Abstract

We analyze a birth, migration, and death stochastic process modeling the dynamics of a finite population, in which individuals transit unidirectionally across successive compartments. The model is formulated as a continuous-time Markov chain, whose transition matrix involves multiscale effects; the whole (or part of the) population affects the rates of individual birth, migration, and death events. Using the slow-fast property of the model, we prove the existence and uniqueness of the limit model in the framework of stochastic singular perturbations. The derivation of the limit model is based on compactness and coupling arguments. The uniqueness is handled by applying the ergodicity theory and studying a dedicated Poisson equation. The limit model consists of an ordinary differential equation ruling the dynamics of the first (slow) compartment, coupled with a quasi-stationary distribution in the remaining (fast) compartments, which averages the contribution of the fast component of the Markov chain on the slow one. We illustrate numerically the convergence and highlight the relevance of dealing with nonlinear event rates for our application in reproductive biology. The numerical simulations involve a simple integration scheme for the deterministic part, coupled with the nested algorithm to sample the quasi-stationary distribution.

Keywords

  1. continuous-time Markov chain
  2. singular perturbations
  3. stochastic coupling techniques
  4. Foster--Lyapunov criterion
  5. nested algorithm
  6. reproductive biology

MSC codes

  1. 92D25
  2. 60G10
  3. 60J28

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Supplementary Material


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Index of Supplementary Materials

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

Authors: G. Ballif, F. Clement, and R. Yvinec

File: M140961SupMat.pdf

Type: PDF

Contents: Technical parts of the proofs and additional figures.

References

1.
C. Bonnet, K. Chahour, F. Clément, M. Postel, and R. Yvinec, Multiscale population dynamics in reproductive biology: Singular perturbation reduction in deterministic and stochastic models, ESAIM Proc. Surveys, 67 (2020), pp. 72--99.
2.
F. Clément and D. Monniaux, Mathematical modeling of ovarian follicle development: A population dynamics viewpoint, Curr. Opin. Endocr. Metab. Res., 18 (2021), pp. 54--61.
3.
J. E. Coxworth and K. Hawkes, Ovarian follicle loss in humans and mice: Lessons from statistical model comparison, Hum. Reprod., 25 (2010), pp. 1796--1805.
4.
J. Donnez and M.-M. Dolmans, Transplantation of ovarian tissue, Best Pract. Res. Clin. Obstet. Gynaecol., 28 (2014), pp. 1188--1197.
5.
P. Dürrenberger, A. Gupta, and M. Khammash, A finite state projection method for steady-state sensitivity analysis of stochastic reaction networks, Int. J. Chem. Phys., 150 (2019), 134101.
6.
S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence, Wiley Ser. Probab. Stat. 282, Wiley-Interscience, New York, 2009.
7.
M. J. Faddy and R. G. Gosden, Physiology: A mathematical model of follicle dynamics in the human ovary, Hum. Reprod., 10 (1995), pp. 770--775.
8.
M. J. Faddy, E. C. Jones, and R. G. Edwards, An analytical model for ovarian follicle dynamics, J. Exp. Zool., 197 (1976), pp. 173--185.
9.
J. K. Findlay, K. J. Hutt, M. Hickey, and R. A. Anderson, How is the number of primordial follicles in the ovarian reserve established, Biol. Reprod., 93 (2015), pp. 111, 1--7.
10.
D. T. Gillespie, A general method for numerically simulating the stochastic time evolution of coupled chemical reactions, J. Comput. Phys., 22 (1976), pp. 403--434.
11.
P. W. Glynn and S. P. Meyn, A Liapounov bound for solutions of the Poisson equation, Ann. Probab., 24 (1996), pp. 916--931.
12.
A. Gougeon, Regulation of ovarian follicular development in primates: Facts and hypotheses, Endocr. Rev., 11 (1996), pp. 121--155.
13.
H.-W. Kang and T. G. Kurtz, Separation of time-scales and model reduction for stochastic reaction networks, Ann. Appl. Probab., 23 (2013), pp. 529--583.
14.
T. G. Kurtz, Averaging for martingale problems and stochastic approximation, in Applied Stochastic Analysis, I. Karatzas and D. Ocone, eds., Springer, Cham, 1992, pp. 186--209.
15.
T. Lindvall, Lectures on the Coupling Method, Courier Corporation, New York, 2002.
16.
S. P. Meyn and R. L. Tweedie, Stability of Markovian processes \textIII: Foster-Lyapunov criteria for continuous-time processes, Adv. in Appl. Probab., 25 (1993), pp. 518--548.
17.
S. Méléard and V. Bansaye, Some Stochastic Models for Structured Populations: Scaling Limits and Long Time Behavior, Springer, Cham, 2015.
18.
M. F. Neuts and F. P. Kelly, Reversibility and stochastic networks, J. Amer. Statist. Assoc., 76 (1981), p. 492.
19.
G. B. Schaalje and H. R. Van der Vaart, Relationships among recent models for insect population dynamics with variable rates of development, J. Math. Biol., 27 (1989), pp. 399--428.
20.
M. Strugarek, L. Dufour, N. Vauchelet, L. Almeida, B. Perthame, and D. A. M. Villela, Oscillatory regimes in a mosquito population model with larval feedback on egg hatching, J. Biol. Dyn., 13 (2019), pp. 269--300.
21.
C. Thibault, M.-C. Levasseur, and R. H. F. Hunter, Reproduction in Mammals and Man, INRA Editions, Versailles, France, 2001.
22.
W. H. Wallace and T. W. Kelsey, Human ovarian reserve from conception to the menopause, PLoS One, 5 (2010), e8772.
23.
E. Weinan, D. Liu, and E. Vanden-Eijnden, Nested stochastic simulation algorithms for chemical kinetic systems with multiple time scales, J. Comput. Phys., 221 (2007), pp. 158--180.
24.
X. Zhu, D. B. Finlay, M. Glass, and S. B. Duffull, Model-free and kinetic modelling approaches for characterising non-equilibrium pharmacological pathway activity: Internalisation of cannabinoid CB\text1 receptors, British J. Pharmacol., 176 (2019), pp. 2593--2607.

Information & Authors

Information

Published In

cover image SIAM Journal on Applied Mathematics
SIAM Journal on Applied Mathematics
Pages: 359 - 380
ISSN (online): 1095-712X

History

Submitted: 5 April 2021
Accepted: 1 November 2021
Published online: 28 February 2022

Keywords

  1. continuous-time Markov chain
  2. singular perturbations
  3. stochastic coupling techniques
  4. Foster--Lyapunov criterion
  5. nested algorithm
  6. reproductive biology

MSC codes

  1. 92D25
  2. 60G10
  3. 60J28

Authors

Affiliations

Frédérique Clément

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