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Evolutionary stable strategies and cubic vector fields

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Abstract

The introduction of concepts of Game Theory and Ordinary Differential Equations into Biology gave birth to the field of Evolutionary Stable Strategies, with applications in Biology, Genetics, Politics, Economics and others. In special, the model composed by two players having two pure strategies each results in a planar cubic vector field with an invariant octothorpe. Therefore, in this paper we study such class of vector fields, suggesting the notion of genericity and providing the global phase portraits of the generic systems with a singularity at the central region of the octothorpe.

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References

  1. Accinelli, E., Martins, F., Oviedo, J., Pinto, A., Quintas, L.: Who controls the controller? A dynamical model of corruption. J. Math. Sociol. 41, 220–247 (2017)

    MathSciNet  Google Scholar 

  2. Artés, J., Llibre, J.: Quadratic Hamiltonian vector fields. J. Differ. Equ. 107, 80–95 (1994)

    MathSciNet  Google Scholar 

  3. Artés, J., Llibre, J.: Quadratic vector fields with a weak focus of third order. Publ. Mat. 41, 7–39 (1997)

    MathSciNet  Google Scholar 

  4. Artés, J., Kooij, R., Llibre, J.: Structurally Stable Quadratic Vector Fields, vol. 639. American Mathematical Society, Providence (1998)

    Google Scholar 

  5. Artés, J., Rezende, A., Llibre, J.: Structurally Unstable Quadratic Vector Fields of Codimension One. Birkhäuser, Cham (2018)

    Google Scholar 

  6. Artés, J., Oliveira, R., Rezende, A.: Structurally unstable quadratic vector fields of codimension two: families possessing either a cusp point or two finite saddle-nodes. J. Dyn. Differ. Equ. 33, 1779–1821 (2021)

    MathSciNet  Google Scholar 

  7. Artés, J., Mota, M., Rezende, A.: Structurally unstable quadratic vector fields of codimension two: families possessing a finite saddle-node and an infinite saddle-node. Electron. J. Qual. Theory Differ. Equ. 35, 1–89 (2021)

    MathSciNet  Google Scholar 

  8. Bomze, I.: Lotka–Volterra equation and replicator dynamics: a two-dimensional classification. Biol. Cybern. 48, 201–211 (1983)

    Google Scholar 

  9. Buendía, J., López, V.: On the Markus–Neumann theorem. J. Differ. Equ. 265, 6036–6047 (2018)

    MathSciNet  Google Scholar 

  10. Cairó, L., Llibre, J.: Phase portraits of quadratic polynomial vector fields having a rational first integral of degree 2. Nonlinear Anal. Theory Methods Appl. 67, 327–348 (2007)

    MathSciNet  Google Scholar 

  11. Cao, F., Jiang, J.: The classification on the global phase portraits of two-dimensional Lotka–Volterra system. J. Dyn. Differ. Equ. 20, 797–830 (2008)

    MathSciNet  Google Scholar 

  12. Cherkas, L.: The stability of singular cycles. Differ. Uravn. 4, 1012–1017 (1968)

    MathSciNet  Google Scholar 

  13. Cozma, D., Suba, A.: Solution of the problem of the centre for, a cubic differential system with three invariant straight lines. Qual. Theory Dyn. Syst. 2, 129–143 (2001)

    MathSciNet  Google Scholar 

  14. Cozma, D., Suba, A.: The solution of the problem of center for cubic differential systems with four invariant straight lines. Sci. Ann. Al. I. Cuza Univ. Math. 44, 517–530 (1998)

    MathSciNet  Google Scholar 

  15. Dumortier, F., Llibre, J., Artés, J.: Qualitative theory of planar differential systems. In: Universitext. Springer, Berlim (2006)

  16. Gouveia, M., Llibre, J., Roberto, L.: Phase portraits of the quadratic polynomial Liénard differential systems. Proc. R. Soc. Edinb. Sect. A Math. 151, 202–216 (2020)

    Google Scholar 

  17. Hines, W.: Evolutionary stable strategies: a review of basic theory. Theor. Popul. Biol. 31(2), 195–272 (1987)

    MathSciNet  Google Scholar 

  18. Hofbauer, J.: On the occurrence of limit cycles in the Volterra–Lotka equation. Nonlinear Anal. Theory Methods Appl. 5, 1003–1007 (1981)

    MathSciNet  Google Scholar 

  19. Kertész, V., Kooij, R.: Degenerate Hopf bifurcation in two dimensions. Nonlinear Anal. Theory Methods Appl. 17, 267–283 (1991)

