Modelapproximatie in transportdynamica

Mai Nguyen

Begeleider: Dr. Sander Hille

11 mei 2026

Recap: Transportdynamica

Huidige staat (positie en snelheid)

Positie \(\mathbf{x} \in \Omega \subseteq \mathbb{R}^n\)
Snelheidsvector \(\mathbf{v} \in V \subseteq \mathbb{R}^n\)

Regels voor veranderingen van de huidige staat

Poisson proces met parameter \(\lambda\)

Recap: Velocity-jump

|v| λ

Recap: Velocity-jump model

\(p(\mathbf{x},\mathbf{v},t)\): dichtheid van deeltjes op positie \(\mathbf{x}\), velocity \(\mathbf{v}\), tijdstip \(t\)

\(T(\mathbf{v},\mathbf{v}')\): turning kernel (kans op sprong van \(\mathbf{v}'\) naar \(\mathbf{v}\))

\[ \begin{aligned} \class{fragment}{\underbrace{\frac{\partial}{\partial t} p(\mathbf{x},\mathbf{v},t)}_{\text{verandering in tijd}} \;+\; \underbrace{\mathbf{v} \cdot \nabla_\mathbf{x} p(\mathbf{x},\mathbf{v},t)}_{\text{ruimtelijk transport}}} & \class{fragment}{= \underbrace{-\lambda\, p(\mathbf{x},\mathbf{v},t)}_{\text{sprong vanaf } \mathbf{v}} + \underbrace{\int_V \lambda\, T(\mathbf{v},\mathbf{v}')\, p(\mathbf{x},\mathbf{v}',t)\, d\mathbf{v}'}_{\text{sprongen naar } \mathbf{v}}} \\ & \class{fragment}{=: \mathcal{L} p} \end{aligned} \]

Recap: Velocity-jump op macroschaal

|v| λ

Recap: Velocity-jump op macroschaal

\[ \frac{\partial}{\partial t}q(\mathbf{x},t) = \nabla_\mathbf{x}\cdot \left(D\nabla_\mathbf{x} q(\mathbf{x},t)\right) \]

\(D\): diffusiecoëfficiënt

Onderzoeksvraag

Macroschaal: meetbaar
Microschaal: informatie over proces

Onderzoeksvraag
Hoe kunnen stochastische transportmodellen voor velocity-jump processen formeel gerelateerd worden aan deterministische PDE modellen?

Context: foretische interacties

Hitchhikers model

Hitchhikers: twee toestanden

Bewegingstoestand: \(s_m(\mathbf{x},\mathbf{v},t)\) dichtheid van individuen die bewegen met velocity \(\mathbf{v} \in V\)
Rusttoestand: \(s_i(\mathbf{x},\mathbf{v},t)\) dichtheid van individuen in pauze na een jump van velocity \(\mathbf{v}\in V\)

Bewegingstoestand

Rusttoestand

Hitchhikers model: algemeen

\(u(\mathbf{x},\mathbf{v},t)\): dichtheid van de transporters

\[\begin{equation} \begin{cases} \frac{\partial}{\partial t}s_i(\mathbf{x}, \mathbf{v},t) = \nu s_m (\mathbf{x}, \mathbf{v},t)- \mu(u) s_i(\mathbf{x}, \mathbf{v},t),\\[4pt] \frac{\partial}{\partial t}s_m(\mathbf{x}, \mathbf{v},t) + \mathbf{v} \cdot \nabla_\mathbf{x} s_m(\mathbf{x}, \mathbf{v},t)= -\nu s_m(\mathbf{x}, \mathbf{v},t) + \mu(u) s_i(\mathbf{x}, \mathbf{v},t) + \mathcal{L}s_m(\mathbf{x}, \mathbf{v},t), \end{cases}\end{equation}\]

met

\(\mathcal{L} p = -\lambda p + \lambda\int_V T(\mathbf{v},\mathbf{v}')~ d\mathbf{v}\)

