Skip to main content
SpringerLink
Log in

Convergence of an infinite dimensional stochastic process to a spatially structured trait substitution sequence

  • Published:
Stochastics and Partial Differential Equations: Analysis and Computations Aims and scope Submit manuscript

Abstract

We consider an individual-based spatially structured population for Darwinian evolution in an asexual population. The individuals move randomly on a bounded continuous space according to a reflected brownian motion. The dynamics involves also a birth rate, a density-dependent logistic death rate and a probability of mutation at each birth event. We study the convergence of the microscopic process in a long time scale when the population size grows to \(+\infty \) and the mutation probability decreases to 0. We prove the convergence towards a jump process that jumps in the infinite dimensional space containing the monomorphic stable spatial distributions. The proof requires specific studies of the microscopic model. First, we study the extinction time of the branching diffusion processes that approximate small size populations. Then, we examine the upper bound of large deviation principle around the deterministic large population limit of the microscopic process. Finally, we find a lower bound on the exit time of a neighborhood of a stationary spatial distribution.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Arnold, A., Desvillettes, L., Prévost, C.: Existence of nontrivial steady states for populations structured with respect to space and a continuous trait. Commun. Pure Appl. Anal. 11(1), 83–96 (2012)

    MathSciNet  MATH  Google Scholar 

  2. Berestycki, H., Hamel, F., Roques, L.: Analysis of periodically fragmented environment model: I-species persistence. J. Math. Biol. 51, 75–113 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Champagnat, N.: A microscopic interpretation for adaptive dynamics trait substitution sequence models. Stoch. Process. Appl. 116(8), 1127–1160 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Champagnat, N., Jabin, P.E., Méléard, S.: Adaptation in a stochastic multi-resources chemostat model. Journal de Mathématiques Pures et Appliquées 101(6), 755–788 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Champagnat, N., Méléard, S.: Invasion and adaptative evolution for individual-based spatially structured populations. J. Math. Biol. 55(2), 147–188 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Costa, M., Hauzy, C., Loeuille, N., Méléard S.: Stochastic eco-evolutionary model of a prey-predator community. arXiv preprint arXiv:1407.3069 (2014)

  7. Coville, J.: Convergence to equilibrium for positive solutions of some mutation-selection model. arXiv preprint arXiv:1308.6471 (2013)

  8. Dawson, D., Gartner, J.: Large deviations from the Mc-Kean–Vlasov limit for weakly interacting diffusions. Stochastics 20, 247–308 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications. Stochastic Modelling and Applied Probability. Springer, Berlin (1998)

  10. Desvillettes, L., Ferriére, R., Prévost, C.: Infinite dimensional reaction-diffusion for population dynamics. Preprint 4(3), 529–605 (2004)

  11. Doebeli, M., Dieckmann, U.: Speciation along environmental gradients. Nature 421, 259–263 (2003)

    Article  Google Scholar 

  12. Durrett, R., Levin, S.: Spatial aspects of interspecific competition. Theor. Popul. Biol. 53(1), 30–43 (1998)

    Article  MATH  Google Scholar 

  13. Endler, J.A.: Geographic Variation, Speciation, and Clines, vol. 2. Princeton University Press, Princeton (1977)

    Google Scholar 

  14. Fontbona, J.: Uniqueness for a weak nonlinear evolution equation and large deviations for diffusing particles with electrostatic repulsion. Stoch. Process. Appl. 112(1), 119–144 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  15. Freidlin, M., Wentzell, A.: Random Perturbations. Springer, Berlin (1984)

    Book  MATH  Google Scholar 

  16. Friedman, A.: Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs (1964)

    MATH  Google Scholar 

  17. Futuyma, D., Moreno, G.: The evolution of ecological specialization. Ann. Rev. Ecol. Syst. 19, 207–233 (1988)

    Article  Google Scholar 

  18. Graham, C., Méléard, S.: An upper bound of large deviations for a generalized star-shaped loss network. Markov Process. Relat. Fields 3(2), 199–224 (1997)

