Invasion and adaptive evolution for individual-based spatially structured populations



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.


Spatially structured population Adaptive evolution Stochastic individual-based model Birth-and-death point process Reflected diffusion Mutation and selection Nonlinear reaction–diffusion equation Nonlocal interaction and local interaction Clustering Polymorphism Invasion and evolution 

Mathematics Subject Classification (2000)

Primary 60J85 60K35 92D15 Secondary 92D25 35K60 


  1. 1.
    Aldous D. (1978). Stopping times and tightness. Ann. Probab. 6: 335–340 MATHMathSciNetGoogle Scholar
  2. 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 MATHCrossRefGoogle Scholar
  3. 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 CrossRefGoogle Scholar
  4. 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 MATHCrossRefMathSciNetGoogle Scholar
  5. 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 MATHCrossRefGoogle Scholar
  6. 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)Google Scholar
  7. 7.
    Desvillettes L., Ferrière R. and Prévost C. (2004). Infinite dimensional reaction–diffusion for population dynamics. Preprint CMLA, ENS Cachan, Google Scholar
  8. 8.
    Dieckmann U. and Doebeli M. (1999). On the origin of species by sympatric speciation. Nature 400: 354–357 CrossRefGoogle Scholar
  9. 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. 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. 11.
    Doebeli M. and Dieckmann U. (2003). Speciation along environmental gradients. Nature 421: 259–263 CrossRefGoogle Scholar
  12. 12.
    Durrett R. and Levin S. (1994). The importance of being discrete (and spatial). Theor. Popul. Biol. 46: 363–394 MATHCrossRefGoogle Scholar
  13. 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 CrossRefGoogle Scholar
  14. 14.
    Endler J.A. (1977). Geographic Variation, Speciation and Clines. Princeton university Press, Princeton Google Scholar
  15. 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 CrossRefGoogle Scholar
  16. 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 MATHCrossRefGoogle Scholar
  17. 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 CrossRefGoogle Scholar
  18. 18.
    Gobet E. (2001). Euler schemes and half-space approximations for the simulation of diffusion in a domain. ESAIM PS 5: 261–293 MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Grant P.R. and Grant B.R. (2002). Unpredictable evolution in a 30-year study of Darwin’s finches. Science 296: 707–711 CrossRefGoogle Scholar
  20. 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 CrossRefGoogle Scholar
  21. 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 CrossRefGoogle Scholar
  22. 22.
    Jacod J. and Shiryaev A.N. (1987). Limit Theorems for Stochastic Processes. Springer, Heidelberg MATHGoogle Scholar
  23. 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 MATHCrossRefGoogle Scholar
  24. 24.
    Lépingle D. (1995). Euler scheme for reflected stochastic differential equations. Math. Comp. Simul. 38: 119–126 MATHCrossRefGoogle Scholar
  25. 25.
    Lewis M.A. and Pacala S. (2000). Modeling and analysis of stochastic invasion processes. J. Math. Biol. 41: 387–429 MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    McGlade J. (1999). Advanced Ecological Theory: Principles and Applications. Blackwell, Oxford Google Scholar
  27. 27.
    Mayr E. (1963). Animal Species and Evolution. Harvard University Press, Cambridge Google Scholar
  28. 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 MATHGoogle Scholar
  29. 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. 30.
    Mollison D. (1977). Spatial contact models for ecological and epidemic spread. J. R. Stat. Soc. B 39: 283–326 MATHMathSciNetGoogle Scholar
  31. 31.
    Murray J.D. (1989). Mathematical Biology. Biomathematics texts 19. Springer, Berlin Google Scholar
  32. 32.
    Niwa H.S. (1994). Self-organizing dynamic-model of fish schooling. J. Theor. Biol. 171: 123–136 CrossRefGoogle Scholar
  33. 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 CrossRefGoogle Scholar
  34. 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)Google Scholar
  35. 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 CrossRefGoogle Scholar
  36. 36.
    Roelly-Coppoletta S. (1986). A criterion of convergence of measure-valued processes: application to measure-valued branching processes. Stochastics 17: 43–65 MATHMathSciNetGoogle Scholar
  37. 37.
    Roelly S. and Rouault A. (1990). Construction et propriétés de martingales des branchements spatiaux interactifs. Int. Stat. Rev. 58(2): 173–189 MATHCrossRefGoogle Scholar
  38. 38.
    Roughgarden J. (1972). Evolution of niche width. Am. Nat. 106: 683–718 CrossRefGoogle Scholar
  39. 39.
    Rudin W. (1987). Real and Complex Analysis, 3rd edn. McGraw-Hill, New York Google Scholar
  40. 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 MATHMathSciNetGoogle Scholar
  41. 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 MATHCrossRefMathSciNetGoogle Scholar
  42. 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 CrossRefGoogle Scholar
  43. 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. 44.
    Young W.R., Roberts A.J. and Stuhne G. (2001). Reproductive pair correlations and the clustering of organisms. Nature 412: 328–331 CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Institut National de Recherche en Informatique et en Automatique (INRIA)Sophia Antipolis cedexFrance
  2. 2.CMAP, ECOLE POLYTECHNIQUE, CNRSPalaiseau CedexFrance

Personalised recommendations