Hitchhikers model: diffunderende transporters

Transporters diffunderen met diffusieconstante \(D\)

\[\begin{equation} \frac{\partial}{\partial t} u(\mathbf{x},t) = \nabla_\mathbf{x}(D\nabla_\mathbf{x}u(\mathbf{x},t)) \end{equation}\]

\[s_m(\mathbf{x},t) = \int_V s_m(\mathbf{x},\mathbf{v},t) ~ d\mathbf{v}, \quad s_i(\mathbf{x},t) = \int_V s_i(\mathbf{x},\mathbf{v},t) ~ d\mathbf{v},\quad u(\mathbf{x},t) = \int_V u(\mathbf{x},\mathbf{v},t) ~ d\mathbf{v} \]

\[\begin{equation} \begin{cases} \frac{\partial}{\partial t}s_i = \nu s_m - \mu(u) s_i,\\[4pt] \frac{\partial}{\partial t}s_m = -\nu s_m + \mu(u) s_i + \nabla_\mathbf{x} \left(D \nabla_\mathbf{x} s_m \right) \end{cases}\end{equation}\]

Parabolische schaling

\[\begin{equation} \chi = \varepsilon \mathbf{x}, \quad \tau = \varepsilon^2 t. \end{equation}\]

\[\begin{equation} \overline{s_m}(\chi, \tau) = s_m(\mathbf{x}, t),\quad \overline{s_i}(\chi, \tau) = s_i(\mathbf{x}, t), \quad \overline{u}(\chi, \tau) = u(\mathbf{x}, t) \end{equation}\]

\[\begin{equation} \begin{cases} \varepsilon^2 \frac{\partial}{\partial \tau} s_i= \nu\overline{s_i} - \mu (\bar{u})\bar{s_i},\\ \varepsilon^2 \frac{\partial}{\partial \tau} s_m = -\nu\overline{s_m} + \mu (\bar{u})\bar{s_i} + \varepsilon^2\nabla_\chi (D\nabla_\chi \overline{s_m}) \end{cases} \end{equation}\]

Expansies

\[\begin{equation} \overline{s_i} = \sum_{j=0}^k \varepsilon^j \overline{s_i}^{(j)}, \qquad \overline{s_m} = \sum_{j=0}^k \varepsilon^j \overline{s_m}^{(j)} \end{equation}\]

Invullen in

\[\begin{equation} \begin{cases} \varepsilon^2 \frac{\partial}{\partial \tau} s_i = \nu\overline{s_i} - \mu (\bar{u})\bar{s_i},\\ \varepsilon^2 \frac{\partial}{\partial \tau}s_m= -\nu\overline{s_m} + \mu (\bar{u})\bar{s_i} + \varepsilon^2\nabla_\chi (D\nabla_\chi \overline{s_m}) \end{cases} \end{equation}\]

Expansies

\[ \begin{aligned} \varepsilon^0: &\quad 0=\nu\overline{s_m}^{(0)} - \mu(\overline{u}) \overline{s_i}^{(0)} \\[2pt] \varepsilon^1: &\quad 0=\nu \overline{s_m}^{(1)} -\mu(\overline{u}) \overline{s_i}^{(1)}\\[2pt] \varepsilon^2: &\quad \frac{\partial}{\partial \tau} \overline{s_i}^{(0)} = \nu \overline{s_m}^{(2)} - \mu(\overline{u}) \overline{s_i}^{(2)}\\[2pt] \vdots \end{aligned} \qquad \begin{aligned} 0 &= -\nu\overline{s_m}^{(0)} - \mu(\overline{u}) \overline{s_i}^{(0)} \\[2pt] 0 &= -\nu \overline{s_m}^{(1)} - \mu(\overline{u}) \overline{s_i}^{(1)}\\[2pt] \frac{\partial}{\partial \tau} \overline{s_m}^{(0)} &= -\nu \overline{s_m}^{(2)} + \mu(\overline{u}) \overline{s_i}^{(2)} + \nabla_\chi \left(D \nabla_\chi \overline{s_m}^{(0)}\right) \\[2pt] \vdots \end{aligned} \]

