# American Institute of Mathematical Sciences

April  2015, 2(2): 157-185. doi: 10.3934/jdg.2015.2.157

## Conservative and dissipative polymatrix replicators

 1 Departamento de Matemática, Instituto de Ciências Exatas, Universidade Federal de Minas Gerais, Belo Horizonte, 31270-901, Brazil 2 Departamento de Matemática and CMAF-CIO, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, Edificio C6, Piso 2, 1749-016 Lisboa, Portugal 3 Departamento de Matemática, Instituto Superior de Economia e Gestão and CMAF-CIO, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, Edificio C6, Piso 2, 1749-016 Lisboa, Portugal

Received  July 2015 Revised  October 2015 Published  December 2015

In this paper we address a class of replicator dynamics, referred as polymatrix replicators, that contains well known classes of evolutionary game dynamics, such as the symmetric and asymmetric (or bimatrix) replicator equations, and some replicator equations for $n$-person games. Polymatrix replicators form a simple class of algebraic o.d.e.'s on prisms (products of simplexes), which describe the evolution of strategical behaviours within a population stratified in $p\geq 1$ social groups.
In the 80's Raymond Redheffer et al. developed a theory on the class of stably dissipative Lotka-Volterra systems. This theory is built around a reduction algorithm that infers'' the localization of the system' s attractor in some affine subspace. It was later proven that the dynamics on the attractor of such systems is always embeddable in a Hamiltonian Lotka-Volterra system.
In this paper we extend these results to polymatrix replicators.
Citation: Hassan Najafi Alishah, Pedro Duarte, Telmo Peixe. Conservative and dissipative polymatrix replicators. Journal of Dynamics and Games, 2015, 2 (2) : 157-185. doi: 10.3934/jdg.2015.2.157
##### References:
 [1] H. N. Alishah and P. Duarte, Hamiltonian evolutionary games, Journal of Dynamics and Games, 2 (2015), 33-49. doi: 10.3934/jdg.2015.2.33. [2] H. N. Alishah, P. Duarte and T. Peixe, Asymptotic poincaré maps along the edges of polytopes,, preprint, (). [3] H. N. Alishah, P. Duarte and T. Peixe, Assymptotic poincaré maps for polymatrix games,, work in progress., (). [4] W. Brannath, Heteroclinic networks on the tetrahedron, Nonlinearity, 7 (1994), 1367-1384, Available from: http://stacks.iop.org/0951-7715/7/i=5/a=006. doi: 10.1088/0951-7715/7/5/006. [5] L. Brenig, Complete factorisation and analytic solutions of generalized Lotka-Volterra equations, Phys. Lett. A, 133 (1988), 378-382. doi: 10.1016/0375-9601(88)90920-6. [6] L. Brenig and A. Goriely, Universal canonical forms for time-continuous dynamical systems, Phys. Rev. A, 40 (1989), 4119-4122. doi: 10.1103/PhysRevA.40.4119. [7] L. A. Bunimovich and B. Z. Webb, Isospectral compression and other useful isospectral transformations of dynamical networks, Chaos: An Interdisciplinary Journal of Nonlinear Science, 22 (2012), 033118, 14 pp. doi: 10.1063/1.4739253. [8] L. A. Bunimovich and B. Z. Webb, Isospectral Transformations, Springer-Verlag, New York, 2014. doi: 10.1007/978-1-4939-1375-6. [9] T. Chawanya, A new type of irregular motion in a class of game dynamics systems, Progr. Theoret. Phys., 94 (1995), 163-179. doi: 10.1143/PTP.94.163. [10] T. Chawanya, Infinitely many attractors in game dynamics system, Progr. Theoret. Phys., 95 (1996), 679-684. doi: 10.1143/PTP.95.679. [11] P. Duarte, Hamiltonian systems on polyhedra, in Dynamics, games and science. II (Springer Proc. Math. 2) Springer, Heidelberg (2011), 257-274. doi: 10.1007/978-3-642-14788-3_21. [12] P. Duarte, R. L. Fernandes and W. M. Oliva, Dynamics of the attractor in the Lotka-Volterra equations, J. Differential Equations, 149 (1998), 143-189. doi: 10.1006/jdeq.1998.3443. [13] P. Duarte and T. Peixe, Rank of stably dissipative