Multiscale population dynamics in reproductive biology: singular perturbation reduction in deterministic and stochastic models

Abstract

In this study, we describe different modeling approaches for ovarian follicle population dynamics, based on either ordinary (ODE), partial (PDE) or stochastic (SDE) differential equations, and accounting for interactions between follicles. We put a special focus on representing the population-level feedback exerted by growing ovarian follicles onto the activation of quiescent follicles. We take advantage of the timescale difference existing between the growth and activation processes to apply model reduction techniques in the framework of singular perturbations. We first study the linear versions of the models to derive theoretical results on the convergence to the limit models. In the nonlinear cases, we provide detailed numerical evidence of convergence to the limit behavior. We reproduce the main semi-quantitative features characterizing the ovarian follicle pool, namely a bimodal distribution of the whole population, and a slope break in the decay of the quiescent pool with aging.

Averaging of a Stochastic Slow-Fast Model for Population Dynamics: Application to the Development of Ovarian Follicles

“…In each compartment i, individuals may either progress to the next compartment i + 1, at rate λ i (except in the last compartment), or die at rate µ i . In addition, we complete the initial formulation of [1] by adding a birth term, at rate r 0 in the slow compartment X. After introducing a small parameter ε, rescaling time from t to t ε and the reserve size from X to εX, and multiplying by ε the rates λ 0 , µ 0 and r 0 , the model can be fully characterized by the following state transitions:…”

“…These immigration events occur at rate λ 0 (•)x > 0 (Hypothesis 2.2). Note that, by definition of M , M ≥ w k −v k for any k, and v (1)…”

“…The first formulation of the model, as a death and migration process without birth, was introduced in [1] to follow the changes in the population of ovarian follicles throughout lifespan. Ovarian follicles are dynamic endocrine units of the ovaries, enclosing a single germ cell.…”

“…In [1], taking advantage of the timescale difference between follicle activation and follicle growth, the authors conjectured the existence of a reduced deterministic limit model. In the current study, we derive rigorously the reduction on an extended model including birth events in the reserve.…”

Mathematical modeling approaches of cellular endocrinology within the hypothalamo-pituitary-gonadal axis

“…Recently, a new multiscale model of the whole process of follicle development on a lifespan horizon has been introduced in (Bonnet et al, 2020). This model is implemented in different formulations, all entering the framework of structured population dynamics: as a nonlinear compartmental ODE model, as a nonlocal and nonlinear PDE model based on transport equations, and as a stochastic Continuous time Markov Chain model with nonlinear intensities.…”