Mathematical theories versus biological facts: A debate on mathematical population dynamics in the 1930s

Historical Studies in the Physical and Biological Sciences

Volumn 26, Issue 2, 1996, Pages 346-403

  Millán Gasca, Ana ^a  

^a NONE   (Italy)

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EID: 0042785480     PISSN: 08909997     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (197)

1. On the development of mathematical studies on population ecology in the United States, see S. Kingsland, Modeling nature. Episodes in the history of population ecology (Chicago, 1985); F. Scudo, "The 'Golden Age' of theoretical ecology: A conceptual appraisal," Revue européene des sciences sociales, 22 (1984), 11-64.
2. On the development of mathematical studies on population ecology in the United States, see S. Kingsland, Modeling nature. Episodes in the history of population ecology (Chicago, 1985); F. Scudo, "The 'Golden Age' of theoretical ecology: A conceptual appraisal," Revue européene des sciences sociales, 22 (1984), 11-64.
3. W.B. Provine, The origins of theoretical population genetics (Chicago, 1971); E. Mayr and W.B. Provine, eds., The evolutionary synthesis: Perspectives on the unification of biology (Cambridge, 1980).
4. W.B. Provine, The origins of theoretical population genetics (Chicago, 1971); E. Mayr and W.B. Provine, eds., The evolutionary synthesis: Perspectives on the unification of biology (Cambridge, 1980).
5. T. Vignoli and G.V. Schiaparelli, Peregrinazioni antropologiche e fisiche. Studio comparativo fra le forme organiche naturali e le forme geometriche pure (Milan, 1898); D'A. W. Thompson, On growth and form (Cambridge, 1917).
6. T. Vignoli and G.V. Schiaparelli, Peregrinazioni antropologiche e fisiche. Studio comparativo fra le forme organiche naturali e le forme geometriche pure (Milan, 1898); D'A. W. Thompson, On growth and form (Cambridge, 1917).
7. B.L. van der Pol and J. Van der Mark, "The heartbeat considered as a relaxation oscillation, and an electrical model of the heart," Philosophical magazine, 6 (1928), 763-775; J. Lotka, Elements of physical biology (Baltimore, 1925).
8. B.L. van der Pol and J. Van der Mark, "The heartbeat considered as a relaxation oscillation, and an electrical model of the heart," Philosophical magazine, 6 (1928), 763-775; J. Lotka, Elements of physical biology (Baltimore, 1925).
9. During the 1928 International Congress of Entomology the distinguished biologist Stephen Forbes, congratulating R.N. Chapman on his work, spoke about the breakthrough that his ideas represented: "Forbes reminded the audience that ecologists had often not known when to pause from the gathering of facts to draw a conclusion from the fragments they studied, confident that it would have general applicability," Kingsland (ref. 1), 129.
10. See G. Israel, "The two paths of the mathematization of the social and economic sciences. The decline of the 'mathématique sociale' and the beginnings of mathematical economics at the turn of the eighteenth century," Physis, 30 (1993), 27-78.
11. Volterra's equations for "hereditary" physical phenomena prompted criticism from colleagues who supported the methods of classical physics; Israel, "Vito Volterra: Un fisico matematico di fronte ai problemi della fisica del Novecento," Revista di storia della scienza, I (1984), 39-72; "Il determinismo e la teoria delle equazioni differenziali ordinarie. Un'analisi retrospettiva a partire dalla meccanica ereditaria," Physis, 28 (1991), 305-358.
12. Volterra's equations for "hereditary" physical phenomena prompted criticism from colleagues who supported the methods of classical physics; Israel, "Vito Volterra: Un fisico matematico di fronte ai problemi della fisica del Novecento," Revista di storia della scienza, I (1984), 39-72; "Il determinismo e la teoria delle equazioni differenziali ordinarie. Un'analisi retrospettiva a partire dalla meccanica ereditaria," Physis, 28 (1991), 305-358.
13. Speeches delivered at a dinner held in University College, London, in honour of Professor Karl Pearson. 23 April 1934 (Cambridge, 1934), 22-23.
14. Israel, "The origins of mathematical modelling in biology: A revival of mechanical reductionism and Darwinism," Science in context, 6 (1993), 469-509; "Volterra e la dinamica delle popolazioni biologiche," in A. Roccheggiani, ed., Il pensiero scientifico di Vito Volterra (Ancona, 1990), 87-113; "Volterra's analytical mechanics of biological associations," Archives internationales d'histoire des sciences, 41 (1991), 57-104, 307-352; "Le equazioni di Volterra e Lotka: una questione di prioritè" in O. Montaldo and L. Grugnetti, eds., La storia delle matematiche in Italia (Cagliari, 1982), 495-502; and "On the contribution of Volterra and Lotka to the development of modern biomathematics," History and philosophy of life sciences, 10 (1988), 37-49.
15. Israel, "The origins of mathematical modelling in biology: A revival of mechanical reductionism and Darwinism," Science in context, 6 (1993), 469-509; "Volterra e la dinamica delle popolazioni biologiche," in A. Roccheggiani, ed., Il pensiero scientifico di Vito Volterra (Ancona, 1990), 87-113; "Volterra's analytical mechanics of biological associations," Archives internationales d'histoire des sciences, 41 (1991), 57-104, 307-352; "Le equazioni di Volterra e Lotka: una questione di prioritè" in O. Montaldo and L. Grugnetti, eds., La storia delle matematiche in Italia (Cagliari, 1982), 495-502; and "On the contribution of Volterra and Lotka to the development of modern biomathematics," History and philosophy of life sciences, 10 (1988), 37-49.
16. Israel, "The origins of mathematical modelling in biology: A revival of mechanical reductionism and Darwinism," Science in context, 6 (1993), 469-509; "Volterra e la dinamica delle popolazioni biologiche," in A. Roccheggiani, ed., Il pensiero scientifico di Vito Volterra (Ancona, 1990), 87-113; "Volterra's analytical mechanics of biological associations," Archives internationales d'histoire des sciences, 41 (1991), 57-104, 307-352; "Le equazioni di Volterra e Lotka: una questione di prioritè" in O. Montaldo and L. Grugnetti, eds., La storia delle matematiche in Italia (Cagliari, 1982), 495-502; and "On the contribution of Volterra and Lotka to the development of modern biomathematics," History and philosophy of life sciences, 10 (1988), 37-49.
