Abstract
The general analysis of the growth of mixed populations of any number of species in mutual interdependence of any kind which has “been given by the writer in prior publications1 covers many special cases. It is of interest to note how it applies to and readily furnishes the solution of a special case that has since been separately discussed by Volterra,2 namely that of two species competing for a common food supply.
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Footnotes
Among these may be mentioned Physical Review, 1912, 34: 235; Proc. American Academy Arts and Sciences 55: 137; 1920; American Journal of Hygiene January Supplement; 1923; Elements of Physical Biology, Baltimore, 1925. This last contains references to the author’s other publications relative to this subject.
Memorie della R. Acad. Nazion. dei Lincei 1926 ser 6, vol. 2, part 3, page 5; Lecons sur la theorie mathêmatique de la lutte-pour la vie, Paris, Gauthiers-Villars, 1931, page 9.
See, for example, Lotka, A. J., Elements of Physical Biology, 1925, page64.
This special case, as Volterra has shovn, can be integrated in finite terms. It is, however, of minor interest, since such an exact relation between the coefficients p1 and p2 represents, in concrete cases, an infinitely improbable condition.
A considerable number of significant figures has been retained in these constants and throughout the computations, in order to furnish an arithmetical check on the series solution (19). This check was obtained by substituting the solution (l9) separately in the left hand member and the right hand member of equations (4) or (12). In the absence of a special investigation of the conditions of convergence of the series (l9) this arithmetical check is necessary, and was found to be well satisfied within the limits of the curves shown in Figure 1.
A similar conclusion had been previously reached by J.B.S. Haldane regarding the competition between two Mendelian phenotypes. Trans. Cambridge Philos. Soc. 23: 39;1924.
Compare H. Hotelling, Economic Journal (London) 41: 41; 1929
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Lotka, A.J. (1978). The growth of mixed populations: Two species competing for a common food supply. In: The Golden Age of Theoretical Ecology: 1923–1940. Lecture Notes in Biomathematics, vol 22. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-50151-7_12
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