graphs, Linear Algebra Appl., 437 (2012), 2573-2586. doi: 10.1016/j.laa.2012.06.015. [14] J. Eldering, Normally Hyperbolic Invariant Manifolds, Atlantis Press, Paris, 2013. doi: 10.2991/978-94-6239-003-4. [15] Z. M. Guo, Z. M. Zhou and S. S. Wang, Volterra multipliers of $3\times 3$ real matrices, Math. Practice Theory, 1 (1995), 47-54. doi: 10.1016/j.laa.2012.06.015. [16] B. Hernández-Bermejo and V. Fairén, Lotka-Volterra representation of general nonlinear systems, Math. Biosci., 140 (1997), 1-32. doi: 10.1016/S0025-5564(96)00131-9. [17] M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Springer-Verlag, Berlin-New York, 1977. [18] M. W. Hirsch, Systems of differential equations which are competitive or cooperative. I. Limit sets, SIAM J. Math. Anal., 13 (1982), 167-179. doi: 10.1137/0513013. [19] M. W. Hirsch, Systems of differential equations that are competitive or cooperative. {II}. Convergence almost everywhere, SIAM J. Math. Anal., 16 (1985), 423-439. doi: 10.1137/0516030. [20] M. W. Hirsch, Systems of differential equations which are competitive or cooperative. III. Competing species, Nonlinearity, 1 (1988), 51-71. Available from: http://stacks.iop.org/0951-7715/1/51 doi: 10.1088/0951-7715/1/1/003. [21] J. Hofbauer and J. W.-H. So, Multiple limit cycles for three-dimensional Lotka-Volterra equations, Appl. Math. Lett., 7 (1994), 65-70. doi: 10.1016/0893-9659(94)90095-7. [22] J. Hofbauer, On the occurrence of limit cycles in the Volterra-Lotka equation, Nonlinear Anal., 5 (1981), 1003-1007. doi: 10.1016/0362-546X(81)90059-6. [23] J. Hofbauer, Heteroclinic cycles on the simplex, in Proceedings of the Eleventh International Conference on Nonlinear Oscillations (Budapest János Bolyai Math. Soc., Budapest, (1987), 828-831. [24] J. Hofbauer, Heteroclinic cycles in ecological differential equations, Tatra Mt. Math. Publ., 4 (1994), 105-116. [25] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998. doi: 10.1017/CBO9781139173179. [26] J. T. Howson, Jr., Equilibria of polymatrix games,, Management Sci., 18 (): 312. [27] W. Jansen, A permanence theorem for replicator and Lotka-Volterra systems, J. Math. Biol., 25 (1987), 411-422. doi: 10.1007/BF00277165. [28] G. Karakostas, Global stability in job systems, J. Math. Anal. Appl., 131 (1988), 85-96. doi: 10.1016/0022-247X(88)90191-6. [29] V. Kirk and M. Silber, A competition between heteroclinic cycles, Nonlinearity, 7 (1994), 1605-1621, Available from http://stacks.iop.org/0951-7715/7/1605. doi: 10.1088/0951-7715/7/6/005. [30] J. P. LaSalle, Stability theory for ordinary differential equations, J. Differential Equations, 4 (1968), 57-65. doi: 10.1016/0022-0396(68)90048-X. [31] A. J. Lotka, Elements of Mathematical Biology. (Formerly Published Under the Title Elements of Physical Biology), Dover Publications, Inc., New York, 1958. [32] J. M. Smith, The logic of animal conflicts, Nature, 246 (1973), 15-18. doi: 10.1038/246015a0. [33] G. Palm, Evolutionary stable strategies and game dynamics for $n$-person games, J. Math. Biol., 19 (1984), 329-334. doi: 10.1007/BF00277103. [34] M. Plank, Some qualitative differences between the replicator dynamics of two player and $n$ player games, in Proceedings of the Second World Congress of Nonlinear Analysts, Part 3 (Athens 30 (1997), 1411-1417. doi: 10.1016/S0362-546X(97)00202-2. [35] L. G. Quintas, A note on polymatrix games, Internat. J. Game Theory, 18 (1989), 261-272. doi: 10.1007/BF01254291. [36] R. Redheffer, Volterra multipliers. I, II, SIAM J. Algebraic Discrete Methods, 6 (1985), 592-611, 612-623. doi: 10.1137/0606059. [37] R. Redheffer, A new class of Volterra differential equations for which the solutions are globally asymptotically stable, J. Differential Equations, 82 (1989), 251-268. doi: 10.1016/0022-0396(89)90133-2. [38] R. Redheffer and