    MathSciNet  MATH  Google Scholar 

  19. Grant, P., Grant, B.: Unpredictable evolution in a 30-year study of darwin’s finches. Science 296, 707–711 (2002)

    Article  Google Scholar 

  20. Hastings, A.: Dynamics of a single species in a spatially varying environment: the stabilizing role of high dispersal rates. J. Math. Biol. 16(1), 49–55 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hastings, A.: Can spatial variation alone lead to selection for dispersal? Theor. Popul. Biol. 24(3), 244–251 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  22. Johnson, M., Gaines, M.: Evolution of dispersal: theoretical models and empirical tests using birds and mammals. Annu. Rev. Ecolo. Syst. 21, 449–480 (1990)

    Article  Google Scholar 

  23. Kassen, R.: The experimental evolution of specialists, generalists, and the maintenance of diversity. J. Evol. Biol. 15, 173–190 (2002)

    Article  Google Scholar 

  24. Leimar, O., Doebeli, M., Dieckmann, U.: Evolution of phenotypic clusters through competition and local adaptation along an environmental gradient. Evolution 62(4), 807–822 (2008)

    Article  Google Scholar 

  25. Leman, H., Méléard, S., Mirrahimi, S.: Influence of a spatial structure on the long time behavior of a competitive Lotka–Volterra type system. Discret. Contin. Dyn. Syst. Ser. B 20, 469–493 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  26. Leonard, C.: Large deviations for long range interacting particle systems with jumps. Annales de l’IHP Probabilités et statistiques 31, 289–323 (1995a)

    MathSciNet  MATH  Google Scholar 

  27. Léonard, C.: On large deviations for particle systems associated with spatially homogeneous boltzmann type equations. Probab. Theory Relat. Fields 101(1), 1–44 (1995b)

    Article  MathSciNet  MATH  Google Scholar 

  28. Léonard, C.: Convex conjugates of integral functionals. Acta Math. Hungar. 93(4), 253–280 (2001a)

    Article  MathSciNet  MATH  Google Scholar 

  29. Léonard, C.: Minimizers of energy functionals. Acta Math. Hungar. 93(4), 281–325 (2001b)

    Article  MathSciNet  MATH  Google Scholar 

  30. Metz, J.A.J., Geritz, S., Meszéna, G., Jacobs, F., Van Heerwaarden, J.S.: Adaptive dynamics, a geometrical study of the consequences of nearly faithful reproduction. Stoch. Spat. Struct. Dyn. Syst. 45, 183–231 (1996)

    MathSciNet  MATH  Google Scholar 

  31. Perthame, B., Souganidis, P.: Rare mutations limit of a steady state dispersion trait model. arXiv preprint arXiv:1505.03420 (2015)

  32. Polechová, J., Barton, N.H.: Speciation through competition: a critical review. Evolution 59(6), 1194–1210 (2005)

    Article  Google Scholar 

  33. Rao, M.M., Ren, Z.D.: Applications of Orlicz Spaces, vol. 250. CRC Press, New York (2002)

    MATH  Google Scholar 

  34. Roques, L.: Modéles de réaction-diffusion pour l’écologie spatiale. Editions Quae (2013)

  35. Tilman, D., Kareiva, P.M.: Spatial Ecology: The Role of Space in Population Dynamics and Interspecific Interactions, vol. 30. Princeton University Press, Princeton (1997)

    Google Scholar 

  36. Tran, V.: Large population limit and time behaviour of a stochastic particle model describing an age-structured population. ESAIM 12, 345–386 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  37. Wang, F., Yan, L.: Gradient estimate on convex domains and applications. Proc. Am. Math. Soc. 141(3), 1067–1081 (2013)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

I would like to thank S. Méléard for her guidance during my work. I also thank C. Léonard, G. Raoul, C. Tran and A. Veber for their help. I acknowledge partial support by the “Chaire Modélisation Mathématique et Biodiversité” of VEOLIA—École Polytechnique—MNHN—F.X.

Author information

Authors and Affiliations

  1. CMAP, Ecole Polytechnique, UMR 7641, Université Paris-Saclay, Route de Saclay, 91128, Palaiseau Cedex, France

    Hélène Leman

Authors
  1. Hélène Leman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hélène Leman.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Leman, H. Convergence of an infinite dimensional stochastic process to a spatially structured trait substitution sequence. Stoch PDE: Anal Comp 4, 791–826 (2016). https://doi.org/10.1007/s40072-016-0077-y

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40072-016-0077-y

Keywords

Mathematics Subject Classification

Search

Navigation