# 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 & 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. doi: 10.3934/jdg.2015.2.33. Google Scholar [2] H. N. Alishah, P. Duarte and T. Peixe, Asymptotic poincaré maps along the edges of polytopes,, preprint, (). Google Scholar [3] H. N. Alishah, P. Duarte and T. Peixe, Assymptotic poincaré maps for polymatrix games,, work in progress., (). Google Scholar [4] W. Brannath, Heteroclinic networks on the tetrahedron,, Nonlinearity, 7 (1994), 1367. doi: 10.1088/0951-7715/7/5/006. Google Scholar [5] L. Brenig, Complete factorisation and analytic solutions of generalized Lotka-Volterra equations,, Phys. Lett. A, 133 (1988), 378. doi: 10.1016/0375-9601(88)90920-6. Google Scholar [6] L. Brenig and A. Goriely, Universal canonical forms for time-continuous dynamical systems,, Phys. Rev. A, 40 (1989), 4119. doi: 10.1103/PhysRevA.40.4119. Google Scholar [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). doi: 10.1063/1.4739253. Google Scholar [8] L. A. Bunimovich and B. Z. Webb, Isospectral Transformations,, Springer-Verlag, (2014). doi: 10.1007/978-1-4939-1375-6. Google Scholar [9] T. Chawanya, A new type of irregular motion in a class of game dynamics systems,, Progr. Theoret. Phys., 94 (1995), 163. doi: 10.1143/PTP.94.163. Google Scholar [10] T. Chawanya, Infinitely many attractors in game dynamics system,, Progr. Theoret. Phys., 95 (1996), 679. doi: 10.1143/PTP.95.679. Google Scholar [11] P. Duarte, Hamiltonian systems on polyhedra,, in Dynamics, 2 (2011), 257. doi: 10.1007/978-3-642-14788-3_21. Google Scholar [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. doi: 10.1006/jdeq.1998.3443. Google Scholar [13] P. Duarte and T. Peixe, Rank of stably dissipative graphs,, Linear Algebra Appl., 437 (2012), 2573. doi: 10.1016/j.laa.2012.06.015. Google Scholar [14] J. Eldering, Normally Hyperbolic Invariant Manifolds,, Atlantis Press, (2013). doi: 10.2991/978-94-6239-003-4. Google Scholar [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. doi: 10.1016/j.laa.2012.06.015. Google Scholar [16] B. Hernández-Bermejo and V. Fairén, Lotka-Volterra representation of general nonlinear systems,, Math. Biosci., 140 (1997), 1. doi: 10.1016/S0025-5564(96)00131-9. Google Scholar [17] M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds,, Springer-Verlag, (1977). Google Scholar [18] M. W. Hirsch, Systems of differential equations which are competitive or cooperative. I. Limit sets,, SIAM J. Math. Anal., 13 (1982), 167. doi: 10.1137/0513013. Google Scholar [19] M. W. Hirsch, Systems of differential equations that are competitive or cooperative. {II}. Convergence almost everywhere,, SIAM J. Math. Anal., 16 (1985), 423. doi: 10.1137/0516030. Google Scholar [20] M. W. Hirsch, Systems of differential equations which are competitive or cooperative. III. Competing species,, Nonlinearity, 1 (1988), 51. doi: 10.1088/0951-7715/1/1/003. Google Scholar [21] J. Hofbauer and J. W.-H. So, Multiple limit cycles for three-dimensional Lotka-Volterra equations,, Appl. Math. Lett., 7 (1994), 65. doi: 10.1016/0893-9659(94)90095-7. Google Scholar [22] J. Hofbauer, On the occurrence of limit cycles in the Volterra-Lotka equation,, Nonlinear Anal., 5 (1981), 1003. doi: 10.1016/0362-546X(81)90059-6. Google Scholar [23] J. Hofbauer, Heteroclinic cycles on the simplex,, in Proceedings of the Eleventh International Conference on Nonlinear Oscillations (Budapest János Bolyai Math. Soc., (1987), 828. Google Scholar [24] J. Hofbauer, Heteroclinic cycles in ecological differential equations,, Tatra Mt. Math. Publ., 4 (1994), 105. Google Scholar [25] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics,, Cambridge University Press, (1998). doi: 10.1017/CBO9781139173179. Google Scholar [26] J. T. Howson, Jr., Equilibria of polymatrix games,, Management Sci., 18 (): 312. Google Scholar [27] W. Jansen, A permanence theorem for replicator and Lotka-Volterra systems,, J. Math. Biol., 