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Invariant structures on Lie groups
Conservative replicator and Lotka-Volterra equations in the context of Dirac\big-isotropic structures
Departamento de Matemática, Instituto de Ciências Exatas, Universidade Federal de Minas Gerais, Belo Horizonte, 31270-901, Brazil |
We introduce an algorithm to find possible constants of motion for a given replicator equation. The algorithm is inspired by Dirac geometry and a Hamiltonian description for the replicator equations with such constants of motion, up to a time re-parametrization, is provided using Dirac$ \backslash $big-isotropic structures. Using the equivalence between replicator and Lotka-Volterra (LV) equations, the set of conservative LV equations is enlarged. Our approach generalizes the well-known use of gauge transformations to skew-symmetrize the interaction matrix of a LV system. In the case of predator-prey model, our method does allow interaction between different predators and between different preys.
References:
[1] |
H. N. Alishah and R. de la Llave,
Tracing KAM tori in presymplectic dynamical systems, J. Dynam. Differential Equations, 24 (2012), 685-711.
doi: 10.1007/s10884-012-9265-2. |
[2] |
H. N. Alishah and P. Duarte,
Hamiltonian evolutionary games, J. Dyn. Games, 2 (2015), 33-49.
doi: 10.3934/jdg.2015.2.33. |
[3] |
H. N. Alishah and J. Lopes Dias,
Realization of tangent perturbations in discrete and continuous time conservative systems, Discrete Contin. Dyn. Syst., 34 (2014), 5359-5374.
doi: 10.3934/dcds.2014.34.5359. |
[4] |
P. Antoniou and A. Pitsillides, Congestion control in autonomous decentralized networks based on the Lotka-Volterra competition model, Artificial Neural Networks-ICANN 2009. ICANN 2009, (2009), 986–996.
doi: 10.1007/978-3-642-04277-5_99. |
[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] |
H. Bursztyn, A brief introduction to Dirac manifolds, Geometric and Topological Methods for Quantum Field Theory, Cambridge Univ. Press, Cambridge, (2013), 4–38.
doi: 10.1017/CBO9781139208642.002. |
[7] |
T. J. Courant,
Dirac manifolds, Trans. Amer. Math. Soc., 319 (1990), 631-661.
doi: 10.1090/S0002-9947-1990-0998124-1. |
[8] |
T. Courant and A. Weinstein, Beyond Poisson structures, Action Hamiltoniennes de Groupes, Troisième Théorème de Lie (Lyon 1988), Travaux en Cours, Hermann, Paris, 27 (1988), 39–49, Available from: https://math.berkeley.edu/~alanw/Beyond.pdf. |
[9] |
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. |
[10] |
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. |
[11] |
J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, 1998.
doi: 10.1017/CBO9781139173179.![]() ![]() |
[12] |
M. Jotz and T. S. Ratiu,
Dirac structures, nonholonomic systems and reduction, Rep. Math. Phys., 69 (2012), 5-56.
doi: 10.1016/S0034-4877(12)60016-0. |
[13] |
A. J. Lotka, Elements of mathematical biology, Dover Publications, Inc., New York, N. Y., (1958). |
[14] |
S. Smale,
On the differential equations of species in competition, J. Math. Biol., 3 (1976), 5-7.
doi: 10.1007/BF00307854. |
[15] |
I. Vaisman,
Isotropic subbundles of $TM\oplus T^*M$, Int. J. Geom. Methods Mod. Phys., 4 (2007), 487-516.
doi: 10.1142/S0219887807002156. |
[16] |
I. Vaisman, Weak-Hamiltonian dynamical systems, J. Math. Phys., 48 (2007), 082903, 13 pp.
doi: 10.1063/1.2769145. |
[17] |
A. van der Schaft,
Port-Hamiltonian systems: An introductory survey, International Congress of Mathematicians, Eur. Math. Soc., Zürich, 3 (2006), 1339-1365.