    MathSciNet  Google Scholar 

  20. Kooij, R.: Cubic systems with four real line invariants. Math. Proc. Camb. Philos. Soc. 118, 7–19 (1995)

    MathSciNet  Google Scholar 

  21. Kooij, R.: Real polynomial systems of degree n with n+1 line invariants. J. Differ. Equ. 116, 249–264 (1995)

    MathSciNet  Google Scholar 

  22. Li, T., Llibre, J.: Phase portraits of separable quadratic systems and a bibliographical survey on quadratic systems. Expo. Math. 39, 540–565 (2021)

    MathSciNet  Google Scholar 

  23. Liang, H., Zhao, Y.: Limit cycles bifurcated from a class of quadratic reversible center of genus one. J. Math. Anal. Appl. 391, 240–254 (2012)

    MathSciNet  Google Scholar 

  24. Llibre, J., Oliveira, R.: Phase portraits of quadratic polynomial vector fields having a rational first integral of degree 3. Nonlinear Anal. Theory Methods Appl. 70, 3549–3560 (2009)

    MathSciNet  Google Scholar 

  25. Llibre, J., Pereira, W., Pessoa, C.: Phase portraits of Bernoulli quadratic polynomial differential systems. Electron. J. Differ. Equ. 48, 1–19 (2020)

    MathSciNet  Google Scholar 

  26. Llibre, J., Rodrigues, A.: Periodic orbits for real planar polynomial vector fields of degree n having n invariant straight lines taking into account their multiplicities. Electron. J. Qual. Theory Differ. Equ. 55, 1–15 (2015)

    MathSciNet  Google Scholar 

  27. Llibre, J., Silva, M.: Phase portraits of integrable quadratic systems with an invariant parabola and an invariant straight line. C.R. Math. 357, 143–166 (2019)

    MathSciNet  Google Scholar 

  28. Llibre, J., Valls, C.: Global phase portraits of quadratic systems with a complex ellipse as invariant algebraic curve. Acta. Math. Sin. Engl. Ser. 35, 801–811 (2018)

    MathSciNet  Google Scholar 

  29. Markus, L.: Global structure of ordinary differential equations in the plane. Trans. Am. Math. Soc. 76, 127–148 (1954)

    MathSciNet  Google Scholar 

  30. Neumann, D.: Classification of continuous flows on 2-manifolds. Proc. Am. Math. Soc. 48, 73–81 (1975)

    MathSciNet  Google Scholar 

  31. Ozkan-Canbolat, E., Beraha, A.: Evolutionary stable strategies for business innovation and knowledge transfer. Int. J. Innov. Stud. 3, 55–70 (2019)

    Google Scholar 

  32. Peixoto, M.C., Peixoto, M.M.: Structural stability in the plane with enlarged boundary conditions. An. Acad. Bras. Ciênc. 31, 135–160 (1959)

    MathSciNet  Google Scholar 

  33. Peixoto, M.M.: On structural stability. Ann. Math. 69, 199–222 (1959)

    MathSciNet  Google Scholar 

  34. Peixoto, M.M.: Structural stability on two-dimensional manifolds. Topology 1, 101–120 (1961)

    MathSciNet  Google Scholar 

  35. Peixoto, M.M.: On the classification of flows on 2-manifolds. In: Dynamical Systems (Proceedings of a Symposium Held at the University of Bahia, Salvador, 1971), pp. 389–419 (1973)

  36. Pereira, W., Pessoa, C.: A class of reversible quadratic polynomial vector fields on \({\mathbb{S} }^2\). J. Math. Anal. Appl. 371, 203–209 (2010)

    MathSciNet  Google Scholar 

  37. Pereira, W., Pessoa, C.: On the reversible quadratic polynomial vector fields on \({\mathbb{S} }^2\). J. Math. Anal. Appl. 396, 455–465 (2012)

    MathSciNet  Google Scholar 

  38. Perko, L.: Differential Equations and Dynamical Systems, vol. 7 of Texts in Applied Mathematics, 3rd edn. Springer, New York (2001)

  39. Reyn, J.: Phase portraits of quadratic systems without finite critical points. Nonlinear Anal. Theory Methods Appl. 27, 207–222 (1996)

    MathSciNet  Google Scholar 

  40. Reyn, J.: Phase portraits of quadratic systems with finite multiplicity one. Nonlinear Anal. Theory Methods Appl. 28, 755–778 (1997)

    MathSciNet  Google Scholar 

  41. Reyn, J.: Phase Portraits of Planar Quadratic Systems, 1st edn., p. 334+XVI. Springer, New York (2007)

    Google Scholar 

  42. Schlomiuk, D., Vulpe, N.: The full study of planar quadratic differential systems possessing a line of singularities at infinity. J. Dyn. Differ. Equ. 20, 737–775 (2008)

