- Access by Xinjiang University
Phase transition and selection in a four-species cyclic predator-prey model
Phys. Rev. E 69, 031911 – Published 31 March, 2004
DOI: https://doi.org/10.1103/PhysRevE.69.031911
Abstract
We study a four-species ecological system with cyclic dominance whose individuals are distributed on a square lattice. Randomly chosen individuals migrate to one of the neighboring sites if it is empty or invade this site if occupied by their prey. The cyclic dominance maintains the coexistence of all four species if the concentration of vacant sites is lower than a threshold value. Above the threshold, a symmetry breaking ordering occurs via growing domains containing only two neutral species inside. These two neutral species can protect each other from the external invaders (predators) and extend their common territory. According to our Monte Carlo simulations the observed phase transition seems to be equivalent to those found in spreading models with two equivalent absorbing states although the present model has continuous sets of absorbing states with different portions of the two neutral species. The selection mechanism yielding symmetric phases is related to the domain growth process with wide boundaries where the four species coexist.
References (32)
- V. Volterra, Lecon sur la Theorie Mathematique de la Lutte pour la Vie (Gouthier-Villars, Paris, 1931).
- A.J. Lotka, Proc. Natl. Acad. Sci. U.S.A. 6, 410 (1920).
- J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics (Cambridge University Press, Cambridge, 1998).
- B. Drossel, Adv. Phys. 50, 209 (2001).
- K. Tainaka, Phys. Rev. E 50, 3401 (1994).
- L. Frachebourg and P.L. Krapivsky, J. Phys. A 31, L287 (1998).
- K. Tainaka, Phys. Rev. Lett. 63, 2688 (1989).
- G. Szabó, M.A. Santos, and J.F.F. Mendes, Phys. Rev. E 60, 3776 (1999).
- G. Szabó and A. Szolnoki, Phys. Rev. E 65, 036115 (2002).
- M.C. Boerlijst and P. Hogeweg, Physica D 48, 17 (1991).
- G. Szabó and T. Czárán, Phys. Rev. E 63, 061904 (2001).
- G. Szabó and T. Czárán, Phys. Rev. E 64, 042902 (2001).
- K. Sato, N. Yoshida, and N. Konno, Appl. Math. Comput. 126, 255 (2002).
- B. Kerr, M.A. Riley, M.W. Feldman, and B.J.M. Bohannan, Nature (London) 418, 171 (2002).
- J. Huisman and F.J. Weissing, Ecology 82, 2682 (2001).
- L. Frachebourg, P.L. Krapivsky, and E. Ben-Naim, Phys. Rev. Lett. 77, 2125 (1996).
- R. Dickman and A.Y. Tretyakov, Phys. Rev. E 52, 3218 (1995).
- H. Hinrichsen, Phys. Rev. E 55, 219 (1997).
- I. Dornic, H. Chaté, J. Chave, and H. Hinrichsen, Phys. Rev. Lett. 87, 045701 (2001).
- A. Mehta and J.M. Luck, Phys. Rev. E 60, 5218 (1999).
- A. Lipowski and M. Droz, Phys. Rev. E 65, 056114 (2002).
- J. Marro and R. Dickman, Nonequilibrium Phase Transitions in Lattice Models (Cambridge University Press, Cambridge, 1999).
- H. Hinrichsen, Adv. Phys. 49, 815 (2000).
- J.F.F. Mendes, R. Dickman, M. Henkel, and M.C. Marques, J. Phys. A 27, 3019 (1994).
- M.A. Muñoz, G. Grinstein, and Y. Tu, Phys. Rev. E 56, 5101 (1997).
- I. Jensen and R. Dickman, Phys. Rev. E 48, 1710 (1993).
- M.C. Marques and J.F.F. Mendes, Eur. Phys. J. B 12, 123 (1999).
- A. Lipowski, Phys. Rev. E 62, 3356 (2000).
- M. Droz, A.L. Ferreira, and A. Lipowski, Phys. Rev. E 67, 056108 (2003).
- J.E. Satulovsky and T. Tomé, Phys. Rev. E 49, 5073 (1994).
- R.D. Holt, Ecol. Res. 17, 261 (2002).
- D. Chowdhury and D. Stauffer, Phys. Rev. E 68, 041901 (2003).