17. Israel, "The origins of mathematical modelling in biology: A revival of mechanical reductionism and Darwinism," Science in context, 6 (1993), 469-509; "Volterra e la dinamica delle popolazioni biologiche," in A. Roccheggiani, ed., Il pensiero scientifico di Vito Volterra (Ancona, 1990), 87-113; "Volterra's analytical mechanics of biological associations," Archives internationales d'histoire des sciences, 41 (1991), 57-104, 307-352; "Le equazioni di Volterra e Lotka: una questione di prioritè" in O. Montaldo and L. Grugnetti, eds., La storia delle matematiche in Italia (Cagliari, 1982), 495-502; and "On the contribution of Volterra and Lotka to the development of modern biomathematics," History and philosophy of life sciences, 10 (1988), 37-49.
18. Israel, "The origins of mathematical modelling in biology: A revival of mechanical reductionism and Darwinism," Science in context, 6 (1993), 469-509; "Volterra e la dinamica delle popolazioni biologiche," in A. Roccheggiani, ed., Il pensiero scientifico di Vito Volterra (Ancona, 1990), 87-113; "Volterra's analytical mechanics of biological associations," Archives internationales d'histoire des sciences, 41 (1991), 57-104, 307-352; "Le equazioni di Volterra e Lotka: una questione di prioritè" in O. Montaldo and L. Grugnetti, eds., La storia delle matematiche in Italia (Cagliari, 1982), 495-502; and "On the contribution of Volterra and Lotka to the development of modern biomathematics," History and philosophy of life sciences, 10 (1988), 37-49.
19. Israel, "L'interminabile crisi del meccanicismo," Rivista di storia della scienza, 1 (1984), 73-99; "Approccio descrittivo e approccio ermeneutico nella modellistica matematica contemporanea," in S. Petruccioli, ed., Immagini, linguaggi, concetti (Rome, 1991), and La mathématisation du réel (Paris, 1996). On economics, see B. Ingrao and G. Israel, The invisible hand. Economic equilibrium in the history of science (Cambridge, 1990), and on biology, Israel, "The origins" (ref. 9).
20. Israel, "L'interminabile crisi del meccanicismo," Rivista di storia della scienza, 1 (1984), 73-99; "Approccio descrittivo e approccio ermeneutico nella modellistica matematica contemporanea," in S. Petruccioli, ed., Immagini, linguaggi, concetti (Rome, 1991), and La mathématisation du réel (Paris, 1996). On economics, see B. Ingrao and G. Israel, The invisible hand. Economic equilibrium in the history of science (Cambridge, 1990), and on biology, Israel, "The origins" (ref. 9).
21. Israel, "L'interminabile crisi del meccanicismo," Rivista di storia della scienza, 1 (1984), 73-99; "Approccio descrittivo e approccio ermeneutico nella modellistica matematica contemporanea," in S. Petruccioli, ed., Immagini, linguaggi, concetti (Rome, 1991), and La mathématisation du réel (Paris, 1996). On economics, see B. Ingrao and G. Israel, The invisible hand. Economic equilibrium in the history of science (Cambridge, 1990), and on biology, Israel, "The origins" (ref. 9).
22. Israel, "L'interminabile crisi del meccanicismo," Rivista di storia della scienza, 1 (1984), 73-99; "Approccio descrittivo e approccio ermeneutico nella modellistica matematica contemporanea," in S. Petruccioli, ed., Immagini, linguaggi, concetti (Rome, 1991), and La mathématisation du réel (Paris, 1996). On economics, see B. Ingrao and G. Israel, The invisible hand. Economic equilibrium in the history of science (Cambridge, 1990), and on biology, Israel, "The origins" (ref. 9).
23. Israel, "L'interminabile crisi del meccanicismo," Rivista di storia della scienza, 1 (1984), 73-99; "Approccio descrittivo e approccio ermeneutico nella modellistica matematica contemporanea," in S. Petruccioli, ed., Immagini, linguaggi, concetti (Rome, 1991), and La mathématisation du réel (Paris, 1996). On economics, see B. Ingrao and G. Israel, The invisible hand. Economic equilibrium in the history of science (Cambridge, 1990), and on biology, Israel, "The origins" (ref. 9).
24. See G. Israel, "Volterra archive at the Accademia Nazionale dei Lincei," Historia mathematica, 9 (1982), 229-238; G. Paoloni, ed., Vito Volterra e il suo tempo (1860-1940) (Rome, 1990). The correspondence Kostitzin has been published in G. Israel and A. Millán Gasca, "La correspondencia entre Vladimir A. Kostitzin y Vito Volterra (1933-1962) y los inicios de la biomatemática," Llull, Revista de la Sociedad Española de Historia de las Ciencias y de las Técnicas, 16 (1993), 159-224.
25. See G. Israel, "Volterra archive at the Accademia Nazionale dei Lincei," Historia mathematica, 9 (1982), 229-238; G. Paoloni, ed., Vito Volterra e il suo tempo (1860-1940) (Rome, 1990). The correspondence Kostitzin has been published in G. Israel and A. Millán Gasca, "La correspondencia entre Vladimir A. Kostitzin y Vito Volterra (1933-1962) y los inicios de la biomatemática," Llull, Revista de la Sociedad Española de Historia de las Ciencias y de las Técnicas, 16 (1993), 159-224.
26. See G. Israel, "Volterra archive at the Accademia Nazionale dei Lincei," Historia mathematica, 9 (1982), 229-238; G. Paoloni, ed., Vito Volterra e il suo tempo (1860-1940) (Rome, 1990). The correspondence Kostitzin has been published in G. Israel and A. Millán Gasca, "La correspondencia entre Vladimir A. Kostitzin y Vito Volterra (1933-1962) y los inicios de la biomatemática," Llull, Revista de la Sociedad Española de Historia de las Ciencias y de las Técnicas, 16 (1993), 159-224.
27. Volterra, "Variazioni e fluttuazioni del numero d'individui in specie animali conviventi," Accademia dei Lincei, Memorie, 2 (1926), 31-113.
28. Volterra began his work on D'Ancona's problem in late 1925; both of them arrived independently to the same result. Volterra, Leçons sur la théorie mathématique de la lutte pour la vie, ed. Marcel Brelot (Paris, 1931), préface. D'Ancona's statistical studies appear in his Dell'influenza della stasi peschereccia del periodo 1914-1918 sul patrimonio ittico dell'Alto Adriatico, Comitato Talassografico Italiano, Memorie, 126 (Venice, 1926).
29. Volterra began his work on D'Ancona's problem in late 1925; both of them arrived independently to the same result. Volterra, Leçons sur la théorie mathématique de la lutte pour la vie, ed. Marcel Brelot (Paris, 1931), préface. D'Ancona's statistical studies appear in his Dell'influenza della stasi peschereccia del periodo 1914-1918 sul patrimonio ittico dell'Alto Adriatico, Comitato Talassografico Italiano, Memorie, 126 (Venice, 1926).
