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

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.).

Appendices

Appendix A. Proof of Proposition 2

Let $f(t,\xi ,a)$ be a function of class $\mathcal C ^1$. From (7), we obtain

$$\begin{aligned} \langle Z_t,f\rangle&= \sum _{i\in V_0}f(t,\Xi ^i(t ; 0,\bar{Z}_0,S_0),a^i_0+t)+\int \limits _0^t \int \limits _\mathbb{N ^*\times \mathbb R _+} Q(ds,di,d\theta )\ {\small 1}\!\!1_{i\in V_{s_-}}\Big [\nonumber \\&{\small 1}\!\!1_{\theta \le m_1(i,s_-,\bar{Z}_{s_-},S_{s})} \Big (f(t,\Xi (t; s,\xi _0,\bar{Z}_{s}+\delta _{(I_{s_-}+1,\xi _0,0)},S_s),t-s)\nonumber \\&+ \sum _{j\in V_{s_-}}\big (f(t,\Xi ^j(t ; s,\bar{Z}_{s_-}+\delta _{(I_{s_-}+1,\xi _0,0)},S_s),a^j_s+(t-s))\nonumber \\&-f(t,\Xi ^j(t ; s,\bar{Z}_{s_-},S_s),a^j_s+(t-s))\big )\Big )\nonumber \\&+\, {\small 1}\!\!1_{m_1(i,s_-,\bar{Z}_{s_-},S_s)<\theta \le m_2(i,s_-,\bar{Z}_{s_-},S_{s})}\Big (\!-\! f(t,\Xi ^i(t ; s,\bar{Z}_{s_-},S_s),a^i_s\!+\!(t-s))\nonumber \\&+\, \sum _{j\in V_{s_-}, j\not = i}\big (f(t,\Xi ^j(t ; s,\bar{Z}_{s_-}-\delta _{(i,\xi ^i_{s_-},a^i_{s_-})},S_s),a^j_s+(t-s))\nonumber \\&-\, f(t,\Xi ^j(t ; s, \bar{Z}_{s_-},S_s),a^j_s+(t-s))\big ) \Big )\Big ]. \end{aligned}$$

(23)

Using (2), we have for any $s<t$:

$$\begin{aligned}&f(t,\Xi ^i(t ; s,\bar{Z}_s,S_s),a^i_s+(t-s))\\&\qquad \quad = f(s,\xi ^i_s,a^i_s)+\int \limits _s^t \Big (\frac{\partial f}{\partial u}+\frac{\partial f}{\partial a}(u,\Xi ^i(u ; s,\bar{Z}_s,S_s),a^i_s+u-s)\nonumber \\&\qquad \quad \quad + g(\Xi ^i(u ; s,\bar{Z}_s,S_s),S_u)\frac{\partial f}{\partial x}(u,\Xi ^i(u ; s,\bar{Z}_s,S_s),a^i_s+u-s)\Big )du \end{aligned}$$

Recall that we denoted by $T_k$, $k\ge 1$ the birth and death events in the population. By convention, we let $T_0=0$. Let us consider an individual $i$. Let $t_0\in \{T_k, k\ge 0\}$ be the birth time of the individual (or 0 if the individual is alive at time $0$) and $a^i_{t_0}$ be its age at time $t_0$ (0 if $t_0$ is the birth time). The sum of the terms in the r.h.s. of (23) associated with individual $i$ is equal to:

$$\begin{aligned}&f(t_0,\xi ^i_{t_0},a^i_{t_0})+\sum _{k\ge 0} \int \limits _{t\wedge T_k \vee t_0}^{t\wedge T_{k+1} \vee t_0} \Big (\frac{\partial f}{\partial u}+\frac{\partial f}{\partial a}(s,\Xi ^i(s ; T_k,\bar{Z}_{T_k},S_{T_k}),a^i_{t_0}+s-t_0)\\&\quad + g(s,\Xi ^i(s ; T_k,\bar{Z}_{T_k},S_{T_k}),S_s)\frac{\partial f}{\partial x}(s,\Xi ^i(s ; T_k,\bar{Z}_{T_k},S_{T_k}),a^i_{t_0}+s-t_0)\Big )ds\\&\quad -\int \limits _0^t \int \limits _\mathbb{N ^*\times \mathbb R _+} {\small 1}\!\!1_{j=i ; i\in V_{s_-}}{\small 1}\!\!1_{m_1(i,s_-,\bar{Z}_{s_-},S_{s})<\theta \le m_2(i,s_-,\bar{Z}_{s_-},S_{s})}f(s,\xi ^i_{s_-},a^i_{s_-})\Big ]dQ. \end{aligned}$$

