Abstract
The Lotka-Volterra equations for the population growth of a system of interacting species are linearized around equilibrium. The linear equations are solved for a special choice of the coefficients. It is shown that, in the limit of a large number of species, each one of them obeys a Langevin equation without memory. Subsequently, following Zwanzig, one species is added that is treated without linearization. The character of the equation governing its population is materially dependent on the special choice of the interaction coefficients. It is concluded that no general statement can be made concerning the stochastic behavior of the solutions of the Lotka-Volterra equations without being more specific about the coefficients than has been customary so far.
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References
N. S. Goel, S. C. Maitra, and E. W. Montroll,Rev. Mod. Phys. 43:231 (1971).
R. Zwanzig,Proc. Nat. Acad. Sci. USA 70:3048 (1973).
R. Zwanzig,J. Stat. Phys. 9:215 (1973).
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van Kampen, N.G. Remark on the Lotka-Volterra model. J Stat Phys 11, 475–480 (1974). https://doi.org/10.1007/BF01008890
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DOI: https://doi.org/10.1007/BF01008890