30. Volterra, "Fluctuations in the abundance of species considered mathematically," Nature, 118 (1926), 558-560, on 559.
31. Volterra quoted from Darwin, The origin of species by means of natural selection, or the preservation of favoured races in the struggle for life (6th edn., London, 1882), 53-54; Volterra (ref. 14), 559.
32. Volterra quoted from Darwin, The origin of species by means of natural selection, or the preservation of favoured races in the struggle for life (6th edn., London, 1882), 53-54; Volterra (ref. 14), 559.
33. Ibid., 559-560.
34. Volterra (ref. 14).
35. Draft, Volterra to Larmor, 11 June 1926, and to Pearson, 21 Sep 1926. In his letter to Pearson, Volterra was more cautious: "I have tried to provide a mathematical theory of fluctuations of animal species by deriving laws that seemed confirmed by statistics."
36. Volterra, "Sulle fluttuazioni biologiche," Accademia dei Lincei, Rendiconti, 5 (1927), 3-10; "Leggi sulle fluttuazioni biologiche," ibid., 61-67; "Sulla periodicità delle fluttuazioni biologiche," ibid., 465-470.
37. Volterra, "Sulle fluttuazioni biologiche," Accademia dei Lincei, Rendiconti, 5 (1927), 3-10; "Leggi sulle fluttuazioni biologiche," ibid., 61-67; "Sulla periodicità delle fluttuazioni biologiche," ibid., 465-470.
38. Volterra, "Sulle fluttuazioni biologiche," Accademia dei Lincei, Rendiconti, 5 (1927), 3-10; "Leggi sulle fluttuazioni biologiche," ibid., 61-67; "Sulla periodicità delle fluttuazioni biologiche," ibid., 465-470.
39. Volterra, Variazioni e fluttuazioni del numero d'individui in specie animali conviventi, Comitato Talassografico, Memorie, 131 (Venezia, 1927); Essai mathématique sur les fluctuations biologiques (Paris, 1927); "Una teoria matematica sulla lotta per l'esistenza," Scientia, 41 (1927), 85-102, and French translation, ibid., Supplément, 33-48; "Variations and fluctuations of the number of individuals in animal species living together," Conseil international pour l'exploration de la mer, Journal, 3 (1928), 3-51; and Mathematical theory of the struggle for existence (in Russian) (Moscow, 1928). Volterra's results were discussed at the meeting of the Association française pour l'avancement des sciences of 1927 and by the French mathematician Joseph Pérés in the Revue génerale des sciences pures et appliquées.
40. Volterra, Variazioni e fluttuazioni del numero d'individui in specie animali conviventi, Comitato Talassografico, Memorie, 131 (Venezia, 1927); Essai mathématique sur les fluctuations biologiques (Paris, 1927); "Una teoria matematica sulla lotta per l'esistenza," Scientia, 41 (1927), 85-102, and French translation, ibid., Supplément, 33-48; "Variations and fluctuations of the number of individuals in animal species living together," Conseil international pour l'exploration de la mer, Journal, 3 (1928), 3-51; and Mathematical theory of the struggle for existence (in Russian) (Moscow, 1928). Volterra's results were discussed at the meeting of the Association française pour l'avancement des sciences of 1927 and by the French mathematician Joseph Pérés in the Revue génerale des sciences pures et appliquées.
41. Volterra, Variazioni e fluttuazioni del numero d'individui in specie animali conviventi, Comitato Talassografico, Memorie, 131 (Venezia, 1927); Essai mathématique sur les fluctuations biologiques (Paris, 1927); "Una teoria matematica sulla lotta per l'esistenza," Scientia, 41 (1927), 85-102, and French translation, ibid., Supplément, 33-48; "Variations and fluctuations of the number of individuals in animal species living together," Conseil international pour l'exploration de la mer, Journal, 3 (1928), 3-51; and Mathematical theory of the struggle for existence (in Russian) (Moscow, 1928). Volterra's results were discussed at the meeting of the Association française pour l'avancement des sciences of 1927 and by the French mathematician Joseph Pérés in the Revue génerale des sciences pures et appliquées.
42. Volterra, Variazioni e fluttuazioni del numero d'individui in specie animali conviventi, Comitato Talassografico, Memorie, 131 (Venezia, 1927); Essai mathématique sur les fluctuations biologiques (Paris, 1927); "Una teoria matematica sulla lotta per l'esistenza," Scientia, 41 (1927), 85-102, and French translation, ibid., Supplément, 33-48; "Variations and fluctuations of the number of individuals in animal species living together," Conseil international pour l'exploration de la mer, Journal, 3 (1928), 3-51; and Mathematical theory of the struggle for existence (in Russian) (Moscow, 1928). Volterra's results were discussed at the meeting of the Association française pour l'avancement des sciences of 1927 and by the French mathematician Joseph Pérés in the Revue génerale des sciences pures et appliquées.
43. Volterra, Variazioni e fluttuazioni del numero d'individui in specie animali conviventi, Comitato Talassografico, Memorie, 131 (Venezia, 1927); Essai mathématique sur les fluctuations biologiques (Paris, 1927); "Una teoria matematica sulla lotta per l'esistenza," Scientia, 41 (1927), 85-102, and French translation, ibid., Supplément, 33-48; "Variations and fluctuations of the number of individuals in animal species living together," Conseil international pour l'exploration de la mer, Journal, 3 (1928), 3-51; and Mathematical theory of the struggle for existence (in Russian) (Moscow, 1928). Volterra's results were discussed at the meeting of the Association française pour l'avancement des sciences of 1927 and by the French mathematician Joseph Pérés in the Revue génerale des sciences pures et appliquées.
44. Volterra, Variazioni e fluttuazioni del numero d'individui in specie animali conviventi, Comitato Talassografico, Memorie, 131 (Venezia, 1927); Essai mathématique sur les fluctuations biologiques (Paris, 1927); "Una teoria matematica sulla lotta per l'esistenza," Scientia, 41 (1927), 85-102, and French translation, ibid., Supplément, 33-48; "Variations and fluctuations of the number of individuals in animal species living together," Conseil international pour l'exploration de la mer, Journal, 3 (1928), 3-51; and Mathematical theory of the struggle for existence (in Russian) (Moscow, 1928). Volterra's results were discussed at the meeting of the Association française pour l'avancement des sciences of 1927 and by the French mathematician Joseph Pérés in the Revue génerale des sciences pures et appliquées.
45. Volterra to Richard Gregory, editor of Nature, draft, enclosed in a letter from Volterra to D'Arcy W. Thompson, 28 Nov 1926. He continued to call attention to the fact that he did not know Ross' and Lotka's work as late as 1936; Volterra, "La théorie mathématique de la lutte pour la vie et l'expérience (à propos de deux ouvrages de C.F. Gause)," Scientia, 60 (1936), 169-174). The prey-predator equations are known as the Volterra-Lotka equations. On the priority controversy, see Israel, "Le equazioni" and "On the contribution" (ref. 9).