Expansies

\[\begin{aligned} \varepsilon^0: &\quad 0=\nu\overline{s_m}^{(0)} - \mu(\overline{u})\overline{s_i}^{(0)} \\[2pt] \color{gray}{\varepsilon^1:} &\quad \color{gray}{0=\nu\overline{s_m}^{(1)} - \mu(\overline{u})\overline{s_i}^{(1)}} \\[2pt] \color{gray}{\varepsilon^2:} &\quad \color{gray}{\frac{\partial}{\partial \tau}\overline{s_i}^{(0)} = \nu\overline{s_m}^{(2)} - \mu(\overline{u})\overline{s_i}^{(2)}}\\[2pt] \color{gray}{\vdots} \end{aligned} \qquad \begin{aligned} 0 &= -\nu\overline{s_m}^{(0)} - \mu(\overline{u})\overline{s_i}^{(0)} \\[2pt] \color{gray}{0} &= \color{gray}{-\nu\overline{s_m}^{(1)} - \mu(\overline{u})\overline{s_i}^{(1)}} \\[2pt] \color{gray}{\frac{\partial}{\partial \tau}\overline{s_m}^{(0)}} &= \color{gray}{-\nu\overline{s_m}^{(2)} + \mu(\overline{u})\overline{s_i}^{(2)} + \nabla_\chi(D\nabla_\chi\overline{s_m}^{(0)})}\\[2pt] \color{gray}{\vdots} \end{aligned}\]

Voor \(s := \overline{s_i}^{(0)} + \overline{s_m}^{(0)}\) geldt:

\[\begin{equation} \begin{pmatrix} \overline{s_i}^{(0)}\\ \overline{s_m}^{(0)} \end{pmatrix}\left(\mathbf{\chi}, \tau\right) = \begin{pmatrix} \nu \\ \mu(\overline{u}(\mathbf{\chi}, \tau)) \end{pmatrix} \frac{s(\mathbf{\chi}, \tau)}{\mu(\overline{u} (\mathbf{\chi}, \tau))+\nu}. \end{equation}\]

\[\begin{aligned} \implies\overline{s_m}^{(0)}(\chi,\tau) & \class{fragment}{ =\frac{\mu(\overline{u}(\chi,\tau))}{\mu(\overline{u}(\chi,\tau)) + \nu} s(\chi,\tau)}\\ & \class{fragment}{=: \beta(\chi,\tau) s(\chi,\tau)} \end{aligned}\]

Expansies

\[\begin{aligned} \color{gray}{\varepsilon^0:} &\quad \color{gray}{0=\nu\overline{s_m}^{(0)} - \mu(\overline{u})\overline{s_i}^{(0)}} \\[2pt] \color{gray}{\varepsilon^1:} &\quad \color{gray}{0=\nu\overline{s_m}^{(1)} - \mu(\overline{u})\overline{s_i}^{(1)}} \\[2pt] {\varepsilon^2:} &\quad {\frac{\partial}{\partial \tau}\overline{s_i}^{(0)} = \nu\overline{s_m}^{(2)} - \mu(\overline{u})\overline{s_i}^{(2)}}\\[2pt] \color{gray}{\vdots} \end{aligned} \qquad \begin{aligned} \color{gray}{0} &= \color{gray}{-\nu\overline{s_m}^{(0)} - \mu(\overline{u})\overline{s_i}^{(0)}} \\[2pt] \color{gray}{0} &= \color{gray}{-\nu\overline{s_m}^{(1)} - \mu(\overline{u})\overline{s_i}^{(1)}} \\[2pt] {\frac{\partial}{\partial \tau}\overline{s_m}^{(0)}} &= {-\nu\overline{s_m}^{(2)} + \mu(\overline{u})\overline{s_i}^{(2)} + \nabla_\chi(D\nabla_\chi\overline{s_m}^{(0)})}\\[2pt] \color{gray}{\vdots} \end{aligned}\]

