Skip to main content
SpringerLink
Log in

Invasion and adaptive evolution for individual-based spatially structured populations

  • Published:
Journal of Mathematical Biology Aims and scope Submit manuscript

Abstract

The interplay between space and evolution is an important issue in population dynamics, that is particularly crucial in the emergence of polymorphism and spatial patterns. Recently, biological studies suggest that invasion and evolution are closely related. Here, we model the interplay between space and evolution starting with an individual-based approach and show the important role of parameter scalings on clustering and invasion. We consider a stochastic discrete model with birth, death, competition, mutation and spatial diffusion, where all the parameters may depend both on the position and on the phenotypic trait of individuals. The spatial motion is driven by a reflected diffusion in a bounded domain. The interaction is modelled as a trait competition between individuals within a given spatial interaction range. First, we give an algorithmic construction of the process. Next, we obtain large population approximations, as weak solutions of nonlinear reaction–diffusion equations. As the spatial interaction range is fixed, the nonlinearity is nonlocal. Then, we make the interaction range decrease to zero and prove the convergence to spatially localized nonlinear reaction–diffusion equations. Finally, a discussion of three concrete examples is proposed, based on simulations of the microscopic individual-based model. These examples illustrate the strong effects of the spatial interaction range on the emergence of spatial and phenotypic diversity (clustering and polymorphism) and on the interplay between invasion and evolution. The simulations focus on the qualitative differences between local and nonlocal interactions.

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

Similar content being viewed by others

References

  1. Aldous D. (1978). Stopping times and tightness. Ann. Probab. 6: 335–340

    MATH  MathSciNet  Google Scholar 

  2. Bolker B. and Pacala S.W. (1997). Using moment equations to understand stochastically driven spatial pattern formation in ecological systems. Theor. Popul. Biol. 52: 179–197

    Article  MATH  Google Scholar 

  3. Bolker B.M. and Pacala S.W. (1999). Spatial moment equations for plant competition: understanding spatial strategies and the advantages of short dispersal. Am. Nat. 153: 575–602

    Article  Google Scholar 

  4. Bossy M., Gobet E. and Talay D. (2004). A symmetrized Euler scheme for an efficient approximation of reflected diffusions. J. Appl. Probab. 41: 877–889

    Article  MATH  MathSciNet  Google Scholar 

  5. Champagnat N., Ferrière R. and Méléard S. (2006). Unifying evolutionary dynamics: from individual stochastic processes to macroscopic models. Theor. Popul. Biol. 69: 297–321

    Article  MATH  Google Scholar 

  6. Champagnat, N., Ferrière, R., Méléard, S.: Individual-based probabilistic models and various time scaling approximations in adaptive evolution. In: Proceedings of the 5th Conference on Stochastic Analysis, Random Fields and Applications, Ascona 2005, Prog. Prob. Ser., Birkauser (to appear) (2006)

  7. Desvillettes L., Ferrière R. and Prévost C. (2004). Infinite dimensional reaction–diffusion for population dynamics. Preprint CMLA, ENS Cachan,

    Google Scholar 

  8. Dieckmann U. and Doebeli M. (1999). On the origin of species by sympatric speciation. Nature 400: 354–357

    Article  Google Scholar 

  9. Dieckmann U. and Law R. (2000). Relaxation projections and the method of moments. In: Dieckmann, U., Law, R. and Metz, J.A.J. (eds) The Geometry of Ecological Interactions: Symplifying Spatial Complexity, pp 412–455. Cambridge University Press, Cambridge

    Google Scholar 

  10. Dieckmann U., Law R. and Metz J.A.J. (2000). The Geometry of Ecological Interactions: Symplifying Spatial Complexity. Cambridge University Press, Cambridge

    Google Scholar 

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

    Article  Google Scholar 

  12. Durrett R. and Levin S. (1994). The importance of being discrete (and spatial). Theor. Popul. Biol. 46: 363–394

    Article  MATH  Google Scholar 

  13. Durrett R. and Levin S. (1994). Stochastic spatial models: a user’s guide to ecological applications. Phil. Trans. R. Soc. Lond. B 343: 329–350

    Article  Google Scholar 

  14. Endler J.A. (1977). Geographic Variation, Speciation and Clines. Princeton university Press, Princeton

    Google Scholar 

  15. Flierl G., Grünbaum D., Levin S. and Olson D. (1999). From individuals to aggregations: the interplay between behaviour and physics. J. Theor. Biol. 196: 397–454

    Article  Google Scholar 

  16. Fournier N. and Méléard S. (2004). A microscopic probabilistic description of a locally regulated population and macroscopic approximations. Ann. Appl. Prob. 14: 1880–1919

    Article  MATH  Google Scholar 

  17. Geritz S.A.H., Metz J.A.J., Kisdi E. and Meszena G. (1997). The dynamics of adaptation and evolutionary branching. Phys. Rev. Lett. 78: 2024–2027

    Article  Google Scholar 

  18. Gobet E. (2001). Euler schemes and half-space approximations for the simulation of diffusion in a domain. ESAIM PS 5: 261–293

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  20. Hassel M.P. and May R.M. (1974). Aggregation in predators and insect parasites and its effects on stability. J. Anim. Ecol. 43: 567–594