46. Volterra to Richard Gregory, editor of Nature, draft, enclosed in a letter from Volterra to D'Arcy W. Thompson, 28 Nov 1926. He continued to call attention to the fact that he did not know Ross' and Lotka's work as late as 1936; Volterra, "La théorie mathématique de la lutte pour la vie et l'expérience (à propos de deux ouvrages de C.F. Gause)," Scientia, 60 (1936), 169-174). The prey-predator equations are known as the Volterra-Lotka equations. On the priority controversy, see Israel, "Le equazioni" and "On the contribution" (ref. 9).
47. E. Pearson. "The application of the theory of differential equations to the solution of problems connected with the interdependence of species," Biometrika, 19 (1927), 216-222, 442; F.S. Bodenheimer, "Welche Faktorem reguheren die Individuenzahl einer Insektenart in der Natur?," Biologisches Zentralblatt, 48 (1928), 714.
48. E. Pearson. "The application of the theory of differential equations to the solution of problems connected with the interdependence of species," Biometrika, 19 (1927), 216-222, 442; F.S. Bodenheimer, "Welche Faktorem reguheren die Individuenzahl einer Insektenart in der Natur?," Biologisches Zentralblatt, 48 (1928), 714.
49. Thompson to Volterra, 2 Jul 1926: "I am only a biologist, but I have some little interest in mathematics of an elementary kind. One of my little papers I should like to offer you."
50. See M. Mazzocut-Mis, "D'Arcy Thompson, la forme et le vivant," Alliage, 22 (1995), 24-31, on 24: "Morphological differentiation, that is, the indefinite variety of forms, arouses no feeling of ecstatic astonishment in D'Arcy Thompson. It is rather the possibility of relating morphological plurality to a procedure governed by a small number of formalizable laws which stimulates his solitary research work" (p. 24).
51. Monod Herzen to Volterra, 10 Dec 1927; Mémoire CXXXI is Volterra, "Variazioni" (ref. 20). Cf. E. Monod-Herzen, Science et esthétique. Principes de morphologie générale (2 vols., Paris, 1927).
52. Monod Herzen to Volterra, 10 Dec 1927; Mémoire CXXXI is Volterra, "Variazioni" (ref. 20). Cf. E. Monod-Herzen, Science et esthétique. Principes de morphologie générale (2 vols., Paris, 1927).
53. Thompson to Volterra, 29 Jul 1927.
54. Chapman to Volterra, 9 Aug 1928. R.N. Chapman, "The quantitative analysis of environmental factors," Ecology, 9 (1928), 111-122.
55. Chapman to Volterra, 9 Aug 1928. R.N. Chapman, "The quantitative analysis of environmental factors," Ecology, 9 (1928), 111-122.
56. Kingsland (ref. 1), 95-96; Chapman to Volterra, 28 Jan 1929. Kingsland shows that Pearl was interested in a qualitative appraisal of the influence of inherited and environmental factors in the evolution of populations.
57. Ibid., "I am greatly interested in the work which you have done. I believe that it is of great importance to the field of biology. In fact, I think it is comparable to what Dr. Willard Gibbs did for the field of physical chemistry and 1 hope that unlike his work, yours will come immediately to the attention of the biologists" (Chapman to Volterra, 26 Sep 1928).
58. Kingsland (ref. 1), 234. K. Friederichs, Die Grundfragen und Gesetzmässigkeiten der land- und forstwirtschaftlichen Zoologie insbesondere der Entomologie (Berlin, 1930), and "Welche Faktoren regulieren die Individuenzahl einer Insektenart in der Natur? Ein kritisches Referat," Anzeiger für Schädlingskunde, 6 (1929), 25.
59. Kingsland (ref. 1), 234. K. Friederichs, Die Grundfragen und Gesetzmässigkeiten der land- und forstwirtschaftlichen Zoologie insbesondere der Entomologie (Berlin, 1930), and "Welche Faktoren regulieren die Individuenzahl einer Insektenart in der Natur? Ein kritisches Referat," Anzeiger für Schädlingskunde, 6 (1929), 25.
60. Kingsland (ref. 1), 234. K. Friederichs, Die Grundfragen und Gesetzmässigkeiten der land- und forstwirtschaftlichen Zoologie insbesondere der Entomologie (Berlin, 1930), and "Welche Faktoren regulieren die Individuenzahl einer Insektenart in der Natur? Ein kritisches Referat," Anzeiger für Schädlingskunde, 6 (1929), 25.
61. Chapman to Volterra, 28 Jan 1929, and infra, §6.
62. Graham to Volterra, 1 Nov 1928.
63. Graham to Volterra, 30 Jan 1929.
64. Elton to Volterra, 6 Jan 1931. In Animal ecology (2nd edn., 1925), Elton had mentioned biomathematical studies of population dynamics but pointed out the lack of effective confirmation by experimental data.
65. See E.G. Leigh, "The ecological role of Volterra's equations" in Some mathematical problems in biology (Providence, 1968), 1-61. Leigh rightly observes (p. 2) that figure 1, which derive from data in D.A. MacLulich, Fluctuation in the number of the varying hare (Toronto 1937), "is not a prey-predator oscillation" because it is not periodic. However, as the figure shows, the oscillatory trend represents; one of the best empirical fits of Volterra's law.
66. See E.G. Leigh, "The ecological role of Volterra's equations" in Some mathematical problems in biology (Providence, 1968), 1-61. Leigh rightly observes (p. 2) that figure 1, which derive from data in D.A. MacLulich, Fluctuation in the number of the varying hare (Toronto 1937), "is not a prey-predator oscillation" because it is not periodic. However, as the figure shows, the oscillatory trend represents; one of the best empirical fits of Volterra's law.
67. Volterra, Leçons (ref. 13), 3.
68. Ibid., 210.
69. See infra §4.
70. Volterra, Leçons (ref. 13), 198-99.
71. Pearl developed a deep analysis of the biological meaning of the logistic curve. He first stated the influence of the density of the population on reproduction; then, in 1932, he explained that "encounters" among individuals altered the food intake, energy output in muscular activity, and oviposition. R. Pearl, "The influence of density of population upon egg production in Drosophila melanogaster," Journal of experimental zoology, 63 (1932), 57-84; see Kingsland (ref. 1 ), 77-83.
72. Pearl developed a deep analysis of the biological meaning of the logistic curve. He first stated the influence of the density of the population on reproduction; then, in 1932, he explained that "encounters" among individuals altered the food intake, energy output in muscular activity, and oviposition. R. Pearl, "The influence of density of population upon egg production in Drosophila melanogaster," Journal of experimental zoology, 63 (1932), 57-84; see Kingsland (ref. 1 ), 77-83.