W. Walter, Solution of the stability problem for a class of generalized Volterra prey-predator systems, J. Differential Equations, 52 (1984), 245-263. doi: 10.1016/0022-0396(84)90179-7. [39] R. Redheffer and Z. M. Zhou, Global asymptotic stability for a class of many-variable Volterra prey-predator systems, Nonlinear Anal., 5 (1981), 1309-1329. doi: 10.1016/0362-546X(81)90108-5. [40] R. Redheffer and Z. M. Zhou, A class of matrices connected with Volterra prey-predator equations, SIAM J. Algebraic Discrete Methods, 3 (1982), 122-134. doi: 10.1137/0603012. [41] K. Ritzberger and J. Weibull, Evolutionary selection in normal-form games, Econometrica, 63 (1995), 1371-1399. doi: 10.2307/2171774. [42] T. M. Rocha Filho, I. M. Gléria and A. Figueiredo, A novel approach for the stability problem in non-linear dynamical systems, Comput. Phys. Comm., 155 (2003), 21-30. doi: 10.1016/S0010-4655(03)00295-9. [43] P. Schuster and K. Sigmund, Coyness, philandering and stable strategies, Animal Behaviour, 29 (1981), 186-192. doi: 10.1016/S0003-3472(81)80165-0. [44] P. Schuster, K. Sigmund and R. Wolff, Self-regulation of behaviour in animal societies. II. Games between two populations without self-interaction, Biol. Cybernet., 40 (1981), 9-15. doi: 10.1007/BF00326676. [45] M. Shub, Global Stability of Dynamical Systems, Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4757-1947-5. [46] G. Karakostas, On the differential equations of species in competition, J. Math. Biol., 3 (1976), 5-7. doi: 10.1007/BF00307854. [47] H. L. Smith, On the asymptotic behavior of a class of deterministic models of cooperating species, SIAM J. Appl. Math., 46 (1986), 368-375. doi: 10.1137/0146025. [48] L. B. Taylor and L. B. Jonker, Evolutionarily stable strategies and game dynamics, Math. Biosci., 40 (1978), 145-156. doi: 10.1016/0025-5564(78)90077-9. [49] P. van den Driessche and M. L. Zeeman, Three-dimensional competitive Lotka-Volterra systems with no periodic orbits, SIAM J. Appl. Math., 58 (1998), 227-234. doi: 10.1137/S0036139995294767. [50] V. Volterra, Leçons sur la Théorie Mathématique de la Lutte Pour la vie, Éditions Jacques Gabay, Sceaux, 1990. [51] E. B. Yanovskaya, Equilibrium situations in multi-matrix games (in russian), Litovsk. Mat. Sb., 8 (1968), 381-384. [52] M. L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems, Dynam. Stability Systems, 8 (1993), 189-217. doi: 10.1080/02681119308806158. [53] M. L. Zeeman, Extinction in competitive Lotka-Volterra systems, Proc. Amer. Math. Soc., 123 (1995), 87-96. doi: 10.1090/S0002-9939-1995-1264833-2. [54] X. Zhao and J. Luo, Classification and dynamics of stably dissipative lotka-volterra systems, International Journal of Non-Linear Mechanics, 45 (2010), 603-607. doi: 10.1016/j.ijnonlinmec.2009.07.006.

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##### References:
 [1] H. N. Alishah and P. Duarte, Hamiltonian evolutionary games, Journal of Dynamics and Games, 2 (2015), 33-49. doi: 10.3934/jdg.2015.2.33. [2] H. N. Alishah, P. Duarte and T. Peixe, Asymptotic poincaré maps along the edges of polytopes,, preprint, (). [3] H. N. Alishah, P. Duarte and T. Peixe, Assymptotic poincaré maps for polymatrix games,, work in progress., (). [4] W. Brannath, Heteroclinic networks on the tetrahedron, Nonlinearity, 7 (1994), 1367-1384, Available from: http://stacks.iop.org/0951-7715/7/i=5/a=006. doi: 10.1088/0951-7715/7/5/006. [5] L. Brenig, Complete factorisation and analytic solutions of generalized Lotka-Volterra equations, Phys. Lett. A, 133 (1988), 378-382. doi: 10.1016/0375-9601(88)90920-6. [6] L. Brenig and A. Goriely, Universal canonical forms for time-continuous dynamical systems, Phys. Rev. A, 40 (1989), 4119-4122. doi: 10.1103/PhysRevA.40.4119. [7] L. A. Bunimovich and B. Z. Webb, Isospectral compression and other useful isospectral