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June  2020, 12(2): 149-164. doi: 10.3934/jgm.2020008

Conservative replicator and Lotka-Volterra equations in the context of Dirac\big-isotropic structures

Hassan Najafi Alishah

Departamento de Matemática, Instituto de Ciências Exatas, Universidade Federal de Minas Gerais, Belo Horizonte, 31270-901, Brazil

Received  August 2019 Published  June 2020 Early access  March 2020

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.

Keywords: Replicator equation, Lotka-Volterra equations, constants of motion, Dirac structure, big-isotropic structure.
Mathematics Subject Classification: Primary: 70H33, 92D25; Secondary: 37J15.
Citation: Hassan Najafi Alishah. Conservative replicator and Lotka-Volterra equations in the context of Dirac\big-isotropic structures. Journal of Geometric Mechanics, 2020, 12 (2) : 149-164. doi: 10.3934/jgm.2020008
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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