25 (1987), 411. doi: 10.1007/BF00277165. Google Scholar [28] G. Karakostas, Global stability in job systems,, J. Math. Anal. Appl., 131 (1988), 85. doi: 10.1016/0022-247X(88)90191-6. Google Scholar [29] V. Kirk and M. Silber, A competition between heteroclinic cycles,, Nonlinearity, 7 (1994), 1605. doi: 10.1088/0951-7715/7/6/005. Google Scholar [30] J. P. LaSalle, Stability theory for ordinary differential equations,, J. Differential Equations, 4 (1968), 57. doi: 10.1016/0022-0396(68)90048-X. Google Scholar [31] A. J. Lotka, Elements of Mathematical Biology. (Formerly Published Under the Title Elements of Physical Biology),, Dover Publications, (1958). Google Scholar [32] J. M. Smith, The logic of animal conflicts,, Nature, 246 (1973), 15. doi: 10.1038/246015a0. Google Scholar [33] G. Palm, Evolutionary stable strategies and game dynamics for $n$-person games,, J. Math. Biol., 19 (1984), 329. doi: 10.1007/BF00277103. Google Scholar [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, 30 (1997), 1411. doi: 10.1016/S0362-546X(97)00202-2. Google Scholar [35] L. G. Quintas, A note on polymatrix games,, Internat. J. Game Theory, 18 (1989), 261. doi: 10.1007/BF01254291. Google Scholar [36] R. Redheffer, Volterra multipliers. I, II,, SIAM J. Algebraic Discrete Methods, 6 (1985), 592. doi: 10.1137/0606059. Google Scholar [37] R. Redheffer, A new class of Volterra differential equations for which the solutions are globally asymptotically stable,, J. Differential Equations, 82 (1989), 251. doi: 10.1016/0022-0396(89)90133-2. Google Scholar [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. doi: 10.1016/0022-0396(84)90179-7. Google Scholar [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. doi: 10.1016/0362-546X(81)90108-5. Google Scholar [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. doi: 10.1137/0603012. Google Scholar [41] K. Ritzberger and J. Weibull, Evolutionary selection in normal-form games,, Econometrica, 63 (1995), 1371. doi: 10.2307/2171774. Google Scholar [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. doi: 10.1016/S0010-4655(03)00295-9. Google Scholar [43] P. Schuster and K. Sigmund, Coyness, philandering and stable strategies,, Animal Behaviour, 29 (1981), 186. doi: 10.1016/S0003-3472(81)80165-0. Google Scholar [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. doi: 10.1007/BF00326676. Google Scholar [45] M. Shub, Global Stability of Dynamical Systems,, Springer-Verlag, (1987). doi: 10.1007/978-1-4757-1947-5. Google Scholar [46] G. Karakostas, On the differential equations of species in competition,, J. Math. Biol., 3 (1976), 5. doi: 10.1007/BF00307854. Google Scholar [47] H. L. Smith, On the asymptotic behavior of a class of deterministic models of cooperating species,, SIAM J. Appl. Math., 46 (1986), 368. doi: 10.1137/0146025. Google Scholar [48] L. B. Taylor and L. B. Jonker, Evolutionarily stable strategies and game dynamics,, Math. Biosci., 40 (1978), 145. doi: 10.1016/0025-5564(78)90077-9. Google Scholar [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. doi: 10.1137/S0036139995294767. Google Scholar [50] V. Volterra, Leçons sur la Théorie Mathématique de la Lutte Pour la vie,, Éditions Jacques Gabay, (1990). Google Scholar [51] E. B. Yanovskaya, Equilibrium situations in multi-matrix games (in russian),, Litovsk. Mat. Sb., 8 (1968), 381. Google Scholar [52] M. L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems,, Dynam. Stability Systems, 8 (1993), 189. doi: 10.1080/02681119308806158. Google Scholar [53] M. L. Zeeman, Extinction in competitive Lotka-Volterra systems,, Proc. Amer. Math. Soc., 123 (1995), 87. doi: 10.1090/S0002-9939-1995-1264833-2. Google Scholar [54] X. Zhao and J. Luo, Classification and dynamics of stably dissipative lotka-volterra systems,, International Journal of Non-Linear Mechanics, 45 (2010), 603. doi: 10.1016/j.ijnonlinmec.2009.07.006. Google Scholar

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##### References:
 [1] H. N. Alishah and P. Duarte, Hamiltonian evolutionary games,, Journal of Dynamics and Games, 2 (2015), 33. doi: 10.3934/jdg.2015.2.33. Google Scholar [2] H. N. Alishah, P. Duarte and T. Peixe, Asymptotic poincaré maps along the edges of polytopes,, preprint, (). Google Scholar [3] H. N. Alishah, P. Duarte and T. Peixe, Assymptotic poincaré maps for polymatrix games,, work in progress., (). Google Scholar [4] W. Brannath, Heteroclinic networks on the tetrahedron,, Nonlinearity, 7 (1994), 1367. doi: 10.1088/0951-7715/7/5/006. Google Scholar [5] L. Brenig, Complete factorisation and analytic solutions of generalized Lotka-Volterra equations,, Phys. Lett. A, 133 (1988), 378. doi: 10.1016/0375-9601(88)90920-6. Google Scholar [6] L. Brenig and A. Goriely, Universal canonical forms for time-continuous dynamical systems,, Phys. Rev. A, 40 (1989), 4119. doi: 10.1103/PhysRevA.40.4119. Google Scholar [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). doi: 10.1063/1.4739253. Google Scholar [8] L. A. Bunimovich and B. Z. Webb, Isospectral Transformations,, Springer-Verlag, (2014). doi: 10.1007/978-1-4939-1375-6. Google Scholar [9] T. Chawanya, A new type of irregular motion in a class of game dynamics systems,, Progr. Theoret. Phys., 94 (1995), 163. doi: 10.1143/PTP.94.163. Google Scholar [10] T. Chawanya, Infinitely many attractors in game dynamics system,, Progr. Theoret. Phys., 95 (1996), 679. doi: 10.1143/PTP.95.679. Google Scholar [11] P. Duarte, Hamiltonian systems on polyhedra,, in Dynamics, 2 (2011), 257. doi: 10.1007/978-3-642-14788-3_21. Google Scholar [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. doi: 10.1006/jdeq.1998.3443. Google Scholar [13] P. Duarte and T. Peixe, Rank of stably dissipative graphs,, Linear Algebra Appl., 437 (2012), 2573. doi: 10.1016/j.laa.2012.06.015. Google Scholar [14] J. Eldering, Normally Hyperbolic Invariant Manifolds,, Atlantis Press, (2013). doi: 10.2991/978-94-6239-003-4. Google Scholar [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. doi: 10.1016/j.laa.2012.06.015. Google Scholar [16] B. Hernández-Bermejo and V. Fairén, Lotka-Volterra representation of general nonlinear systems,, Math. Biosci., 140 (1997), 1. doi: 10.1016/S0025-5564(96)00131-9. Google Scholar [17] M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds,, Springer-Verlag, (1977). Google Scholar [18] M. W. Hirsch, Systems of differential equations which are competitive or cooperative. I. Limit sets,, SIAM J. Math. Anal., 13 (1982), 167. doi: 10.1137/0513013. Google Scholar [19] M. W. Hirsch, Systems of differential equations that are competitive or cooperative. {II}. Convergence almost everywhere,, SIAM J. Math. Anal., 16 (1985), 423. doi: 10.1137/0516030. Google Scholar [20] M. W. Hirsch, Systems of differential equations which are competitive or cooperative. III. Competing species,, Nonlinearity, 1 (1988), 51. doi: 10.1088/0951-7715/1/1/003. Google Scholar [21] J. Hofbauer and J. W.-H. So, Multiple limit cycles for three-dimensional Lotka-Volterra equations,, Appl. Math. Lett., 7 (1994), 65. doi: 10.1016/0893-9659(94)90095-7. Google Scholar [22] J. Hofbauer, On the occurrence of limit cycles in the Volterra-Lotka equation,, Nonlinear Anal., 5 (1981), 1003. doi: 10.1016/0362-546X(81)90059-6. Google Scholar [23] J. Hofbauer, Heteroclinic cycles on the simplex,, in Proceedings of the Eleventh International Conference on Nonlinear Oscillations (Budapest János Bolyai Math. Soc., (1987), 828. Google Scholar [24] J. Hofbauer, Heteroclinic cycles in ecological differential equations,, Tatra Mt. Math. Publ., 4 (1994), 105. Google Scholar [25] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics,, Cambridge University Press, (1998). doi: 10.1017/CBO9781139173179. Google Scholar [26] J. T. Howson, Jr., Equilibria of polymatrix games,, Management Sci., 18 (): 312. Google Scholar [27] W. Jansen, A permanence theorem for replicator and Lotka-Volterra systems,, J. Math. Biol., 