transformations of dynamical networks, Chaos: An Interdisciplinary Journal of Nonlinear Science, 22 (2012), 033118, 14 pp. doi: 10.1063/1.4739253. [8] L. A. Bunimovich and B. Z. Webb, Isospectral Transformations, Springer-Verlag, New York, 2014. doi: 10.1007/978-1-4939-1375-6. [9] T. Chawanya, A new type of irregular motion in a class of game dynamics systems, Progr. Theoret. Phys., 94 (1995), 163-179. doi: 10.1143/PTP.94.163. [10] T. Chawanya, Infinitely many attractors in game dynamics system, Progr. Theoret. Phys., 95 (1996), 679-684. doi: 10.1143/PTP.95.679. [11] P. Duarte, Hamiltonian systems on polyhedra, in Dynamics, games and science. II (Springer Proc. Math. 2) Springer, Heidelberg (2011), 257-274. doi: 10.1007/978-3-642-14788-3_21. [12] P. Duarte, R. L. Fernandes and W. M. Oliva, Dynamics of the attractor in the Lotka-Volterra equations, J. Differential Equations, 149 (1998), 143-189. doi: 10.1006/jdeq.1998.3443. [13] P. Duarte and T. Peixe, Rank of stably dissipative graphs, Linear Algebra Appl., 437 (2012), 2573-2586. doi: 10.1016/j.laa.2012.06.015. [14] J. Eldering, Normally Hyperbolic Invariant Manifolds, Atlantis Press, Paris, 2013. doi: 10.2991/978-94-6239-003-4. [15] Z. M. Guo, Z. M. Zhou and S. S. Wang, Volterra multipliers of $3\times 3$ real matrices, Math. Practice Theory, 1 (1995), 47-54. doi: 10.1016/j.laa.2012.06.015. [16] B. Hernández-Bermejo and V. Fairén, Lotka-Volterra representation of general nonlinear systems, Math. Biosci., 140 (1997), 1-32. doi: 10.1016/S0025-5564(96)00131-9. [17] M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Springer-Verlag, Berlin-New York, 1977. [18] M. W. Hirsch, Systems of differential equations which are competitive or cooperative. I. Limit sets, SIAM J. Math. Anal., 13 (1982), 167-179. doi: 10.1137/0513013. [19] M. W. Hirsch, Systems of differential equations that are competitive or cooperative. {II}. Convergence almost everywhere, SIAM J. Math. Anal., 16 (1985), 423-439. doi: 10.1137/0516030. [20] M. W. Hirsch, Systems of differential equations which are competitive or cooperative. III. Competing species, Nonlinearity, 1 (1988), 51-71. Available from: http://stacks.iop.org/0951-7715/1/51 doi: 10.1088/0951-7715/1/1/003. [21] J. Hofbauer and J. W.-H. So, Multiple limit cycles for three-dimensional Lotka-Volterra equations, Appl. Math. Lett., 7 (1994), 65-70. doi: 10.1016/0893-9659(94)90095-7. [22] J. Hofbauer, On the occurrence of limit cycles in the Volterra-Lotka equation, Nonlinear Anal., 5 (1981), 1003-1007. doi: 10.1016/0362-546X(81)90059-6. [23] J. Hofbauer, Heteroclinic cycles on the simplex, in Proceedings of the Eleventh International Conference on Nonlinear Oscillations (Budapest János Bolyai Math. Soc., Budapest, (1987), 828-831. [24] J. Hofbauer, Heteroclinic cycles in ecological differential equations, Tatra Mt. Math. Publ., 4 (1994), 105-116. [25] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998. doi: 10.1017/CBO9781139173179. [26] J. T. Howson, Jr., Equilibria of polymatrix games,, Management Sci., 18 (): 312. [27] W. Jansen, A permanence theorem for replicator and Lotka-Volterra systems, J. Math. Biol., 25 (1987), 411-422. doi: 10.1007/BF00277165. [28] G. Karakostas, Global stability in job systems, J. Math. Anal. Appl., 131 (1988), 85-96. doi: 10.1016/0022-247X(88)90191-6. [29] V. Kirk and M. Silber, A competition between heteroclinic cycles, Nonlinearity, 7 (1994), 1605-1621, Available from http://stacks.iop.org/0951-7715/7/1605. doi: 10.1088/0951-7715/7/6/005. [30] J. P. LaSalle, Stability theory for ordinary differential equations, J. Differential Equations, 4 (1968), 57-65. doi: 10.1016/0022-0396(68)90048-X. [31] A. J. Lotka, Elements of Mathematical Biology. (Formerly Published Under the Title Elements of Physical Biology), Dover Publications, Inc., New York, 