    Article  Google Scholar 

  21. Hassel M.P. and Pacala S.W. (1990). Heterogeneity and the dynamics of host parasitoid interactions. Phil. Trans. R. Soc. Lond. B 330: 203–220

    Article  Google Scholar 

  22. Jacod J. and Shiryaev A.N. (1987). Limit Theorems for Stochastic Processes. Springer, Heidelberg

    MATH  Google Scholar 

  23. Joffe A. and Métivier M. (1986). Weak convergence of sequences of semimartingales with applications to multitype branching processes. Adv. Appl. Probab. 18: 20–65

    Article  MATH  Google Scholar 

  24. Lépingle D. (1995). Euler scheme for reflected stochastic differential equations. Math. Comp. Simul. 38: 119–126

    Article  MATH  Google Scholar 

  25. Lewis M.A. and Pacala S. (2000). Modeling and analysis of stochastic invasion processes. J. Math. Biol. 41: 387–429

    Article  MATH  MathSciNet  Google Scholar 

  26. McGlade J. (1999). Advanced Ecological Theory: Principles and Applications. Blackwell, Oxford

    Google Scholar 

  27. Mayr E. (1963). Animal Species and Evolution. Harvard University Press, Cambridge

    Google Scholar 

  28. Méléard S. and Roelly S. (1993). Sur les convergences étroite ou vague de processus à valeurs mesures. C. R. Acad. Sci. Paris STr. I Math. 317: 785–788

    MATH  Google Scholar 

  29. Metz J.A.J., Geritz S.A.H., Meszéna G., Jacobs F.A.J. and Heerwaarden J.S. (1996). Adaptive dynamics, a geometrical study of the consequences of nearly faithful reproduction. In: Verduyn Lunel, S.M. (eds) Stochastic and Spatial Structures of Dynamical Systems., pp 183–231. North Holland, Amsterdam

    Google Scholar 

  30. Mollison D. (1977). Spatial contact models for ecological and epidemic spread. J. R. Stat. Soc. B 39: 283–326

    MATH  MathSciNet  Google Scholar 

  31. Murray J.D. (1989). Mathematical Biology. Biomathematics texts 19. Springer, Berlin

    Google Scholar 

  32. Niwa H.S. (1994). Self-organizing dynamic-model of fish schooling. J. Theor. Biol. 171: 123–136

    Article  Google Scholar 

  33. Phillips B.L., Brown G.P., Webb J.K. and Shine R. (2006). Invasion and the evolution of speed in toads. Nature 439: 803

    Article  Google Scholar 

  34. Prévost, C.: Applications des équations aux dérivées partielles aux problèmes de dynamique des populations et traitement numérique. PhD thesis, Université d’Orléans, France (2004)

  35. Rand D.A., Keeling M.J. and Wilson H.B. (1995). Invasion, stability and evolution to criticality in spatially extended, artificial host–pathogen ecologies. Proc. R. Soc. Lond. B 259: 55–63

    Article  Google Scholar 

  36. Roelly-Coppoletta S. (1986). A criterion of convergence of measure-valued processes: application to measure-valued branching processes. Stochastics 17: 43–65

    MATH  MathSciNet  Google Scholar 

  37. Roelly S. and Rouault A. (1990). Construction et propriétés de martingales des branchements spatiaux interactifs. Int. Stat. Rev. 58(2): 173–189

    Article  MATH  Google Scholar 

  38. Roughgarden J. (1972). Evolution of niche width. Am. Nat. 106: 683–718

    Article  Google Scholar 

  39. Rudin W. (1987). Real and Complex Analysis, 3rd edn. McGraw-Hill, New York

    Google Scholar 

  40. Sato K. and Ueno T. (1965). Multi-dimensional diffusion and the Markov process on the boundary. J. Math. Kyoto Univ. 4(3): 529–605

    MATH  MathSciNet  Google Scholar 

  41. Shepp L.A. (1979). The joint density of the maximum and its location for a Wiener process with drift. J. Appl. Probab. 16: 423–427

    Article  MATH  MathSciNet  Google Scholar 

  42. Thomas C.D., Bodsworth E.J., Wilson R.J., Simmons A.D., Davies Z.G., Musche M. and Conradt L. (2001). Ecological and evolutionary processes at expanding range margins. Nature 411: 577–581

    Article  Google Scholar 

  43. Tilman D. and Kareiva P. (1996). Spatial Ecology: The Role of Space in Population Dynamics and Interspecific Interactions. Princeton University Press, Princeton

    Google Scholar 

  44. Young W.R., Roberts A.J. and Stuhne G. (2001). Reproductive pair correlations and the clustering of organisms. Nature 412: 328–331

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut National de Recherche en Informatique et en Automatique (INRIA), 2004 route des Lucioles, BP 93, 06902, Sophia Antipolis cedex, France

    Nicolas Champagnat

  2. CMAP, ECOLE POLYTECHNIQUE, CNRS, Route de Saclay, 91128, Palaiseau Cedex, France

    Sylvie Méléard

Authors
  1. Nicolas Champagnat
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sylvie Méléard
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sylvie Méléard.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Champagnat, N., Méléard, S. Invasion and adaptive evolution for individual-based spatially structured populations. J. Math. Biol. 55, 147–188 (2007). https://doi.org/10.1007/s00285-007-0072-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00285-007-0072-z

Keywords

Mathematics Subject Classification (2000)

Search

Navigation