73. Volterra, Leçons (ref. 13), 32. This same reasoning was introduced for the case of any number of species, 65.
74. Ibid., 201-202. A draft of a letter to Pearl written by Volterra in 1931 shows that they were already in touch before the publication of the Leçons. The correspondence resumed only around 1937.
75. Brelot to Volterra, 25 Oct 1929.
76. W.R. Thompson to Volterra, 4 May 1930.
77. Volterra, "Una teoria matematica" (ref. 20), 100.
78. Brelot to Volterra, 12 Aug 1929.
79. Draft, Volterra to Brelot, 20 Aug 1929.
80. Volterra, "Una teoria matematica" (ref. 20), 101.
81. Ref. 47.
82. D'Ancona to Volterra, 10 Mar 1930.
83. Volterra, Leçons (ref. 13), 7-8. These comments were not included in the 1935 book.
84. V. Kostitzin, Symbiose, parasitisme et evolution (Paris, 1934), 35-36.
85. D'Ancona to Volterra, 4 Apr 1931.
86. M. Falcone and G. Israel, "Qualitative and numerical analysis of a class of preypredator models," Acta applicandae mathematicae, 4 (1985), 225-258.
87. Draft, Volterra to D'A. Thompson, ca. Jan 1931, and further letters regarding Thompson's contacts with Blackie & Son and Clarendon Press; Volterra to Elton, draft, 20 Jan 1932; Chapman to Volterra, 4 May 1931; and the correspondence with J. Stanley, who put forward the idea of a collaboration for an English book.
88. R.N. Chapman, Animal ecology, with special reference to insects (New York, 1931).
89. Kingsland (ref. 1).
90. See R.N. Chapman, "The causes of fluctuations in populations in insects," Hawaiian Entomological Society, Proceedings, 8 (1933), 279-292.
91. See D. Weiner, Ecology, conservation and cultural revolution in Soviet Russia (Bloomington, 1988); R. Balandine, La vie et l'oeuvre de Vernadski (Moscow, 1982); K. Bailes, Science and Russian culture in the age of revolutions: V.I. Vernadsky and his scientific school, 1863-1945 (Bloomington, 1990); J. Grinevald, "On a holistic concept for deep and global ecology: The Biosphere," Fundamenta scientiae, 8 (1987), 197-226.
92. See D. Weiner, Ecology, conservation and cultural revolution in Soviet Russia (Bloomington, 1988); R. Balandine, La vie et l'oeuvre de Vernadski (Moscow, 1982); K. Bailes, Science and Russian culture in the age of revolutions: V.I. Vernadsky and his scientific school, 1863-1945 (Bloomington, 1990); J. Grinevald, "On a holistic concept for deep and global ecology: The Biosphere," Fundamenta scientiae, 8 (1987), 197-226.
93. See D. Weiner, Ecology, conservation and cultural revolution in Soviet Russia (Bloomington, 1988); R. Balandine, La vie et l'oeuvre de Vernadski (Moscow, 1982); K. Bailes, Science and Russian culture in the age of revolutions: V.I. Vernadsky and his scientific school, 1863-1945 (Bloomington, 1990); J. Grinevald, "On a holistic concept for deep and global ecology: The Biosphere," Fundamenta scientiae, 8 (1987), 197-226.
94. See D. Weiner, Ecology, conservation and cultural revolution in Soviet Russia (Bloomington, 1988); R. Balandine, La vie et l'oeuvre de Vernadski (Moscow, 1982); K. Bailes, Science and Russian culture in the age of revolutions: V.I. Vernadsky and his scientific school, 1863-1945 (Bloomington, 1990); J. Grinevald, "On a holistic concept for deep and global ecology: The Biosphere," Fundamenta scientiae, 8 (1987), 197-226.
95. M.B. Adams, "Svertsov and Schmalhausen: Russian morphology and the evolutionary synthesis," in Mayr and Provine (ref. 2), 193-225, and "Sergei Chetverikov, the Kol'tsov Institute, and the evolutionary synthesis," ibid., 242-278.
96. M.B. Adams, "Svertsov and Schmalhausen: Russian morphology and the evolutionary synthesis," in Mayr and Provine (ref. 2), 193-225, and "Sergei Chetverikov, the Kol'tsov Institute, and the evolutionary synthesis," ibid., 242-278.
97. Kingsland (ref. 1), 146 ff.
98. Gause to Volterra, 12 Nov 1932 (original in English).
99. A. Pavé, Modélisation en biologie et en écologie (Paris, 1994), 98.
100. Régnier to Volterra, 12 Mar 1932. Régnier worked on bacteriology, pharmacodynamics, and pharmaceutical chemical-physics. In 1941 he became Professor of Microbiology of the University of Paris Faculty of Pharmacy, where he created a new graduate degree in Microbiology, Serology, Haematology and Parasitology. S. Lambin, Jean Régnier (1892-1946) (Paris, 1969). Of prime importance in the analysis of Régnier's work were the documents preserved by his collaborator Suzanne Lambin (b. 1902), subsequently the first woman to be appointed as a professor at the Paris Faculty of Pharmacy, and an interview with her at her Paris home, on February 1991.
101. Régnier to Volterra, 12 Mar 1932. Régnier worked on bacteriology, pharmacodynamics, and pharmaceutical chemical-physics. In 1941 he became Professor of Microbiology of the University of Paris Faculty of Pharmacy, where he created a new graduate degree in Microbiology, Serology, Haematology and Parasitology. S. Lambin, Jean Régnier (1892-1946) (Paris, 1969). Of prime importance in the analysis of Régnier's work were the documents preserved by his collaborator Suzanne Lambin (b. 1902), subsequently the first woman to be appointed as a professor at the Paris Faculty of Pharmacy, and an interview with her at her Paris home, on February 1991.
102. D'Ancona to Volterra, 3 Nov 1933.
103. E. Mayr, "The arrival of Neo-darwinism in France," in Mayr and Provine (ref. 2), 321-322; D. Buican, Histoire de la génétique et de l'évolutionisme en France (Paris, 1984).
104. E. Mayr, "The arrival of Neo-darwinism in France," in Mayr and Provine (ref. 2), 321-322; D. Buican, Histoire de la génétique et de l'évolutionisme en France (Paris, 1984).
105. See Israel and Millán Gasca (ref. 11), and F. Scudo and J.R. Ziegler, "Vladimir Alexandrovich Kostitzin and Theoretical Ecology," Theoretical population biology, 10 (1976), 395-412.
106. See Israel and Millán Gasca (ref. 11), and F. Scudo and J.R. Ziegler, "Vladimir Alexandrovich Kostitzin and Theoretical Ecology," Theoretical population biology, 10 (1976), 395-412.