1958. [32] J. M. Smith, The logic of animal conflicts, Nature, 246 (1973), 15-18. doi: 10.1038/246015a0. [33] G. Palm, Evolutionary stable strategies and game dynamics for $n$-person games, J. Math. Biol., 19 (1984), 329-334. doi: 10.1007/BF00277103. [34] M. Plank, Some qualitative differences between the replicator dynamics of two player and $n$ player games, in Proceedings of the Second World Congress of Nonlinear Analysts, Part 3 (Athens 30 (1997), 1411-1417. doi: 10.1016/S0362-546X(97)00202-2. [35] L. G. Quintas, A note on polymatrix games, Internat. J. Game Theory, 18 (1989), 261-272. doi: 10.1007/BF01254291. [36] R. Redheffer, Volterra multipliers. I, II, SIAM J. Algebraic Discrete Methods, 6 (1985), 592-611, 612-623. doi: 10.1137/0606059. [37] R. Redheffer, A new class of Volterra differential equations for which the solutions are globally asymptotically stable, J. Differential Equations, 82 (1989), 251-268. doi: 10.1016/0022-0396(89)90133-2. [38] R. Redheffer and W. Walter, Solution of the stability problem for a class of generalized Volterra prey-predator systems, J. Differential Equations, 52 (1984), 245-263. doi: 10.1016/0022-0396(84)90179-7. [39] R. Redheffer and Z. M. Zhou, Global asymptotic stability for a class of many-variable Volterra prey-predator systems, Nonlinear Anal., 5 (1981), 1309-1329. doi: 10.1016/0362-546X(81)90108-5. [40] R. Redheffer and Z. M. Zhou, A class of matrices connected with Volterra prey-predator equations, SIAM J. Algebraic Discrete Methods, 3 (1982), 122-134. doi: 10.1137/0603012. [41] K. Ritzberger and J. Weibull, Evolutionary selection in normal-form games, Econometrica, 63 (1995), 1371-1399. doi: 10.2307/2171774. [42] T. M. Rocha Filho, I. M. Gléria and A. Figueiredo, A novel approach for the stability problem in non-linear dynamical systems, Comput. Phys. Comm., 155 (2003), 21-30. doi: 10.1016/S0010-4655(03)00295-9. [43] P. Schuster and K. Sigmund, Coyness, philandering and stable strategies, Animal Behaviour, 29 (1981), 186-192. doi: 10.1016/S0003-3472(81)80165-0. [44] P. Schuster, K. Sigmund and R. Wolff, Self-regulation of behaviour in animal societies. II. Games between two populations without self-interaction, Biol. Cybernet., 40 (1981), 9-15. doi: 10.1007/BF00326676. [45] M. Shub, Global Stability of Dynamical Systems, Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4757-1947-5. [46] G. Karakostas, On the differential equations of species in competition, J. Math. Biol., 3 (1976), 5-7. doi: 10.1007/BF00307854. [47] H. L. Smith, On the asymptotic behavior of a class of deterministic models of cooperating species, SIAM J. Appl. Math., 46 (1986), 368-375. doi: 10.1137/0146025. [48] L. B. Taylor and L. B. Jonker, Evolutionarily stable strategies and game dynamics, Math. Biosci., 40 (1978), 145-156. doi: 10.1016/0025-5564(78)90077-9. [49] P. van den Driessche and M. L. Zeeman, Three-dimensional competitive Lotka-Volterra systems with no periodic orbits, SIAM J. Appl. Math., 58 (1998), 227-234. doi: 10.1137/S0036139995294767. [50] V. Volterra, Leçons sur la Théorie Mathématique de la Lutte Pour la vie, Éditions Jacques Gabay, Sceaux, 1990. [51] E. B. Yanovskaya, Equilibrium situations in multi-matrix games (in russian), Litovsk. Mat. Sb., 8 (1968), 381-384. [52] M. L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems, Dynam. Stability Systems, 8 (1993), 189-217. doi: 10.1080/02681119308806158. [53] M. L. Zeeman, Extinction in competitive Lotka-Volterra systems, Proc. Amer. Math. Soc., 123 (1995), 87-96. doi: 10.1090/S0002-9939-1995-1264833-2. [54] X. Zhao and J. Luo, Classification and dynamics of stably dissipative lotka-volterra systems, International Journal of Non-Linear Mechanics, 45 (2010), 603-607. doi: 10.1016/j.ijnonlinmec.2009.07.006.
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