Dynamics of Populations in Extended Systems - Cellular Automata - page 201

contain most of the population, with a slight preference for the region II, since
there will be immigrants from regions I and III. Finally, if the selection pressure
is very strong, the population will die out. In the model the role of the selection
pressure seems to be more important because of its global character. We have
constructed the phase diagram of the final states of the evolving population - an
active state (living population) or an absorbing state (extinct population).
4 Conclusions
The two models treated above have shown that several important aspects of
the dynamics of population cannot be understood at the level of simple mean-
field like equations, overlooking the local fluctuations present in any spatially
extended systems. However, simple stochastic models using discrete variables
evolving with sequential or parallel dynamics can be very powerful in investigat-
ing the properties of these complexes systems.
Acknowledgement. This work was partially supported by the Swiss National
Science Foundation.
References
1. A.J. Lotka, Proc. Natl. Acad. Sci. U.S.A. 6 , 410 (1920).
2. V. Volterra, “Lecons sur la theorie mathematique de la lutte pour la vie”,
(Gauthier-Villars, Paris 1931).
3. N.S. Goel, S.C. Maitra and E.W. Montroll, “Nonlinear models of interacting pop-
ulations”, (Academic Press, New-York, 1971); J. Hofbauer, K. Sigmund, “Evolu-
tionary Games and Population Dynamics”, (Cambridge Univ. Press, 1998).
4. P. Rohani, R.M. May and M.P. Hassell, J. Theor. Biol. 181 , 97 (1996)
5. T. Antal and M. Droz, Phys. Rev 6.E , 1111 (2000).
6. M. Droz and A. Pekalski, Phys. Rev 63E , 051909 (2001).
7. A.T. Bradshaw, L.L. Moseley, Physica A 261 107 (1998).
8. B. Chopard and M. Droz Cellular Automata Modeling of Physical Systems , Cam-
bridge University Press (1998).
9. A. Lipowski and D. Lipowska, Physica A 276 , 456 (2000).
10. H. Taitelbaum, Z. Koza, Y. Yanir and G.H. Weiss, Physica A 266 , 280 (1999)
11. M. Droz and A. Pekalski, Physica A298 , 545 (2001).
12. M. Droz and A. Pekalski, Phys. Rev E , to appear (May 2002),
13. M. Begon, M. Mortimer and D.J. Thompson Population Ecology , Blackwell Science
Ltd, Oxford 1996.
14. A. Fraser and D. Burnell, Computer Models in Genetics, Mc Graw-Hill, New York
1970.
15. R. Burger and M. Lynch, Evolution, 49 , 151 (1995).
16. M. Lynch, J. Connery and R. Burger, Evolution, 49 , 1067 (1995).
17. R. Lande, Evolution, 48 , 1460 (1994).
18. R. Gomulkiewicz and R.D. Holt, Evolution, 49 , 201 (1995).
19. J.D. Murray, Mathematical Biology, Springer, Berlin 1993.