    MathSciNet  Google Scholar 

  43. Schuster, P., Sigmund, K.: Coyness, philandering and stable strategies. Anim. Behav. 29, 186–192 (1981)

    Google Scholar 

  44. Schuster, P., Sigmund, K., Hofbauer, J., et al.: Selfregulation of behaviour in animal societies III. Games between two populations with selfinteraction. Biol. Cybern. 40, 17–25 (1981)

    Google Scholar 

  45. Smith, J.M., Price, G.: The logic of animal conflict. Nature 246, 15–18 (1973)

    Google Scholar 

  46. Sotomayor, J.: Generic one-parameter families of vector fields on two-dimensional manifolds. Publ. Math. Inst. H. É. Sci. 43, 5–46 (1974)

    MathSciNet  Google Scholar 

  47. Taylor, P., Jonker, L.: Evolutionary stable strategies and game dynamics. Math. Biosci. 40, 145–156 (1978)

    MathSciNet  Google Scholar 

  48. Velasco, E.: Generic properties of polynomial vector fields at infinity. Trans. Am. Math. Soc. 143, 201–222 (1969)

    MathSciNet  Google Scholar 

  49. Xian, W., Kooij, R.: Limit Cycles in a Cubic System with a Cusp. J. Math. Anal. 23, 1609–1622 (1992)

    MathSciNet  Google Scholar 

  50. Xiao, T., Yu, G.: Supply chain disruption management and evolutionarily stable strategies of retailers in the quantity-setting duopoly situation with homogeneous goods. Eur. J. Oper. Res. 173, 648–668 (2006)

    MathSciNet  Google Scholar 

  51. Zeeman, E.: Population dynamics from game theory. In: Global Theory of Dynamical Systems pp. 471–497 (1980)

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Acknowledgements

The authors are grateful to professor Armengol Gasull for sharing the idea of applying the tools of Dynamic Systems to study the models of Evolutionary Stable Strategies.

Funding

The authors are partially supported by CNPq, Grant 304798/2019-3 and by São Paulo Research Foundation (FAPESP), Grants 2019/10269-3 and 2021/01799-9.

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Authors and Affiliations

  1. IBILCE–UNESP, São José do Rio Preto, São Paulo, CEP 15054–000, Brazil

    Jefferson Bastos, Claudio Buzzi & Paulo Santana

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  1. Jefferson Bastos
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  2. Claudio Buzzi
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All authors contributed equally to this work.

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Correspondence to Paulo Santana.

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Appendices

Appendix A: Phase portraits under the hypothesis of Theorem 10 (Family 2)

To simplify Table 6, we denote \(\Delta =(b_{10}-a_{01})^2+4b_{01}\) (Figs. 19, 20 and 21).

Table 6 Table of the realizable cases under hypothesis of Theorem 10 (Family 2)
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Fig. 19
figure 19

Phase portraits from Cases 2.1 to 2.4

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Fig. 20
figure 20

Phase portraits from Cases 2.4 to 2.9

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Fig. 21
figure 21

Phase portraits from Cases 2.10 to 2.15

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Appendix B: Phase portraits under the hypothesis of Theorem 11 (Family 3)

See the Table 7 and Fig. 22.

Table 7 Table of the realizable cases under hypothesis of Theorem 11 (Family 3)
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Fig. 22
figure 22

Phase portraits of Family 3

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Appendix C: Phase portraits under the hypothesis of Theorem 12 (Family 4)

To simplify the Table 8, we denote (Figs. 23, 24 and 25),

$$\begin{aligned}{} & {} \det A=b_{01}-a_{01}b_{10}, \quad T=(\alpha -1)\alpha +b_{01}(\beta -1)\beta , \\{} & {} K=a_{01}b_{01}(\beta -1)\beta -b_{10}(\alpha -1)\alpha ,\\{} & {} \Delta =(b_{10}-a_{01})^2+4b_{01}, \quad \delta =\left\{ \begin{array}{ll} 2\frac{|b_{01}|}{a_{01}}+(b_{10}-a_{01}), &{} \text {if}\quad a_{01}\ne 0, \\ +\infty , &{} \text {if}\quad a_{01}=0. \end{array}\right. \end{aligned}$$
Table 8 Table of the realizable cases under hypothesis of Theorem 12 (Family 4)
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Fig. 23
figure 23

Phase portraits from Cases 4.1 to 4.6

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Fig. 24
figure 24

Phase portraits from Cases 4.7 to 4.11

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Fig. 25
figure 25

Phase portraits from Cases 4.12 to 4.14

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Bastos, J., Buzzi, C. & Santana, P. Evolutionary stable strategies and cubic vector fields. Nonlinear Differ. Equ. Appl. 31, 13 (2024). https://doi.org/10.1007/s00030-023-00894-4

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