastically lower bounded by exponential random variables with rate \(\bar{\beta }\). The assumption (ii) on the death rate implies that individuals a.s. have lifetimes bounded by \(A\). Finally the assumptions (iii) and (iv) ensure that there exist unique continuous solutions to (2) and (3) as long as the number of individuals \(|Z_t|\) remains finite, the latter being guaranteed by the fact that \(|Z_t|\) is stochastically bounded by a pure birth process with birth rate \(\bar{\beta }\).

In the next Subsection we introduce the pathwise construction of an IBM with the described dynamics and give an SDE driven by a Poisson point process that admits (4) as infinitesimal generator. This is useful for simulations and for deriving moment conditions and large population approximations (e.g. Champagnat et al. 2006). The evolution is piecewise deterministic: The size of the population is modified at birth or death events. Between these, conditionally on the structure of the population after the last event, the growth of the individuals and the food dynamics are deterministic.

### 2.2 Construction of the IBM and a useful SDE

The above description suggests a simple direct way for simulating the IBM. Starting from a birth or death event, first generate a standard exponentially distributed random number \(\tau \), and then run the differential equations for the states of all Daphnia individuals and for the algae. Simultaneously integrate the sum of the birth and death rates of the Daphnia, starting from zero. When this integral reaches \(\tau \), one of the Daphnia dies or gives birth. Which individual is the culprit and whether the event is a birth or a death is then decided from a single multinomial draw with probabilities proportional to the contributions of all the different events to the total event rate at that time.

Simulations of this type allow dealing with e.g. the fluctuations in cyclic dynamics due to population lows occurring in the course of the cycle, and with the possibility of population extinction during periods with small population size.

- If no event happens during \([0,t]\), then it is sufficient to consider the predicted sizes \((\Xi ^i(t ; 0,\bar{Z}_0,S_0)\ ; \ i\in V_0)\) of individuals at \(t\). The population at time \(t\) isand the food concentration is \(S_t=\Sigma (t ; 0,\bar{Z}_0,S_0)\).$$\begin{aligned} \bar{Z}_t=\sum _{i\in V_0}\delta _{(i,\Xi ^i(t ; 0,\bar{Z}_0,S_0),a^i_0+t)} \end{aligned}$$
- If a birth event occurs at time \(s\in [0,t]\), then \(V_s=V_{s_-}\cup \{I_{s_-}+1\}\), where \(I_{s_-}\) is the number of labels already used so that the new individual gets labelled with the first available number \(j=I_{s_-}+1\). The predicted sizes at time \(t,\, (\Xi ^i(t ; s,\bar{Z}_{s_-},S_{s_-}); \ i\in V_{s_-})\), that we had for the individuals \(i\in V_{s_-}\) before the event on the supposition that it were not to occur, are replaced by \((\Xi ^i(t; s,\bar{Z}_{s_-}+\delta _{(j,\xi _0,0)},S_{s_-}); \ i\in V_{s_-})\); see Fig. 1. Moreover for the new individual with label \(j\), we add a Dirac mass at \((j,\Xi ^{j}(t ; s, \bar{Z}_{s_-}+\delta _{(j,\xi _0,0)},S_{s_-}),t-s)\).
If the individual \(j\in V_{s_-}\) dies at time \(s\in [0,t]\), then \(V_s=V_{s_-}\!\setminus \{j\}\) and the predicted sizes at time \(t,\, (\Xi ^i(t ; s,\bar{Z}_{s_-},S_{s_-}); \, i\in V_{s})\), that we had for the individuals \(i\in V_{s}\) before the event on the supposition that it were not to occur, are replaced by \((\Xi ^i(t ; s,\bar{Z}_{s_-}-\delta _{(j,\xi ^j_{s_-},a^j_{s_-})},S_{s_-})\ ; \ i\in V_{s})\); see Fig. 1. Additionally, we delete the Dirac mass at \((j,\Xi ^j(t ; s,\bar{Z}_{s_-},S_{s_-}),a^j_s+(t-s))\) that corresponds to the predicted size and age at time \(t\) of the dead individual.

**Definition 1**

In this definition the first of the two terms in the square brackets corresponds to the births and the second to the deaths. The first term starts with the addition of one new individual, followed by the corresponding updating of the future course of the trajectories of the other individuals in order to eventually get the right outcome at the final time \(t\). The first part of the second term, dealing with the deaths, starts with removing an individual, followed by a corresponding updating of the growth trajectories.

