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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Ahmad, S., Stamova, I.M.: Asymptotic stability of an \(N\)-dimensional impulsive competitive system. Nonlinear Anal. Real World Appl. 8, 654–663 (2007)
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
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)
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)
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)
Gopalsamy, K.: Exchange of equilibria in two species Lotka-Volterra competition models. J. Aust. Math. Soc. Ser. B 24, 160–170 (1982)
Gopalsamy, K.: Global asymptotic stability in Volterras population systems. J. Math. Biol. 19, 157–168 (1984)
Gopalsamy, K.: Global asymptotic stability in a periodic Lotka-Volterra system. J. Aust. Math. Soc. Ser. B 27, 66–72 (1985)
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)
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)
Hu, H., Jiang, J.: Translation-invariant monotone systems, I: autonomous/periodic case. Nonlinear Anal. Real World Appl. 11(4), 3211–3217 (2010)
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)
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)
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)
Itoh, Y.: Integrals of a Lotka-Volterra system of infinite species. Progr. Theor. Phys. 80(5), 749–751 (1988)
Lotka, A.J.: Contribution to the Theory of Periodic Reaction. J. Phys. Chem. 14, 271–274 (1910)
Paparella, F., Pascali, E.: Existence and unicity of solutions for a non-local relaxation equation. Nonlinear Anal. 70, 1702–1710 (2009)
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)
Shipley, B.: The relationship between dynamic game theory and Lotka Volterra competition equations. J. Theor. Biol. 125, 121–123 (1987)
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)
Volterra, V.: Lecons sur la theorie mathematique de la lutte pour la vie. Gauthier-Villars, Paris (1931)
Yonghui, X.: New results on the global asymptotic stability of a Lotka-Volterra system. J. Appl. Math. Comput. 36, 117–128 (2011)
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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