Abstract
We derive the mean field equations of a simple spatial host-pathogen, or predator-prey, model that has been shown to display interesting evolutionary properties. We compare these equations, and the equations including pair-correlations, with the low-density approximations derived by other authors. We study the process of invasion by a mutant pathogen, both in the mean field and in the pair approximation, and discuss our results with respect to the spatial model. Both the mean field and pair correlation approximations do not capture the key spatial behaviors—the moderation of exploitation due to local extinctions, preventing the pathogen from causing its own extinction. However, the results provide important hints about the mechanism by which the local extinctions occur.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
D. Tilman and P. Kareiva, Spatial Ecology: The Role of Space in Population Dynamics and Interspecific Interactions (Princeton Univ. Press, Princeton, NJ, 1997).
H. Sayama, L. Kaufman, and Y. Bar-Yam, Phys. Rev. E 62:7065(2000)
H. Sayama, M. A. M. de Aguiar, Y. Bar-Yam, and M. Baranger, Phys. Rev. E 65:051919(2002)
E. M. Rauch, H. Sayama, and Y. Bar-Yam, Phys. Rev. Lett. 88:228101(2002)
E. M. Rauch, H. Sayama, and Y. Bar-Yam, J. Theor. Biol. 221655.
Y. Bar-Yam, Dynamics of complex systems (Perseus Books, Cambridge, Mass., 1997).
A. Okubo, Diffusion and ecological problems: Mathematical Models (Springer Verlag, New York, 1980).
M. Gyllenberg, K. Parvinen, and U. Dieckmann, J. Math. Biol. 45:79(2002)
K. Tainaka, J. Phys. Soc. Japan 57:2588(1988).
H. N. Comins, M. P. Hassell, and R. M. May, J. Anim. Ecol. 61:735–748 (1992).
M. P. Hassell, H. N. Comins, and R. M. May, Nature 370:290(1994).
K. Sato, H. Matsuda, and A. Sasaki, J. Math. Biol. 32:251–268 (1994).
D. A. Rand, M. J. Keeling, and H. B. Wilson, Proc. R. Soc. London B 259:55(1995).
Y. Haraguchi and A. Sasaki, J. Theor. Biol. 203:85(2000).
M. J. Keeling, in The geometry of ecological interactions, U. Dieckmann, R. Law, and A. J. Metz, eds. (Cambridge University Press, 2000).
Y. Iwasa, in The geometry of ecological interactions, U. Dieckmann, R. Law, and A. J. Metz, eds. (Cambridge University Press, 2000).
D. A. Rand, in Advanced Ecological Theory, J. MacGlade, ed. (Blackwell Science, 1999).
J. E. Satulovsky and T. Tome, Phys. Rev. E 49:5073(1994).
M. A. M. de Aguiar, E. M. Rauch, and Y. Bar-Yam, Phys. Rev. E 67:047102(2003).
M. P. Hassell, H. N. Comins, and R. M. May, Nature (London) 370:290(1994).
M. Boots and A. Sasaki, Ecol. Lett. 3:181(2000).
N. J. Savill and P. Hogeweg, Proc. R. Soc. London B 265:25(1998).
M. van Baalen, in The geometry of ecological interactions, U. Dieckmann, R. Law, and A. J. Metz, eds. (Cambridge University Press, 2000).
H. Mamada and F. Takano, J. Phys. Soc. Jpn. 25:675(1968).
R. Kikuchi, Phys. Rev. 81:988(1951).
R. Dickman, Phys. Rev. A 34:4246(1986).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
de Aguiar, M.A.M., Rauch, E.M. & Bar-Yam, Y. Invasion and Extinction in the Mean Field Approximation for a Spatial Host-Pathogen Model. Journal of Statistical Physics 114, 1417–1451 (2004). https://doi.org/10.1023/B:JOSS.0000013958.15218.47
Issue Date:
DOI: https://doi.org/10.1023/B:JOSS.0000013958.15218.47