The last integral correspond to the death term when individual $i$ is dead before $t$. Thus, (23) gives:

$$\begin{aligned}&\langle Z_t,f(t,.,.)\rangle =\sum _{i\in V_0} \Big [f(0,\xi ^i_{0},a^i_{0})+ \sum _{k\ge 0} \int \limits _{t\wedge T_k}^{t\wedge T_{k+1}} \Big (\frac{\partial f}{\partial u}+\frac{\partial f}{\partial a}(s,\Xi ^i(s ; T_k,\bar{Z}_{T_k},S_{T_k}),a^i_{0}+s)\\&\quad + g(s,\Xi ^i(s ; T_k,\bar{Z}_{T_k},S_{T_k}),S_s)\frac{\partial f}{\partial x}(s,\Xi ^i(s ; T_k,\bar{Z}_{T_k},S_{T_k}),a^i_{0}+s)\Big )ds \\&\quad - \int \limits _0^t \int \limits _\mathbb{N ^*\times \mathbb R _+} Q(ds,dj,d\theta ) {\small 1}\!\!1_{j=i}{\small 1}\!\!1_{i\in V_{s_-}}{\small 1}\!\!1_{m_1(i,s_-,\bar{Z}_{s_-},S_{s})<\theta \le m_2(i,s_-,\bar{Z}_{s_-},S_{s})}f(s,\xi ^i_{s_-},a^i_{s_-})\Big ]\\&\quad + \int \limits _0^t \int \limits _\mathbb{N ^*\times \mathbb R _+} Q(ds,di,d\theta )\ {\small 1}\!\!1_{i\in V_{s_-}\setminus V_0}\Big [\Big (f(s,\xi _0,0)\\&\quad +\sum _{k\ge 0} \int \limits _{t\wedge T_k\vee s}^{t\wedge T_{k+1}\vee s} \Big (\frac{\partial f}{\partial u}+\frac{\partial f}{\partial a}(u,\Xi ^{I_{s_-}+1}(u ; T_k,\bar{Z}_{T_k},S_{T_k}),u-s)\\&\quad + g(u,\Xi ^{I_{s_-}+1}(u ; T_k,\bar{Z}_{T_k},S_{T_k}),S_u)\frac{\partial f}{\partial x}(u,\Xi ^{I_{s_-}+1}(u ; T_k,\bar{Z}_{T_k},S_{T_k}),u-s)\Big )du\Big )\\&\quad \times {\small 1}\!\!1_{\theta \le m_1(i,s_-,\bar{Z}_{s_-},S_{s})} - f(s,\xi ^i_{s_-},a){\small 1}\!\!1_{m_1(i,s_-,\bar{Z}_{s_-},S_{s})<\theta \le m_2(i,s_-,\bar{Z}_{s_-},S_{s})}\Big ], \end{aligned}$$

where the first bracket corresponds to individuals alive at time $0$ and where the second bracket correspond to individuals born after time $0$. For $s<u$

$$\begin{aligned} \sum _{i\in V_{s_-}} \delta _{(\Xi ^i(u ; s,\bar{Z}_s,S_s),a^i_s+u-s)}(d\xi ,da)=Z_u(d\xi ,da) \end{aligned}$$

provided there has been no jumps between $s$ and $u$. Thus, we have:

