Prog. Theor. Phys. Vol. 88 No. 6 (1992) pp. 1035-1049

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Invited Papers

Statistical Mechanics of Population

— The Lattice Lotka-Volterra Model —

Hirotsugu Matsuda, Naofumi Ogita,^* Akira Sasaki and Kazunori Satō

Department of Biology, Faculty of Science, Kyushu University, Fukuoka 812
^*Fujitsu Limited, Tokyo 144

(Received October 5, 1992)

Abstract:

To derive the consequence of heritable traits of individual organisms upon the feature of their populations, the lattice Lotka-Volterra model is studied which is defined as a Markov process of the state of the lattice space. A lattice site is either vacant or occupied by an individual of a certain type or species. Transition rates of the process are given in terms of parameters representing the traits of an individual such as intrinsic birth and death and migration rate of each type. Density is a variable defined as a probability that a site is occupied by a certain type. Under a given state of a site the conditional probability of its nearest neighbor site being occupied by a certain type is termed environs density of the site. Mutual exclusion of individuals is already taken into account by the basic assumption of the lattice model. Other interaction between individuals can be taken into account by assuming that the actual birth and death and migration rates are dependent on the environs densities. Extending the notion of ordinary Malthusian parameters, we define Malthusians as dynamical variables specifying the time development of the densities. Conditions for the positive stationary densities and for the evolutional stability (ES) against the invasion of mutant types is given in terms of Malthusians. Using the pair approximation (PA), a simplest decoupling approximation to take account of spatial correlation, we obtain analytical results for stationary densities, and critical parameters for ES in the case of two types. Assuming that the death rate is dependent on the environs density, we derive conditions for the evolution of altruism. Comparing with computer simulation, we discuss the validity of PA and its improvement.

URL : http://ptp.ipap.jp/link?PTP/88/1035/
DOI : 10.1143/PTP.88.1035

References:

A. J. Lotka, J. Phys. Chem. 14 (1910), 271[CrossRef]; Proc. Nat. Acad. Sci. U.S.A. 6 (1920), 410.
V. Volterra, J. Cons. Perm. Int. Explor. Mer 3 (1928), 1; Lecon sur la Theorie Mathematique de la Lutte pour le Vie (Gauthier-Villars, Paris, 1931).
E. H. Kerner, Bull. Math. Biophys. 19 (1957), 121; Bull. Math. Biophys. 21 (1959), 217; Bull. Math. Biophys. 23 (1961), 141.
N. S. Goel, S. C. Maitra and E. W. Montroll, Rev. Modern Phys. 43 (1971), 231.
W. D. Hamilton, Science 156 (1967), 477
J. Maynard Smith, Evolution and the Theory of Games (Cambridge Univ. Press, Cambridge, 1982).
T. D. Lee and C. N. Yang, Phys. Rev. 87 (1952), 410[APS].
T. Matsubara and H. Matsuda, Prog. Theor. Phys. 16 (1956), 569[PTP].
H. Matsuda and T.Tsuneto, Prog. Theor. Phys. Suppl. No. 46 (1970), 411[PTP].
M. Kimura and G. H. Weiss, Genetics 49 (1964), 561.
G. Malécot, The Mathematics of Heredity (W. H. Freeman and Company, San Francisco, 1969).
W. O. Kermack and A. G. Mckendrick, Proc. Roy. Soc. Ser. A115 (1927), 700.
J. D. Murray, Mathematical Biology (Springer-Verlag, New York, 1989).
S. Yachi, K. Kawasaki, N. Shigesada and E. Teramoto, Forma 4 (1989), 3.
A. M. De Roos, E. McCauley and W. G. Wilson, Proc. Roy. Soc. London B246 (1991), 117.
R. Durrett, Lecture Notes on Particle Systems and Percolation (Wadsworth & Brooks/Cole, California, 1988).
R. Durrett and G. Swindle, Stoch. Proc. Appl. 37 (1991), 19.
M. Katori and N. Konno, J. Phys. Soc. Jpn. 59 (1990), 877[JPSJ].
N. Konno and M. Katori, J. Phys. Soc. Jpn. 59 (1990), 1581[JPSJ].
T. M. Liggett, Interacting Particle Systems (Springer-Verlag, New York, 1985).
D. Mollison, J. R. Statist. Soc. B39 (1977), 283.
T. Ootsuki and T. Keyes, Phys. Rev. A33 (1986), 1223[APS].
K. Tainaka, J. Phys. Soc. Jpn. 57 (1988), 2588[JPSJ].
K. Tainaka and Y. Itoh, Europhys. Lett. 15 (1991), 399.
H. Matsuda, Prog. Theor. Phys. 66 (1981), 1078[PTP]; Animal Societies: Theories and Facts, ed. Y. Itô, et al. (Japan Sci. Soc. Press, Tokyo, 1987), p. 67.
H. Matsuda, N. Tamachi, A. Sasaki and N. Ogita, Lecture Notes in Biomathematics 71 (1987), 154.
K. Satō, H. Matsuda and A. Sasaki, submitted to J. Math. Biol.
W. D. Hamilton, J. Theor. Biol. 7 (1964), 1; Animal Societies: Theories and Facts, ed. Y. Itô, et al., (Japan Sci. Soc. Press, Tokyo, 1987), p. 81.

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