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

•  “Bifurcation analysis of a size-structured population model : Application to oocyte dynamics and ovarian cycle", F. Clement, L. Fostier, R. Yvinec (2025), in SIAM Journal on Applied Dynamical Systems.
• "Rapid cell turnover to model adipocyte size distribution", L. Fostier, A. Dauger, R. Yvinec, M. Ribot, C. Audebert, H. Soula (2025). Accepted in Journal of Theoretical Biology 
• " 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