Moment estimates obtained from (7) are very important for the proofs. By adapting the proofs of Champagnat et al. (2008) (Th. 2.5) (see also Fournier and Méléard (2004)) where the main ingredient is the boundedness of the birth rate, we can show that:

**Lemma 1**

- (i)
For any \(T>0,\, \mathbb E \big (\sup _{t\in [0,T]}\langle Z_t,1\rangle )<\mathbb E (\langle Z_0,1\rangle )e^{\bar{\beta }T}<+\infty \),

- (ii)If we have additionally that \(\mathbb E (\langle Z_0,1\rangle ^p)<+\infty \), then$$\begin{aligned} \mathbb E \big (\sup _{t\in [0,T]}\langle Z_t,1\rangle ^p)<+\infty . \end{aligned}$$(10)

As a consequence we obtain that (see Champagnat et al. 2008; Tran 2006):

**Proposition 1**

The process \((Z_t,S_t)_{t\in \mathbb R _+}\) is well and uniquely defined on \([0,T]\) for any initial condition such that \(E(\langle Z_0,1\rangle )<+\infty \) and for any Poisson point measure \(Q\). Moreover, the infinitesimal generator of \((Z_t)_{t\in \mathbb R _+}\) is (4).

Below we shall use the just described stochastic model to arrive at the deterministic models of Diekmann et al. (2010). However, it should also be possible to derive other limits like diffusion processes or superprocesses as was done by Méléard et al. (2011) and Méléard and Tran (2012) respectively. Moreover, only by taking limits of properly specified stochastic individual-based models one can build e.g. structured population models in stochastic environments without a considerable risk of erring in one’s model specification.

### 2.3 Martingale problem

We end this section with a martingale problem that will be useful for deriving the large population limits. Heuristically, the decomposition of the process \(\langle Z_t,f\rangle \), for any test function \(f\in \mathcal C ^1\), into a predictable finite variation process and a square integrable martingale can be viewed as a description of the paths as solutions of the evolution equation associated with the generator \(L\) (predictable finite variation part) plus noise (martingale part). The proof is given in Appendix A.

**Proposition 2**

## 3 Large populations

**Proposition 3**

The proof of Proposition 3 is given in Appendix B.

We conclude by showing that Eqs (16)–(17) allow us to recover the equations of Diekmann et al. (2010). We first establish a precise form of the solution \(\zeta _t\). That this measure is not absolutely continuous with respect to the Lebesgue measure on \(\mathbb R _+^2\), even if the initial condition \(\zeta _0\) is, was already noticed by e.g. Metz and Diekmann (1986) and Tran (2006, 2008). There it was stated that age and size were both parameterized by time, but no precise form for the measure for these cases was given.

**Corollary 1**

*Proof of Corollary 1*

For the proof, we start by showing that \(\zeta _t(d\xi ,da)\) admits a density \(m(\xi ,a,t)\) w.r.t. a dominating measure underlying (18). The equations satisfied by \(m(\xi ,a,t)\) are then derived by separating the domain into \(\mathbb R _+\times \{a\ge t\}\) and \(\mathbb R _+\times \{a<t\}\), which corresponds to first studying the individuals born before and after initial time.

First, recall that there is a unique solution \(\varrho \) to (17).

## 4 Conclusion

In this paper we rigorously underpin the long suspected individual-based nature of physiological structured population models as studied by the team of researchers surrounding Odo Diekmann. Such results were already known for purely age-based models (e.g. Oelschläger (1990), Tran (2006, 2008), Ferrière and Tran (2009) , Jagers and Klebaner (2000, 2011), and our present result still applies only to a subclass of the models considered by Odo Diekmann and his co-workers, although a paradigmatic one, to wit their so-called ”Daphnia” models (e.g. Diekmann et al. 2010; Diekmann and Metz 2010). The i-state variables of these models move in a continuous deterministic fashion, dependent on their own value and the environment (food: ”algae”), and births occur to a single fixed birth state at a rate that depends on the i-state and the environment (and are not e.g. coupled to specific i-state transitions). Even for this restricted class of physiologically structured population models some nontrivial twiddling of the existing probabilistic toolbox was needed in order to handle the presence of a dynamic environment influencing the movement speed of the i-states. Yet, overall the toolbox proved its mettle, and the long standing assumption of a solid individual-based foundation of the theory of physiologically structured populations was duly vindicated. The longer term goal is to extend this vindication to the full class of models put forward by Metz and Diekmann (1986) and Diekmann et al. (1998, 2001, 2003).

## Acknowledgments

This work benefitted from the support from the “Chaire Modélisation Mathématique et Biodiversité of Veolia Environnement—Ecole Polytechnique—Museum National d’Histoire Naturelle—Fondation X”.