Stochastic competition between two populations in space

  • Simone Pigolotti
  • Roberto Benzi
  • Mogens H. Jensen
  • Prasad Perlekar
  • Federico Toschi
Part of the CISM International Centre for Mechanical Sciences book series (CISM, volume 553)

Abstract

We present a model describing spatial competition between two biological populations. Individuals belonging to the two populations diffuse in space, reproduce, and die as effect of competitions; all these processes are implemented stochastically. We focus on how the macroscopic equations for the densities of the two species can be derived within the formalism of the chemical master equations. We also compare the case in which the total density of individuals is kept fixed by constraint with a case in which it can fluctuate.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. D. A. Birch and W. A Young. A master equation for a spatial population model with pair interactions. Theo. Pop. Biol., 70(1):2642, 2006.Google Scholar
  2. J.F. Crow and M. Kimura. Introduction to Population Genetics Theory. Harper & Row Publishers, 1970.Google Scholar
  3. C. Doering, C. Mueller, and P. Smereka. Interacting particles, the stochastic fkpp equation, and duality. Physica A, 325:243–259, 2003.Google Scholar
  4. C.W. Gardiner. Handbook of Stochastic Methods. Springer, 2004.Google Scholar
  5. E. Hernandez-Garcia and C. Lopez. Clustering, advection and patterns in a model of population dynamics with neighborhood-dependent rates. Phys. Rev. E, 70(1):016216, 2004.Google Scholar
  6. M. Kimura. ”stepping stone” model of population. Ann. Rept. Nat. Inst. Genetics, 3:62–63, 1953.Google Scholar
  7. M. Kimura and G. H. Weiss. The stepping stone model of population structure and the decrease of genetic correlation with distance. Genetics, 49: 561–576, 1964.Google Scholar
  8. K.S. Korolev, M. Avlund, O. Hallatschek, and D.R. Nelson. Genetic demixing and evolutionary forces in the one-dimensional stepping stone model. Review of Modern Physics, 82:1691–1718, 2009.Google Scholar
  9. R. Law, D. J. Murrell, and U. Dieckmann. Population growth in space and time: the spatial logistic equation. Ecology, 84(1):252–262, 2003.Google Scholar
  10. J. D. Murray. Mathematical Biology: an Introduction. Springer, 2007.Google Scholar
  11. P. Perlekar, R. Benzi, S. Pigolotti, and F. Toschi. Particle algorithms for population dynamics in flows. Journal of Physics: Conference Series, 333:012013, 2011.Google Scholar
  12. S. Pigolotti, R. Benzi, M.H. Jensen, and D.R. Nelson. Population genetics in compressible flows. Physical Review Letters, 108:128102, 2012.Google Scholar
  13. S. Pigolotti, R. Benzi, P. Perlekar, M.H. Jensen, F. Toschi, and D.R. Nelson. Growth, competition and cooperation in spatial population genetics. Theoretical Population Biology, 84:72–86, 2013.Google Scholar
  14. H. Risken. The Fokker-Planck equation: Methods of Solution and Applications. Springer, Berlin, 1989.Google Scholar
  15. M. Vlad, L. L. Cavalli-Sforza, and J. Ross. Enhanced (hydrodynamic) transport induced by population growth in reaction-diffusion systems with application to population genetics. Proceedings of the National Academy of Sciences of the United States of America, 101(28):10249–10253, 2004.Google Scholar

Copyright information

© CISM, Udine 2014

Authors and Affiliations

  • Simone Pigolotti
    • 3
  • Roberto Benzi
    • 4
  • Mogens H. Jensen
    • 5
  • Prasad Perlekar
    • 6
  • Federico Toschi
    • 1
    • 2
  1. 1.Department of Physics, Department of Mathematics and Computer Science, and J.M. BurgerscentrumEindhoven University of TechnologyEindhovenThe Netherlands
  2. 2.CNR-IACRomeItaly
  3. 3.Dept. de Fisica i Eng. NuclearUniversitat Politecnica de Catalunya Edif. GAIATerrassaSpain
  4. 4.Dipartimento di FisicaUniversità di Roma “Tor Vergata” and INFNRomaItaly
  5. 5.The Niels Bohr InstitutUniversity of CopenhagenCopenhagenDenmark
  6. 6.Tata Institute of Fundamental ResearchCentre for Interdisciplinary SciencesHyderabadIndia

Personalised recommendations