Multiscale mathematical modeling of oogenesis in fish

Summary

This thesis focuses on studying the dynamics of size-structured population models, formulated as a transport partial differential equation with nonlocal terms in the boundary condition (recruitment of new cells), the transport velocity (cell growth), and the source term (cell death). We investigate the well-posedness of these models and their long-time behavior. 

Assuming that the growth rate excludes the interaction between the effects of cell size and nonlocal terms reduces the problem to studying abstract semi-linear Cauchy problems. For these problems, the Principle of Linearized Stability and the Hopf bifurcation theorem are already established and allow for the study of long-time behavior through linearization. For a non-separable growth rate, we examine stability using a pseudo-spectral approximation method and numerical bifurcation analysis tools. We apply these theoretical results to two biological contexts:
(i) the dynamics of female germ cells (oogenesis) in fish, for which a bifurcation analysis with respect to the recruitment rate reveals Hopf and saddle-node bifurcations corresponding to oscillatory and bistable solutions, respectively;

(ii) the dynamics of adipose cells, for which we demonstrate the existence of a unique steady state and its local stability. We then develop a nonparametric inference method for growth and recruitment rates based on Physics-Informed Neural Networks (PINNs), illustrated on both synthetic and experimental data from the two aforementioned biological contexts.

Finally, as part of a broader perspective on model inference from data, we explore a method for identifying dynamical systems from observed trajectories.

Keywords : size-structured population dynamics ; partial differential equations ; long-time behavior ; bifurcation analysis ;
; inverse problem ; dynamical systems inference ; reproductive biology

Contact

• Louis Fostier (UMR PRC)
• Linkedin

Publications

• Louis Fostier. Analyse et calibration de modèles de dynamique de populations structurées: application à des populations cellulaires. Mathématiques [math]. Université de Tours, 2025. Français. Thèse ⟨NNT : ⟩. ⟨tel-05400934⟩
• Louis Fostier, Aloïs Dauger, Romain Yvinec, Magali Ribot, Chloe Audebert, et al.. Rapid cell turnover to model adipocyte size distribution. Journal of Theoretical Biology, 2026, 618, pp.112311. ⟨10.1016/j.jtbi.2025.112311⟩. ⟨hal-05419068⟩
• Frédérique Clément, Louis Fostier, Romain Yvinec. Bifurcation analysis of a size-structured population model: Application to oocyte dynamics and ovarian cycle. SIAM Journal on Applied Dynamical Systems, 2025, 24 (Issue 3), pp.2427 - 2472. ⟨10.1137/24M1705147⟩. ⟨hal-04699357v2⟩
• Camilla Fiorini, Clément Flint, Louis Fostier, Emmanuel Franck, Reyhaneh Hashemi, et al.. Generalizing the SINDy approach with nested neural networks. 2024. ⟨hal-04557263v2⟩
• " Generalizing the SINDy approach with nested neural networks." C. Fiorini, C. Flint, L. Fostier, E. Franck, R. Hashemi, V. Michel-Dansac, W. Tenachi (2025). Accepted in ESAIM : Proceedings and Surveys