107. Volterra and D'Ancona, Les associations biologiques étudiées au point de vue mathématique (Paris, 1935), recently translated, Le associazioni biologiche studiate dal punto di vista matematico (Rome, 1995), ed. Giorgio Israel.
108. Volterra and D'Ancona, Les associations biologiques étudiées au point de vue mathématique (Paris, 1935), recently translated, Le associazioni biologiche studiate dal punto di vista matematico (Rome, 1995), ed. Giorgio Israel.
109. D'Ancona to Volterra, 10 Feb 1929.
110. D'Ancona to Volterra, 28 Mar 1930.
111. Régnier to Volterra, 11 June 1933.
112. Régnier to Volterra, 16 Jul 1933.
113. D'Ancona to Volterra, 26 Jul 1933.
114. Régnier to Volterra, 14 Mar 1934. Régnier sent the paper to Volterra on 18 Nov.
115. D'Ancona to Volterra, 2 Dec 1934.
116. Volterra, "Remarques sur la note de M. Régnier et Mlle. Lambin," CRAS, 199 (1934), 1684-1686, which followed J Régnier and S. Lambin, "Etude d'un cas d'antagonisme microbien (Bacillus Coli-Staphylococcus aureus)," CRAS, 199 (1934), 1682-1683. Volterra had already used integro-differential equations for describing the non-immediate effects of biological conditions: Volterra, "The general equations of biological strife in the case of historical actions," Edinburgh Mathematical Society, IIs., Proceedings, 6 (1939), 4-10.
117. Volterra, "Remarques sur la note de M. Régnier et Mlle. Lambin," CRAS, 199 (1934), 1684-1686, which followed J Régnier and S. Lambin, "Etude d'un cas d'antagonisme microbien (Bacillus Coli-Staphylococcus aureus)," CRAS, 199 (1934), 1682-1683. Volterra had already used integro-differential equations for describing the non-immediate effects of biological conditions: Volterra, "The general equations of biological strife in the case of historical actions," Edinburgh Mathematical Society, IIs., Proceedings, 6 (1939), 4-10.
118. Volterra, "Remarques sur la note de M. Régnier et Mlle. Lambin," CRAS, 199 (1934), 1684-1686, which followed J Régnier and S. Lambin, "Etude d'un cas d'antagonisme microbien (Bacillus Coli-Staphylococcus aureus)," CRAS, 199 (1934), 1682-1683. Volterra had already used integro-differential equations for describing the non-immediate effects of biological conditions: Volterra, "The general equations of biological strife in the case of historical actions," Edinburgh Mathematical Society, IIs., Proceedings, 6 (1939), 4-10.
119. V. Kostitzin, Biologie mathématique (Paris, 1937), 67.
120. Volterra and D'Ancona (ref. 68), 21-23, 32-33. Curves in figures 2 and 3 were drawn by D'Ancona, the first from Régnier and Lambin's data and the second from the theoretical curves under Volterra's direction.
121. Régnier to Volterra, 9 Dec 1934.
122. Régnier to Volterra, 19 May 1935.
123. In D'Ancona to Volterra, 2 Dec 1934, Pearl's studies on the development of a single species, and a book of his, probably The biology of population growth (New York, 1925), are mentioned; but they do not seem to have known Pearl's contributions well. Cf. ref. 41.
124. G.F. Gause, The struggle for existence (Baltimore, 1934). Gause later published a volume in Teissier's series: Gause, Vérifications expérimentales de la lutte pour la vie (Paris, 1935).
125. G.F. Gause, The struggle for existence (Baltimore, 1934). Gause later published a volume in Teissier's series: Gause, Vérifications expérimentales de la lutte pour la vie (Paris, 1935).
126. Volterra, "La théorie mathématique" (ref. 21); Gause to Volterra, 23 Dec 1935.
127. Volterra, "La théorie mathématique" (ref. 21); Gause to Volterra, 23 Dec 1935.
128. D'Ancona to Volterra, 31 Jan 1935. Emphasis is not in the original.
129. D'Ancona to Volterra, 6 Feb 1935; Régnier to Volterra, 18 Nov 1934. Cf. D'Ancona to Volterra (ref. 75) and ref. 22.
130. D'Ancona to Volterra, 25 Apr 1935; Volterra to Koshitzin, draft 13 Nov 1935: "M. Chapman vérifie les lois des fluctuations que j'ai données." He referred to Chapman (ref. 56).
131. J. Stanley, "A mathematical theory of the growth of populations of the flour beetle, Tribolium confusum Duval," Canadian journal of research, 6 (1931), 632 ff., 7 (1932), 126 ff.; Stanley to Volterra, 29 Mar 1935.
132. J. Stanley, "A mathematical theory of the growth of populations of the flour beetle, Tribolium confusum Duval," Canadian journal of research, 6 (1931), 632 ff., 7 (1932), 126 ff.; Stanley to Volterra, 29 Mar 1935.
133. J. Stanley, "A mathematical theory of the growth of populations of the flour beetle, Tribolium confusum Duval," Canadian journal of research, 6 (1931), 632 ff., 7 (1932), 126 ff.; Stanley to Volterra, 29 Mar 1935.
134. Volterra and D'Ancona (ref. 68), 73.
135. Ibid., 8-9.
136. Ibid., 83-84.
137. Ibid., 82-83. Cf. V.A. Bailey, "The interaction between host and parasites," Quarterly journal of mathematics, 2 (1931), 68-77; A.J. Nicholson, "The balance of animal populations," Journal of animal ecology, 2 (1933), 132-178; Nicholson and Bailey, "The balance of animal populations," Zoological Society of London, Proceedings, part 3 (1935), 551-598; Kingsland (ref. 1), 139 ff.
138. Ibid., 82-83. Cf. V.A. Bailey, "The interaction between host and parasites," Quarterly journal of mathematics, 2 (1931), 68-77; A.J. Nicholson, "The balance of animal populations," Journal of animal ecology, 2 (1933), 132-178; Nicholson and Bailey, "The balance of animal populations," Zoological Society of London, Proceedings, part 3 (1935), 551-598; Kingsland (ref. 1), 139 ff.
139. Ibid., 82-83. Cf. V.A. Bailey, "The interaction between host and parasites," Quarterly journal of mathematics, 2 (1931), 68-77; A.J. Nicholson, "The balance of animal populations," Journal of animal ecology, 2 (1933), 132-178; Nicholson and Bailey, "The balance of animal populations," Zoological Society of London, Proceedings, part 3 (1935), 551-598; Kingsland (ref. 1), 139 ff.
