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Derivations of Lotka-Volterra algebras

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Abstract

Lotka-Volterra algebras emerge in connection with biological problems and Lotka-Volterra systems for the interactions of neighboring individuals. The purpose of this paper is to study the structure of a Lotka-Volterra algebra \(\mathbf {A}\) and its algebra of derivations. This algebra \(\mathbf {A}\) is defined through antisymmetric matrices, and we give explicitly its derivations space. Finally, we prove that a conjecture of [7] is false. We find a local derivation of a 4-dimensional Lotka-Volterra algebra \(\mathbf {A}\) into itself which is not a derivation of \(\mathbf {A}\).

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References

  1. Boujemaa, H., Rachidi, M., Micali, A.: Sur les algbres de Lotka-Volterra. Algebras Gr. Geom. 10, 169–180 (1993)

    Google Scholar 

  2. Boujemaa, H., Rachidi, M., Micali, A.: Automorphisms in Lotka-Volterra algebras. Algebras Gr. Geom. 18, 203–222 (2001)

    MATH  Google Scholar 

  3. Boujemaa, H., Rachidi, M., Micali, A.: Dérivations dans les algbres de Lotka-Volterra. Algebras Gr. Geom. 2, 471–487 (2004)

    MATH  Google Scholar 

  4. Fernandez, Juan, C.Gutierrez: Solution of the Bernstein problem in the non-regular case. J. Algebra 223, 109–132 (2000)

    Article  MathSciNet  Google Scholar 

  5. Fernandez, Juan, C.Gutierrez: Nuclear Bernstein algebras with a stochastic basis. J. Algebra 217, 300–311 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  6. Fernandez, Juan C. Gutierrez, Garcia, Claudia I.: Antisymetric matrices and Lotka-Volterra algebras, arXiv:1801.08888

  7. Ganikhodzhaev, R., Mukhamedov, F., Pirnapasov, A., Qaralleh, I.: Genetic Volterra algebras and their derivations. Commun. Algebra (2017). https://doi.org/10.1080/00927872.2017.1347663

  8. Holgate, P.: Genetic algebras satisfying Bernstein’s stationary principle. J. London Math. Soc. 9, 621–624 (1975)

    MATH  Google Scholar 

  9. Holgate, P.: The interpretation of derivations in genetic algebras. Linear Algebra Appl. 85, 75–79 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  10. Itoh, Y.: Nonassociative algebra and Lotka-Volterra equation with ternary interaction. Nonlinear Anal. 5, 53–56 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kadison, R.V.: Local derivations. J. Algebra 130, 494–509 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kimura, M.: On the change of population fitness by natural selection. Heredity 12, 145–167 (1958)

    Article  Google Scholar 

  13. Ho-Hon Leung, Derivation on four dimensional genetic Volterra algebra, arXiv:1712.09301v1

  14. Lyubich, Y.: Mathemmatical structures in population genetics. Springer, Berlin (1983)

    Google Scholar 

  15. Mather, K.: Selection through competition. Heredity 24, 529–540 (1969)

    Article  Google Scholar 

  16. Yoon, S.I.: Automorphisms of Lotka-Volterra algebras. Commun. Korean Math. Soc. 12, 45–50 (1997)

    MathSciNet  MATH  Google Scholar 

  17. Yoon, S.I.: Idempotent elements in the Lotka-Volterra algebra. Commun. Korean Math. Soc. 10, 123 (1995)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Juan C. Gutierrez Fernandez was supported by FAPESP, Proc. 2014/09310-5.

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Authors and Affiliations

  1. Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão, 1010, São Paulo, CEP, 05508-090, Brazil

    Juan C. Gutierrez Fernandez

  2. EACH, Universidade de São Paulo, São Paulo, CEP, 03828-000, Brazil

    Claudia I. Garcia

Authors
  1. Juan C. Gutierrez Fernandez
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  2. Claudia I. Garcia
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Correspondence to Juan C. Gutierrez Fernandez.

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Fernandez, J.C.G., Garcia, C.I. Derivations of Lotka-Volterra algebras. São Paulo J. Math. Sci. 13, 292–304 (2019). https://doi.org/10.1007/s40863-018-0090-3

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  • DOI: https://doi.org/10.1007/s40863-018-0090-3

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