$$\begin{aligned}&\langle Z_t,f(t,.,.)\rangle = \langle Z_0,f(0,.,.)\rangle \\&\qquad + \int \limits _0^t ds \int \limits _\mathbb{R _+^2}Z_s(d\xi ,da) \Big (\frac{\partial f}{\partial s}+\frac{\partial f}{\partial a}(s,\xi ,a)+ g(\xi ,S_s)\frac{\partial f}{\partial \xi }(s,\xi ,a)\Big )\\&\qquad + \int \limits _0^t \int \limits _\mathbb{R _+} \big ( f(s,\xi _0,0) \beta (\xi ,a,S_s)-f(s,\xi ,a)\mu (\xi ,a,S_s)\big )Z_s(d\xi ,da)\\&\qquad + \int \limits _0^t \int \limits _\mathbb{N ^*\times \mathbb R _+}{\small 1}\!\!1_{i\in V_{s_-}}\Big [f(s,\xi _0,0){\small 1}\!\!1_{\theta \le m_1(i,s_-,\bar{Z}_{s_-},S_{s})}\\&\qquad - f(s,\xi ^i_{s_-},a){\small 1}\!\!1_{m_1(i,s_-,\bar{Z}_{s_-},S_{s})<\theta \le m_2(i,s_-,\bar{Z}_{s_-},S_{s})}\Big ] \widetilde{Q}(ds,di,d\theta ), \end{aligned}$$

where $\widetilde{Q}(ds,di,d\theta )=Q(ds,di,d\theta )-ds\otimes n(di)\otimes d\theta$ is the compensated Poisson point measure associated with $Q$. The integral with respect to $\widetilde{Q}(ds,di,d\theta )$ provides the martingale $M^f$. This achieves the proof. $\square$

Appendix B. Sketch of the Proof of Proposition 3

When starting from (11) and using controls of moments as in Fournier and Méléard (2004), the proof is similar to the one in Tran (2008); Tran (2006).

Step 1 We start by noticing that under the Assumption (15), we have the following estimate (e.g. Champagnat et al. 2008):

$$\begin{aligned} \sup _{n\in \mathbb N ^*} \mathbb E \Big (\sup _{t\in [0,T]} \langle Z^n_t, 1\rangle ^2 \Big )<+\infty . \end{aligned}$$

(24)

Moreover, from Assumptions 1 and (5), the size of any individual is bounded on $[0,T]$ by $\bar{\xi }=\xi _0+\bar{g}T$ and there exists for every $\varepsilon$ a non random constant $\bar{S}_\varepsilon$ such that:

$$\begin{aligned} \sup _{n\in \mathbb N ^*}\mathbb P \Big (\sup _{t\in [0,T]}S^n_t>\bar{S}_\varepsilon \Big )<\varepsilon . \end{aligned}$$

(25)

From these estimates and Assumption 1 (ii), there exists a constant $A_\varepsilon \in (0,A)$ such that:

$$\begin{aligned} \sup _{n\in \mathbb N ^*}\mathbb P \Big (\sup _{t\in [0,T]}Z^n_t\big ([\xi _0,\bar{\xi }]\times [0,A_\varepsilon ]\big )>\varepsilon \Big )<\varepsilon . \end{aligned}$$

(26)

Step 2 It is easy to see that the limiting values of $(Z^n,S^n)_{n\in \mathbb N ^*}$ are necessary continuous. Let us check the $C$-tightness (e.g. Jacod and Shiryaev 1987) of $(Z^n,S^n)_{n\in \mathbb N ^*}$ in $\mathbb D ([0,T],\mathcal{M }_F(\mathbb R _+^2)\times \mathbb R _+)$. Using a criterion by Méléard and Roelly (1993) and given the compact containment that follows from Step 1, it is sufficient to prove the tightness of $(S^n)_{n\in \mathbb N ^*}$ and of the predictable finite variation part and martingale part of $(\langle Z^n,f\rangle )_{n\in \mathbb N ^*}$ for $f$ in $\mathcal C _b^1(\mathbb R _+^3,\mathbb R )$ (which contains the constant function equal to 1). This is obtained by using Aldous-Rebolledo criteria (e.g. Joffe and Métivier 1986)) and adapting the arguments of, for instance, Champagnat et al. (2008) and Tran (2008) using the estimates of Step 1.

Step 3 The identification of the martingale problem satisfied by the limiting values provides (16)–(17). Uniqueness of the solution of (16)–(17) stems from the Assumptions 1. As a consequence, there is a unique limiting value and we have convergence in distribution of $(Z^n,S^n)_{n\in \mathbb N ^*}$ to the solution $(\zeta ,\varrho )$. Since the latter is deterministic, the convergence is also a convergence in distribution.