140. Ibid., 82-83. Cf. V.A. Bailey, "The interaction between host and parasites," Quarterly journal of mathematics, 2 (1931), 68-77; A.J. Nicholson, "The balance of animal populations," Journal of animal ecology, 2 (1933), 132-178; Nicholson and Bailey, "The balance of animal populations," Zoological Society of London, Proceedings, part 3 (1935), 551-598; Kingsland (ref. 1), 139 ff.
141. Ibid., 82-83. Cf. V.A. Bailey, "The interaction between host and parasites," Quarterly journal of mathematics, 2 (1931), 68-77; A.J. Nicholson, "The balance of animal populations," Journal of animal ecology, 2 (1933), 132-178; Nicholson and Bailey, "The balance of animal populations," Zoological Society of London, Proceedings, part 3 (1935), 551-598; Kingsland (ref. 1), 139 ff.
142. Gause, Vérifications (ref. 82), 9-10; Israel, "The origins" (ref. 9).
143. Gause, Vérifications (ref. 82), 9-10; Israel, "The origins" (ref. 9).
144. Gause, Vérifications (ref. 82), 42-43.
145. Israel, "Volterra's analytical mechanics" (ref. 9).
146. Volterra, "Population growth, equilibria and extinction under specified breeding conditions: A development and extension of the theory of the logistic curve," Human Biology, 10 (1938), 1-11; "Calculus of variations and the logistic curve," ibid., 11 (1939), 173-178.
147. Volterra, "Population growth, equilibria and extinction under specified breeding conditions: A development and extension of the theory of the logistic curve," Human Biology, 10 (1938), 1-11; "Calculus of variations and the logistic curve," ibid., 11 (1939), 173-178.
148. D'Ancona to Volterra. 24 Oct 1937; Volterra to D'Ancona, draft, 5 Nov 1937; D'Ancona to Volterra, 11 Nov 1937. Volterra met Pearl in London in November 1937 and decided to send the paper to his journal. Vladimir Kostitzin pointed out that the basic idea of the first part of this paper had been advanced by Léon Brillouin, Notions élémentaires de mathématiques pour les sciences expérimentales (Paris, 1935), and by himself in "Sur une généralisation des équations biologiques dans le cas d'une population intoxiquée." CRAS, 204 (1937), 1683 (Kostitzin to Volterra, 5 May 1938).
149. D'Ancona to Volterra. 24 Oct 1937; Volterra to D'Ancona, draft, 5 Nov 1937; D'Ancona to Volterra, 11 Nov 1937. Volterra met Pearl in London in November 1937 and decided to send the paper to his journal. Vladimir Kostitzin pointed out that the basic idea of the first part of this paper had been advanced by Léon Brillouin, Notions élémentaires de mathématiques pour les sciences expérimentales (Paris, 1935), and by himself in "Sur une généralisation des équations biologiques dans le cas d'une population intoxiquée." CRAS, 204 (1937), 1683 (Kostitzin to Volterra, 5 May 1938).
150. D'Ancona to Volterra. 24 Oct 1937; Volterra to D'Ancona, draft, 5 Nov 1937; D'Ancona to Volterra, 11 Nov 1937. Volterra met Pearl in London in November 1937 and decided to send the paper to his journal. Vladimir Kostitzin pointed out that the basic idea of the first part of this paper had been advanced by Léon Brillouin, Notions élémentaires de mathématiques pour les sciences expérimentales (Paris, 1935), and by himself in "Sur une généralisation des équations biologiques dans le cas d'une population intoxiquée." CRAS, 204 (1937), 1683 (Kostitzin to Volterra, 5 May 1938).
151. D'Ancona to Volterra. 24 Oct 1937; Volterra to D'Ancona, draft, 5 Nov 1937; D'Ancona to Volterra, 11 Nov 1937. Volterra met Pearl in London in November 1937 and decided to send the paper to his journal. Vladimir Kostitzin pointed out that the basic idea of the first part of this paper had been advanced by Léon Brillouin, Notions élémentaires de mathématiques pour les sciences expérimentales (Paris, 1935), and by himself in "Sur une généralisation des équations biologiques dans le cas d'une population intoxiquée." CRAS, 204 (1937), 1683 (Kostitzin to Volterra, 5 May 1938).
152. D'Ancona to Volterra. 24 Oct 1937; Volterra to D'Ancona, draft, 5 Nov 1937; D'Ancona to Volterra, 11 Nov 1937. Volterra met Pearl in London in November 1937 and decided to send the paper to his journal. Vladimir Kostitzin pointed out that the basic idea of the first part of this paper had been advanced by Léon Brillouin, Notions élémentaires de mathématiques pour les sciences expérimentales (Paris, 1935), and by himself in "Sur une généralisation des équations biologiques dans le cas d'une population intoxiquée." CRAS, 204 (1937), 1683 (Kostitzin to Volterra, 5 May 1938).
153. D'Ancona to Volterra. 24 Oct 1937; Volterra to D'Ancona, draft, 5 Nov 1937; D'Ancona to Volterra, 11 Nov 1937. Volterra met Pearl in London in November 1937 and decided to send the paper to his journal. Vladimir Kostitzin pointed out that the basic idea of the first part of this paper had been advanced by Léon Brillouin, Notions élémentaires de mathématiques pour les sciences expérimentales (Paris, 1935), and by himself in "Sur une généralisation des équations biologiques dans le cas d'une population intoxiquée." CRAS, 204 (1937), 1683 (Kostitzin to Volterra, 5 May 1938).
154. Kostitzin (ref. 52) and Evolution de l'atmosphère, circulation organique, époques glaciaires (Paris, 1935).
155. Kostitzin (ref. 77), translated as Mathematical biology (London, 1939). An English translation of other works by Kostitzin can be found in Scuds and Ziegler, The "Golden Age" of theoretical ecology, 1923-1940 (Berlin, 1978).
156. Kostitzin (ref. 77), translated as Mathematical biology (London, 1939). An English translation of other works by Kostitzin can be found in Scuds and Ziegler, The "Golden Age" of theoretical ecology, 1923-1940 (Berlin, 1978).
157. Kostitzin to Volterra, 13 Aug 1935. The mathematical content of the letter is reproduced without modifications in Kostitzin, "Sur l'intoxication d'un milieu par les produits cataboliques d'une population," CRAS, 201 (1935), 516-518.
158. Kostitzin to Volterra, 13 Aug 1935. The mathematical content of the letter is reproduced without modifications in Kostitzin, "Sur l'intoxication d'un milieu par les produits cataboliques d'une population," CRAS, 201 (1935), 516-518.
159. Kostitzin to Volterra, 24 Aug 1935.
160. Kostitzin (ref. 77), 66-72;
161. Israel and Millan Gasca (ref. 11). Kostizin began to collaborate with Régnier in late 1935: Régnier to Volterra, 3 Nov 1935, and Kostitzin to Volterra, 6 Nov 1935.
