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A model for the problem of the cooperation/competition between infinite continuous species

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Abstract

In this paper we consider a particular type of differential equation that we can consider as a simple model for the problem of the cooperation/competition of infinite species. In this model each of the species meets each of the other species with a degree of competition or cooperation and their arrangements affect the evolution of the species. A first result of the existence of a unique, local-in-time, solution is given.

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References

  1. Ahmad, S., Stamova, I.M.: Asymptotic stability of an \(N\)-dimensional impulsive competitive system. Nonlinear Anal. Real World Appl. 8, 654–663 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Brauer, F., Castillo-Chvez, C.: Mathematical models in population biology and epidemiology. In: Texts in Applied Mathematics, vol. 40. Springer, New York (2001) xxiv+416 pp. ISBN: 0-387-98902-1

  3. Cairó, L., Llibre, J.: Phase portraits of cubic polynomial vector fields of Lotka-Volterra type having a rational first integral of degree 2. J. Phys. A 40(24), 6329–6348 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Evans, S.N.: Infinitely-many-species Lotka-Volterra equations arising from systems of coalescing masses. J. Lond. Math. Soc. 2 60(1), 171–186 (1999)

    Article  Google Scholar 

  5. Hongxiao, H., Wang, K., Wu, D.: Permanence and global stability for nonautonomous \(N\)-species Lotka Volterra competitive system with impulses and infinite delays. J. Math. Anal. Appl. 377, 145–160 (2011)

    Google Scholar 

  6. Gopalsamy, K.: Exchange of equilibria in two species Lotka-Volterra competition models. J. Aust. Math. Soc. Ser. B 24, 160–170 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  7. Gopalsamy, K.: Global asymptotic stability in Volterras population systems. J. Math. Biol. 19, 157–168 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gopalsamy, K.: Global asymptotic stability in a periodic Lotka-Volterra system. J. Aust. Math. Soc. Ser. B 27, 66–72 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hou, J., Teng, Z., Gao, S.: Permanence and global stability for nonautonomous \(N\)-species Lotka Valterra competitive system with impulses. Nonlinear Anal. Real World Appl. 11, 1882–1896 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hu, H., Wang, K., Wu, D.: Permanence and global stability for nonautonomous N-species Lotka-Volterra competitive system with impulses and infinite delays. J. Math. Anal. Appl. 377(1), 145–160 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hu, H., Jiang, J.: Translation-invariant monotone systems, I: autonomous/periodic case. Nonlinear Anal. Real World Appl. 11(4), 3211–3217 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hu, H., Teng, Z., Jiang, H.: Permanence of the nonautonomous competitive systems with infinite delay and feedback controls. Nonlinear Anal. Real World Appl. 10(4), 2420–2433 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hu, H., Teng, Z., Jiang, H.: On the permanence in non-autonomous Lotka-Volterra competitive system with pure-delays and feedback controls. Nonlinear Anal. Real World Appl. 10(3), 1803–1815 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hu, H., Teng, Z., Gao, S.: Extinction in nonautonomous Lotka-Volterra competitive system with pure-delays and feedback controls. Nonlinear Anal. Real World Appl. 10(4), 2508–2520 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Itoh, Y.: Integrals of a Lotka-Volterra system of infinite species. Progr. Theor. Phys. 80(5), 749–751 (1988)

    Article  Google Scholar 

  16. Lotka, A.J.: Contribution to the Theory of Periodic Reaction. J. Phys. Chem. 14, 271–274 (1910)

    Article  Google Scholar 

  17. Paparella, F., Pascali, E.: Existence and unicity of solutions for a non-local relaxation equation. Nonlinear Anal. 70, 1702–1710 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Pertsev, N.V., Pichugina, A.N.: Behavior of solutions of a dissipative integral Lotka-Volterra model. Sib. Zh. Ind. Mat. 6(2), 95–106 (2003)

    MathSciNet  MATH  Google Scholar 

  19. Shipley, B.: The relationship between dynamic game theory and Lotka Volterra competition equations. J. Theor. Biol. 125, 121–123 (1987)

    Article  MathSciNet  Google Scholar 

  20. Volterra, V.: Variations and fluctuations of the number of individuals in animal species living together in animal ecology. In: Chapman, R.N. (ed) McGraw Hill, New York (1931)

  21. Volterra, V.: Lecons sur la theorie mathematique de la lutte pour la vie. Gauthier-Villars, Paris (1931)

    Google Scholar 

  22. Yonghui, X.: New results on the global asymptotic stability of a Lotka-Volterra system. J. Appl. Math. Comput. 36, 117–128 (2011)

    Google Scholar 

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Correspondence to E. Pascali.

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Le, U.V., Pascali, E. A model for the problem of the cooperation/competition between infinite continuous species. Ricerche mat. 62, 139–153 (2013). https://doi.org/10.1007/s11587-013-0148-6

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  • DOI: https://doi.org/10.1007/s11587-013-0148-6

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