Weak Convergence of a Mass-Structured Individual-Based Model

Abstract

We propose a model of chemostat where the bacterial population is individually-based, each bacterium is explicitly represented and has a mass evolving continuously over time. The substrate concentration is represented as a conventional ordinary differential equation. These two components are coupled with the bacterial consumption. Mechanisms acting on the bacteria are explicitly described (growth, division and washout). Bacteria interact via consumption. We set the exact Monte Carlo simulation algorithm of this model and its mathematical representation as a stochastic process. We prove the convergence of this process to the solution of an integro-differential equation when the population size tends to infinity. Finally, we propose several numerical simulations.

Appendix: Skorohod Topology

The space of finite measures ${\mathcal {M}}_{F}(\mathcal {X})$ on ${\mathcal {X}}$ is equipped with the topology of the weak convergence, that is the smallest topology for which the applications $\zeta \rightarrow \langle \zeta ,f \rangle =\int _{{\mathcal {X}}}f(x)\,\zeta ({\mathrm{d }}x)$ are continuous for any $f\in \mathcal {C}(\mathcal {X})$. This topology is metrized by the Prokhorov metric:

$$\begin{aligned} d_\mathrm{\tiny \mathrm PR }(\zeta ,\zeta ')&\mathop {=}\limits ^\mathrm{\tiny def }\inf \Bigl \{ \epsilon >0\quad \zeta (F)\le \zeta '(F^\epsilon )+\epsilon \\&\quad \zeta '(F)\le \zeta (F^\epsilon )+\epsilon \,,\ \mathrm{ for all closed }F\subset {\mathcal {X}}\Bigr \}, \end{aligned}$$

where $F^\epsilon \mathop {=}\limits ^\mathrm{\tiny def }\{x\in {\mathcal {X}};\inf _{y\in F}|x-y|<\epsilon \}$ (see [7, Appendix A2.5]). The Prokhorov distance is bounded by the distance of the total variation $d_\mathrm{\tiny \mathrm TV }(\zeta ,\zeta ')=\Vert \zeta -\zeta ' \Vert _\mathrm{\tiny TV }$ associated with the norm defined by:

$$\begin{aligned} \Vert \zeta \Vert _\mathrm{\tiny TV } \mathop {=}\limits ^\mathrm{\tiny def }\sup _{A\in {\mathcal {B}}({\mathcal {X}})}|\zeta (A)+\zeta (A^c)| = \zeta _{+}({\mathcal {X}})+ \zeta _{-}({\mathcal {X}}) = \sup _{\begin{array}{c} f\mathrm{ continuous }\\ \left\| f \right\| _{\infty }\le 1 \end{array}} |\langle \zeta ,f \rangle | \end{aligned}$$

for any finite and signed measure $\zeta$ where $\zeta =\zeta _{+}-\zeta _{-}$ is the Hahn–Jordan decomposition of $\zeta$.

The space $\mathcal D([0,T],{\mathcal {M}}_{F}(\mathcal {X}))$ is equipped with the Skorohod metric $d_\mathrm{\tiny \mathrm S }$. Instead of giving the definition of this metric [12, Eq. (5.2) p. 117] we recall a characterization of the convergence for this metric given in [12, Proposition 5.3, p. 119].

A sequence $(\zeta ^{n})_{n\in \mathbb N}$ converges to $\zeta$ in ${\mathcal {D}}([0,T],{\mathcal {M}}_{F}(\mathcal {X}))$, i.e. $d_\mathrm{\tiny \mathrm S }(\zeta ^{n},\zeta )\rightarrow 0$, if and only if there exists a sequence $\lambda _{n}(t)$ of time change functions (i.e. strictly increasing bijective functions on $[0,T]$, with $\lambda _{n}(0)=0$ and $\lambda _{n}(T)=T$) satisfying:

$$\begin{aligned} \sup _{0\le t\le T} d_\mathrm{\tiny \mathrm PR }({\zeta _t^{n},\zeta _{\lambda _{n}(t)}}) \xrightarrow [n\rightarrow \infty ]{} 0 \end{aligned}$$

(25)

and

$$\begin{aligned} \sup _{0\le t\le T}|\lambda _{n}(t)-t|\rightarrow 0. \end{aligned}$$

(26)

If $(\zeta ^{n})_{n\in \mathbb N}$ converges to $\zeta$ in ${\mathcal {D}}([0,T],{\mathcal {M}}_{F}(\mathcal {X}))$ and if $\zeta \in {\mathcal {C}}([0,T],{\mathcal {M}}_{F}({\mathcal {X}}))$ then in:

$$\begin{aligned} \sup _{0\le t\le T}d_\mathrm{\tiny \mathrm PR }(\zeta ^{n}_{t},\zeta _{t}) \le \sup _{0\le t\le T}d_\mathrm{\tiny \mathrm PR }(\zeta ^{n}_{t},\zeta _{\lambda _n(t)}) + \sup _{0\le t\le T}d_\mathrm{\tiny \mathrm PR }(\zeta _{\lambda _n(t)},\zeta _{t}) \end{aligned}$$

the first term of the right-hand side tends to 0 because of (25); the second one tends to 0 because of (26) and the uniform continuity of $\zeta$ in $[0,T]$. This proves that $\zeta ^{n}$ converges to $\zeta$ in ${\mathcal {D}}([0,T],{\mathcal {M}}_{F}(\mathcal {X}))$ also for the uniform metric.