25 (1987), 411. doi: 10.1007/BF00277165. Google Scholar [28] G. Karakostas, Global stability in job systems,, J. Math. Anal. Appl., 131 (1988), 85. doi: 10.1016/0022-247X(88)90191-6. Google Scholar [29] V. Kirk and M. Silber, A competition between heteroclinic cycles,, Nonlinearity, 7 (1994), 1605. doi: 10.1088/0951-7715/7/6/005. Google Scholar [30] J. P. LaSalle, Stability theory for ordinary differential equations,, J. Differential Equations, 4 (1968), 57. doi: 10.1016/0022-0396(68)90048-X. Google Scholar [31] A. J. Lotka, Elements of Mathematical Biology. (Formerly Published Under the Title Elements of Physical Biology),, Dover Publications, (1958). Google Scholar [32] J. M. Smith, The logic of animal conflicts,, Nature, 246 (1973), 15. doi: 10.1038/246015a0. Google Scholar [33] G. Palm, Evolutionary stable strategies and game dynamics for $n$-person games,, J. Math. Biol., 19 (1984), 329. doi: 10.1007/BF00277103. Google Scholar [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, 30 (1997), 1411. doi: 10.1016/S0362-546X(97)00202-2. Google Scholar [35] L. G. Quintas, A note on polymatrix games,, Internat. J. Game Theory, 18 (1989), 261. doi: 10.1007/BF01254291. Google Scholar [36] R. Redheffer, Volterra multipliers. I, II,, SIAM J. Algebraic Discrete Methods, 6 (1985), 592. doi: 10.1137/0606059. Google Scholar [37] R. Redheffer, A new class of Volterra differential equations for which the solutions are globally asymptotically stable,, J. Differential Equations, 82 (1989), 251. doi: 10.1016/0022-0396(89)90133-2. Google Scholar [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. doi: 10.1016/0022-0396(84)90179-7. Google Scholar [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. doi: 10.1016/0362-546X(81)90108-5. Google Scholar [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. doi: 10.1137/0603012. Google Scholar [41] K. Ritzberger and J. Weibull, Evolutionary selection in normal-form games,, Econometrica, 63 (1995), 1371. doi: 10.2307/2171774. Google Scholar [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. doi: 10.1016/S0010-4655(03)00295-9. Google Scholar [43] P. Schuster and K. Sigmund, Coyness, philandering and stable strategies,, Animal Behaviour, 29 (1981), 186. doi: 10.1016/S0003-3472(81)80165-0. Google Scholar [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. doi: 10.1007/BF00326676. Google Scholar [45] M. Shub, Global Stability of Dynamical Systems,, Springer-Verlag, (1987). doi: 10.1007/978-1-4757-1947-5. Google Scholar [46] G. Karakostas, On the differential equations of species in competition,, J. Math. Biol., 3 (1976), 5. doi: 10.1007/BF00307854. Google Scholar [47] H. L. Smith, On the asymptotic behavior of a class of deterministic models of cooperating species,, SIAM J. Appl. Math., 46 (1986), 368. doi: 10.1137/0146025. Google Scholar [48] L. B. Taylor and L. B. Jonker, Evolutionarily stable strategies and game dynamics,, Math. Biosci., 40 (1978), 145. doi: 10.1016/0025-5564(78)90077-9. Google Scholar [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. doi: 10.1137/S0036139995294767. Google Scholar [50] V. Volterra, Leçons sur la Théorie Mathématique de la Lutte Pour la vie,, Éditions Jacques Gabay, (1990). Google Scholar [51] E. B. Yanovskaya, Equilibrium situations in multi-matrix games (in russian),, Litovsk. Mat. Sb., 8 (1968), 381. Google Scholar [52] M. L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems,, Dynam. Stability Systems, 8 (1993), 189. doi: 10.1080/02681119308806158. Google Scholar [53] M. L. Zeeman, Extinction in competitive Lotka-Volterra systems,, Proc. Amer. Math. Soc., 123 (1995), 87. doi: 10.1090/S0002-9939-1995-1264833-2. Google Scholar [54] X. Zhao and J. Luo, Classification and dynamics of stably dissipative lotka-volterra systems,, International Journal of Non-Linear Mechanics, 45 (2010), 603. doi: 10.1016/j.ijnonlinmec.2009.07.006. Google Scholar
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