|
[18] |
V. Volterra, Leçons sur la Théorie Mathématique de la Lutte Pour la vie, Les Grands Classiques Gauthier-Villars, Éditions Jacques Gabay, Sceaux, 1990. |
[19] |
H. Yoshimura and J. E. Marsden,
Dirac structures in Lagrangian mechanics. I. Implicit Lagrangian systems, J. Geom. Phys., 57 (2006), 133-156.
doi: 10.1016/j.geomphys.2006.02.009. |
show all references
References:
[1] |
H. N. Alishah and R. de la Llave,
Tracing KAM tori in presymplectic dynamical systems, J. Dynam. Differential Equations, 24 (2012), 685-711.
doi: 10.1007/s10884-012-9265-2. |
[2] |
H. N. Alishah and P. Duarte,
Hamiltonian evolutionary games, J. Dyn. Games, 2 (2015), 33-49.
doi: 10.3934/jdg.2015.2.33. |
[3] |
H. N. Alishah and J. Lopes Dias,
Realization of tangent perturbations in discrete and continuous time conservative systems, Discrete Contin. Dyn. Syst., 34 (2014), 5359-5374.
doi: 10.3934/dcds.2014.34.5359. |
[4] |
P. Antoniou and A. Pitsillides, Congestion control in autonomous decentralized networks based on the Lotka-Volterra competition model, Artificial Neural Networks-ICANN 2009. ICANN 2009, (2009), 986–996.
doi: 10.1007/978-3-642-04277-5_99. |
[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] |
H. Bursztyn, A brief introduction to Dirac manifolds, Geometric and Topological Methods for Quantum Field Theory, Cambridge Univ. Press, Cambridge, (2013), 4–38.
doi: 10.1017/CBO9781139208642.002. |
[7] |
T. J. Courant,
Dirac manifolds, Trans. Amer. Math. Soc., 319 (1990), 631-661.
doi: 10.1090/S0002-9947-1990-0998124-1. |
[8] |
T. Courant and A. Weinstein, Beyond Poisson structures, Action Hamiltoniennes de Groupes, Troisième Théorème de Lie (Lyon 1988), Travaux en Cours, Hermann, Paris, 27 (1988), 39–49, Available from: https://math.berkeley.edu/~alanw/Beyond.pdf. |
[9] |
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. |
[10] |
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. |
[11] |
J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, 1998.
doi: 10.1017/CBO9781139173179.![]() ![]() |
[12] |
M. Jotz and T. S. Ratiu,
Dirac structures, nonholonomic systems and reduction, Rep. Math. Phys., 69 (2012), 5-56.
doi: 10.1016/S0034-4877(12)60016-0. |
[13] |
A. J. Lotka, Elements of mathematical biology, Dover Publications, Inc., New York, N. Y., (1958). |
[14] |
S. Smale,
On the differential equations of species in competition, J. Math. Biol., 3 (1976), 5-7.
doi: 10.1007/BF00307854. |
[15] |
I. Vaisman,
Isotropic subbundles of $TM\oplus T^*M$, Int. J. Geom. Methods Mod. Phys., 4 (2007), 487-516.
doi: 10.1142/S0219887807002156. |
[16] |
I. Vaisman, Weak-Hamiltonian dynamical systems, J. Math. Phys., 48 (2007), 082903, 13 pp.
doi: 10.1063/1.2769145. |
[17] |
A. van der Schaft,
Port-Hamiltonian systems: An introductory survey, International Congress of Mathematicians, Eur. Math. Soc., Zürich, 3 (2006), 1339-1365.
|
[18] |
V. Volterra, Leçons sur la Théorie Mathématique de la Lutte Pour la vie, Les Grands Classiques Gauthier-Villars, Éditions Jacques Gabay, Sceaux, 1990. |
[19] |
H. Yoshimura and J. E. Marsden,
Dirac structures in Lagrangian mechanics. I. Implicit Lagrangian systems, J. Geom. Phys., 57 (2006), 133-156.
doi: 10.1016/j.geomphys.2006.02.009. |
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