Invullen:

\[\begin{equation} \overline{s_m}^{(0)}(\chi,\tau) = \beta(\chi,\tau) s(\chi,\tau), \quad s = \overline{s_m}^{(0)} + \overline{s_i}^{(0)} \end{equation}\]

\[\begin{aligned} \frac{\partial}{\partial \tau} s(\chi,\tau) &\class{fragment}{= \nabla_\chi\left(D \nabla_\chi (\beta s)\right)}\\ &\class{fragment}{= \nabla_\chi\cdot(\underbrace{\beta(\chi,\tau) D\nabla_\chi s(\chi,\tau)}_{\text{Diffusie}} + \underbrace{(D\nabla_\chi \beta(\chi,\tau))s(\chi,\tau))}_{\text{Verplaatsing door meeliften}}} \end{aligned}\]

Simulaties: workflow

Simulaties: resultaten

Error-vergelijking PDE en agent-based

Momentensysteem

Meetbare grootheden: momenten

Momenten berekenen: \[\begin{equation} Q_\rho (\varphi,t) = \int_V\int_\Omega \varphi(\mathbf{x},\mathbf{v})\rho(\mathbf{x}, \mathbf{v},t) ~d\mathbf{x}~d\mathbf{v} \end{equation}\]

met \(\rho \in \{s_m, s_i\}\).

Momenten

Momenten in ruimte en velocity:

\(N_\rho(t) = Q(1,t)\): aantal deeltjes in staat \(\rho\)
\(D_\rho^2(t) = Q_\rho(||\mathbf{x}||^2,t)\): mean squared displacement (MSD) van deeltjes in staat \(\rho\)
\(B_\rho(t)=Q_\rho(\mathbf{v}\cdot \mathbf{x},t)\): mean velocity-displacement van deeltjes in staat \(\rho\)
\(V_\rho^2(t)=Q_\rho(||\mathbf{v}||^2,t)\): mean squared velocity van deeltjes in staat \(\rho\)

Voorbeeld moment-ODE

Algemeen model:

\[\begin{equation} \begin{cases} \frac{\partial}{\partial t}s_i = \nu s_m - \mu(u) s_i,\\[4pt] \frac{\partial}{\partial t}s_m + \mathbf{v} \cdot \nabla_\mathbf{x} s_m= -\nu s_m + \mu(u) s_i + \mathcal{L}s_m, \end{cases} \end{equation}\]

Voor \(u=u_0\) en \(\mu:=\mu(u_0)\):

\[\begin{aligned} \class{fragment}{\frac{d}{dt}D_{s_i}^2(t)} &\class{fragment}{= \frac{d}{dt}\left(\int_V\int_\Omega ||\mathbf{x}||^2 s_i(\mathbf{x},\mathbf{v},t)\, d\mathbf{x}\, d\mathbf{v} \right)}\\[4pt] &\class{fragment}{= \int_V\int_\Omega ||\mathbf{x}||^2 \frac{\partial}{\partial t} s_i(\mathbf{x}, \mathbf{v},t)\, d\mathbf{x}\, d\mathbf{v}}\\[4pt] &\class{fragment}{= \int_V\int_\Omega ||\mathbf{x}||^2\left(\nu s_m(\mathbf{x},\mathbf{v},t) -\mu s_i(\mathbf{x},\mathbf{v},t)\right)\, d\mathbf{x}\, d\mathbf{v}}\\[4pt] &\class{fragment}{= Q_{s_m}(||\mathbf{x}||^2,t) + \mu Q_{s_i}(||\mathbf{x}||^2,t)}\\[4pt] &\class{fragment}{= D_{s_m}^2(t) - \mu D_{s_i}^2(t)} \end{aligned}\]