162. Régnier to Volterra, 13 Apr 1936. Régnier and Lambin were preparing a general exposition of microbial antagonism, that was to relate to their studies of Gause, W. Ludwig, and Oscar W. Richards, and to emphasize quantitive methods.
163. Kostitzin "Sur une généralisation des équations biologiques dans le cas d'une population intoxiquée par les produits de son activité chimique," CRAS, 204 (1937), 1683-1685. Kostitzin to Volterra, 12 June 1937.
164. Kostitzin "Sur une généralisation des équations biologiques dans le cas d'une population intoxiquée par les produits de son activité chimique," CRAS, 204 (1937), 1683-1685. Kostitzin to Volterra, 12 June 1937.
165. Lambin to Volterra, 17 Jul 1938.
166. Volterra a Régnier, draft, n.d.
167. Régnier to Volterra, 11 Oct 1938; Kostitzin to Volterra, 12 Oct 1938; Kostitzin to Volterra, 27 Oct 1938.
168. Kostitzin to Volterra, 2 Nov 1938. He mentions his book Biologie mathématique, cf. ref. 101.
169. Régnier and Lambin, "Etude sur le croít microbien en fonction de la quantité de substance nutritive des milieux de culture," CRAS, 207 (1938), 1263; Kostitzin and Volterra, "Remarques sur l'action toxique du milieu à propos de la note de M. Régnier et Mlle. Lambin," CRAS, 207 (1938), 1146-1148.
170. Régnier and Lambin, "Etude sur le croít microbien en fonction de la quantité de substance nutritive des milieux de culture," CRAS, 207 (1938), 1263; Kostitzin and Volterra, "Remarques sur l'action toxique du milieu à propos de la note de M. Régnier et Mlle. Lambin," CRAS, 207 (1938), 1146-1148.
171. Kostitzin to Volterra, 27 Oct 1938; in later letters, Kostitzin considered the general problem of the integrodifferential equation 3 (Kostitzin to Volterra, 6 Dec 1938, 9 Mar 1939 and 3 June 1939). Cf. Kostitzin, "Sur les équations integro-différentielles de la théorie de l'action toxique du milieu," CRAS, 208 (1939), 1545-1547.
172. Volterra, "Sui tentativi di applicazione delle matematiche alle scienze biologiche e sociali," in Universitá di Roma, Annuario (1901), 3-28.
173. Volterra, "Fluctuations dans la lutte pour la vie: leurs lois fondamentales et de réciprocité," in Opere matematiche. Memorie e note (Rome, 1960), 5, 482-495, on 482; Volterra, "Applications des mathématiques à la biologie," L'Enseignement mathématique, 36 (1937), 297-330. Volterra used Pearl's foreword to Gause's Struggle for existence. Cf. ref. 81-83.
174. Volterra, "Fluctuations dans la lutte pour la vie: leurs lois fondamentales et de réciprocité," in Opere matematiche. Memorie e note (Rome, 1960), 5, 482-495, on 482; Volterra, "Applications des mathématiques à la biologie," L'Enseignement mathématique, 36 (1937), 297-330. Volterra used Pearl's foreword to Gause's Struggle for existence. Cf. ref. 81-83.
175. Volterra, "Fluctuations dans la lutte pour la vie: leurs lois fondamentales et de réciprocité," in Opere matematiche. Memorie e note (Rome, 1960), 5, 482-495, on 482; Volterra, "Applications des mathématiques à la biologie," L'Enseignement mathématique, 36 (1937), 297-330. Volterra used Pearl's foreword to Gause's Struggle for existence. Cf. ref. 81-83.
176. D'Ancona to Volterra, 23 Feb 1935.
177. D'Ancona to Volterra, 28 Feb 1935.
178. D'Ancona to Volterra, 1 May 1936.
179. L'Heritier told E. Mayr that "he and Teisier had no difficulty in accepting the Darwinian interpretation, particularly the noninheritance of acquired characters and the principle of natural selection, because as members of a mathematics department they had not been exposed to the prevailing French ideas and natural selection was an eminently reasonable process for a mathematician" (E. Mayr, "The arrival of neo-Darwinism in France," in Mayr and Provine (ref. 2), 322). These reasons held also for Kostitzin and Volterra.
180. D'Ancona to Volterra, 14 Mar 1937.
181. D'Ancona to Volterra, 18 Mar 1937.
182. Kostitziri to Volterra, 5 Apr 1937.
183. Kostitzin to Volterra, 12 June 1937. A. Sutherland, The origin and growth of the moral instinct (2 vols., London, 1898). On the film, see A. Millán Gasca, "Immagini matematiche della vita," Prometeo, 10 (1993), no. 43, 88-96.
184. Kostitzin to Volterra, 12 June 1937. A. Sutherland, The origin and growth of the moral instinct (2 vols., London, 1898). On the film, see A. Millán Gasca, "Immagini matematiche della vita," Prometeo, 10 (1993), no. 43, 88-96.
185. Kostitzin to Volterra, 12 June 1937.
186. Kostitzin to Volterra, 15 May 1938. Kostitzin, "Sélection naturelle et transformation des espèces du point du vue analytique, statistique et biologique,' CRAS, 206 (1938), 1442-1444.
187. Kostitzin to Volterra, 19 Sep 1939.
188. Kostitzin to Volterra, 31 Dec 1935. D'Ancona had told Volterra that Kostizin gave too much importance to Lotka and did not consider his contributions sufficiently in that first monograph; D'Ancona to Volterra, 30 Nov 1933.
189. Kostitzin to Volterra, 4 Mar 1937.
190. Kostitzin to Mrs. Volterra, 14 Feb 1946. The only paper on the topics that he had studied together with Volterra and Régnier that Kostitzin published after the war was Kostitzin, "Sur le développement des populations bactériennes," CRAS, 242 (1956), 611-612.
191. Robert K. May, "Will a large complex system be stable?," Nature, 238 (1972), 413; and Stability and complexity in model ecosystems (Princeton, 1974).
192. Robert K. May, "Will a large complex system be stable?," Nature, 238 (1972), 413; and Stability and complexity in model ecosystems (Princeton, 1974).
193. E. Mayr, The growth of biological thought. Diversity, evolution and inheritance (Cambridge, 1982), chapt. 1.
194. Ingrao and Israel (ref. 10).
195. S. Levin, ed., Frontiers in mathematical biology (Berlin, 1994), ix.
196. J.M. Smith, Models in ecology (Cambridge, 1974).
197. G. Oster and J. Guckenheimer, "Bifurcation phenomena in population models," in J.E. Marsden, M. McCracken, eds. The Hopf bifurcation and its applications (New York, 1976), 327-328.