Aannames

Constante persistentie: \(\overline{\mathbf{v}}(\mathbf{v}') = \int_V \mathbf{v} T(\mathbf{v}, \mathbf{v}') ~ d\mathbf{v} = \psi \mathbf{v}'\)
Constante gemiddelde mean squared speed: \(S_T^2(\mathbf{v}') = S_T^2 = \int_V ||\mathbf{v}||^2 T(\mathbf{v},\mathbf{v}')~ d\mathbf{v}\)
Massabehoud: \(N_{s_m}(t) + N_{s_i}(t)=N_0\)

Gesloten momentensysteem

Voor \(u=u_0\) en \(\mu:=\mu(u_0)\):

\[\begin{equation} \begin{cases} \frac{dD_{s_i}^2(t)}{dt} & = \nu D_{s_m}(t) - \mu D_{s_i}^2\\[0.5mm] \frac{dD_{s_m}^2(t)}{dt} & = 2B_{s_m}(t)-\frac{dD^2_{s_{i}}(t)}{dt},\\[1mm] \frac{dB_{s_i}(t)}{dt} & = \nu B_{s_m}(t)- \mu B_{s_i}(t),\\[0.5mm] \frac{dB_{s_m}(t)}{dt} & = V_{s_m}^2(t)-\frac{d}{dt}B_{s_i}(t)+\lambda \psi B_{s_m},\\[0.5mm] \frac{dV_{s_i}^2(t)}{dt} & =\nu V_{s_m}^2(t) -\mu V_{s_i}^2(t)\\[0.5mm] \frac{dV_{s_m}^2(t)}{dt} & =-\nu V_{s_m}(t) -\frac{dV^2_{s_i}(t)}{dt}+\lambda S^2_T N_{s_m}(t),\\[0.5mm] \frac{dN_{s_m}(t)}{dt} & = - \frac{dN_{s_i}(t)}{dt} = - \nu N_{s_m}(t) + \mu N_{s_i}(t). \end{cases} \end{equation}\]

Mean squared displacement

Bij diffusie: lineair in tijd met richtingscoëfficiënt \(2nD_{{\text{eff}}}\).

\(n\): aantal dimensies
\(D_{\text{eff}}\): effectieve diffusieconstante (\(\beta \cdot D\))

Overzicht

Vervolg

MSD vergelijken tussen simulaties en momentensysteem
Generaliseren: loslaten aanname Markoviaanse dynamiek

Bedankt voor de aandacht!

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References

Hawkins, Noel. 2024. ‘Crab hitches a ride on the back of a jellyfish’. https://www.bbc.com/news/videos/c0krq7dj840o.

Hillen, Thomas, en Hans G. Othmer. 2000. ‘The Diffusion Limit of Transport Equations Derived from Velocity-Jump Processes’. SIAM Journal on Applied Mathematics 61 (3): 751–75. http://www.jstor.org/stable/3061749.

Kettmayer, Constanza, Enrico Gratton, en Laura C Estrada. 2023. ‘Comparison of MSD analysis from single particle tracking with MSD from images. Getting the best of both worlds’. Methods and Applications in Fluorescence 12 (1): 015001. https://doi.org/10.1088/2050-6120/acfd7e.

Morffew, Andy. 2019. ‘Hummingbird Flower Mites’. https://www.flickr.com/photos/andymorffew/48875780486.

Muok, Alise R, Dennis Claessen, en Ariane Briegel. 2021. ‘Microbial hitchhiking: how Streptomyces spores are transported by motile soil bacteria’. The ISME Journal 15 (9): 2591–2600. https://doi.org/10.1038/s41396-021-00952-8.

Wright, Oliver. 2021. ‘Question mark in the sky’. https://x.com/OW_Photography/status/1471209887287939078/photo/1.