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December  2019, 6(2): 223-237. doi: 10.3934/jcd.2019011

A new class of integrable Lotka–Volterra systems

Helen Christodoulidi 1,2, , Andrew N. W. Hone 2,3, and  Theodoros E. Kouloukas 3,,

1. 

Research Center for Astronomy and Applied Mathematics, Academy of Athens, Athens 11527, Greece
2. 

School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury CT2 7FS, UK
3. 

School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia

* Corresponding author: Theodoros E. Kouloukas

Received  March 2019 Revised  July 2019 Published  November 2019

A parameter-dependent class of Hamiltonian (generalized) Lotka–Volterra systems is considered. We prove that this class contains Liouville integrable as well as superintegrable cases according to particular choices of the parameters. We determine sufficient conditions which result in integrable behavior, while we numerically explore the complementary cases, where these analytically derived conditions are not satisfied.

Keywords: Integrable systems, superintegrability, Lotka–Volterra system, lax representation, chaotic cases.
Mathematics Subject Classification: Primary: 37K10, 70Hxx; Secondary: 70Kxx.
Citation: Helen Christodoulidi, Andrew N. W. Hone, Theodoros E. Kouloukas. A new class of integrable Lotka–Volterra systems. Journal of Computational Dynamics, 2019, 6 (2) : 223-237. doi: 10.3934/jcd.2019011
References:
[1]

Á. Ballesteros, A. Blasco and F. Musso, Integrable deformations of Lotka-Volterra systems, Phys. Lett. A, 375 (2011), 3370-3374.  doi: 10.1016/j.physleta.2011.07.055.

[2]

O. I. Bogoyavlenskij, Some constructions of integrable dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat., 51 (1987), 737-766.  doi: 10.1070/IM1988v031n01ABEH001043.

[3]

O. I. Bogoyavlenskij, Integrable Lotka-Volterra systems, Regul. Chaotic Dyn., 13 (2008), 543-556.  doi: 10.1134/S1560354708060051.

[4]

T. Bountis and P. Vanhaecke, Lotka-Volterra systems satisfying a strong Painlevé property, Phys. Lett. A., 380 (2016), 3977-3982.  doi: 10.1016/j.physleta.2016.09.034.

[5]

S. A. Charalambides, P. A. Damianou and C. A. Evripidou, On generalized Volterra systems, J. Geom. Phys., 87 (2015), 86-105.  doi: 10.1016/j.geomphys.2014.07.007.

[6]

P. A. Damianou, C. A. Evripidou, P. Kassotakis and P. Vanhaecke, Integrable reductions of the Bogoyavlenskij-Itoh Lotka-Volterra systems, J. Math. Phys., 58 (2017), 17pp.  doi: 10.1063/1.4978854.

[7]

C. A. Evripidou, P. Kassotakis and P. Vanhaecke, Integrable deformations of the Bogoyavlenskij-Itoh Lotka-Volterra systems, Regul. Chaotic Dyn., 22 (2017), 721-739.  doi: 10.1134/S1560354717060090.

[8]

C. A. Evripidou, P. Kassotakis and P. Vanhaecke, Integrable reductions of the dressing chain, preprint, arXiv: math/1903.02876.

[9]

M. Hénon and C. Heiles, The applicability of the third integral of motion: Some numerical experiments, Astronom. J., 69 (1964), 73-79.  doi: 10.1086/109234.

[10]

B. Hernández-Bermejo and V. Fairén, Hamiltonian structure and Darboux theorem for families of generalized Lotka-Volterra systems, J. Math. Phys., 39 (1998), 6162-6174.  doi: 10.1063/1.532621.

[11]

Y. Itoh, Integrals of a Lotka-Volterra system of odd number of variables, Progr. Theoret. Phys., 78 (1987), 507-510.  doi: 10.1143/PTP.78.507.

[12]

Y. Itoh, A combinatorial method for the vanishing of the Poisson brackets of an integrable Lotka-Volterra system, J. Phys. A, 42 (2009), 11pp.  doi: 10.1088/1751-8113/42/2/025201.

[13]

P. H. van der Kamp, T. E. Kouloukas, G. R. W. Quispel, D. T. Tran and P. Vanhaecke, Integrable and superintegrable systems associated with multi-sums of products, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 470 (2014), 23pp.  doi: 10.1098/rspa.2014.0481.

[14]

T. E. Kouloukas, G. R. W. Quispel and P. Vanhaecke, Liouville integrability and superintegrability of a generalized Lotka-Volterra system and its Kahan discretization, J. Phys. A, 49 (2016), 13pp.  doi: 10.1088/1751-8113/49/22/225201.

[15]

A. J. Lotka, Analytical Theory of Biological Populations, The Plenum Series on Demographic Methods and Population Analysis, Plenum Press, New York, 1998. doi: 10.1007/978-1-4757-9176-1.

[16]

O. Ragnisco and M. Scalia, The Volterra Integrable case, preprint, arXiv: math/1903.03595.

[17]

Y. B. Suris and O. Ragnisco, What is the relativistic Volterra lattice?, Comm. Math. Phys., 200 (1999), 445-485.  doi: 10.1007/s002200050537.

[18]

J. A. Vano, J. C. Wildenberg, M. B. Anderson, J. K. Noel and J. C. Sprott, Chaos in low-dimensional Lotka-Volterra models of competition, Nonlinearity, 19 (2006), 2391-2404.  doi: 10.1088/0951-7715/19/10/006.

[19]

A. P. Veselov and A. V. Penskoï, On algebro-geometric Poisson brackets for the Volterra lattice, Regul. Chaotic Dyn., 3 (1998), 3-9.  doi: 10.1070/rd1998v003n02ABEH000066.

[20]

V. Volterra, Leçons sur la théorie mathématique de la lutte pour la vie, Les Grands Classiques Gauthier-Villars, Éditions Jacques Gabay, Sceaux, 1990.

show all references

References:
[1]

Á. Ballesteros, A. Blasco and F. Musso, Integrable deformations of Lotka-Volterra systems, Phys. Lett. A, 375 (2011), 3370-3374.  doi: 10.1016/j.physleta.2011.07.055.

[2]

O. I. Bogoyavlenskij, Some constructions of integrable dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat., 51 (1987), 737-766.  doi: 10.1070/IM1988v031n01ABEH001043.

[3]

O. I. Bogoyavlenskij, Integrable Lotka-Volterra systems, Regul. Chaotic Dyn., 13 (2008), 543-556.  doi: 10.1134/S1560354708060051.

[4]

T. Bountis and P. Vanhaecke, Lotka-Volterra systems satisfying a strong Painlevé property, Phys. Lett. A., 380 (2016), 3977-3982.  doi: 10.1016/j.physleta.2016.09.034.

[5]

S. A. Charalambides, P. A. Damianou and C. A. Evripidou, On generalized Volterra systems, J. Geom. Phys., 87 (2015), 86-105.  doi: 10.1016/j.geomphys.2014.07.007.

[6]

P. A. Damianou, C. A. Evripidou, P. Kassotakis and P. Vanhaecke, Integrable reductions of the Bogoyavlenskij-Itoh Lotka-Volterra systems, J. Math. Phys., 58 (2017), 17pp.  doi: 10.1063/1.4978854.

[7]

C. A. Evripidou, P. Kassotakis and P. Vanhaecke, Integrable deformations of the Bogoyavlenskij-Itoh Lotka-Volterra systems, Regul. Chaotic Dyn., 22 (2017), 721-739.  doi: 10.1134/S1560354717060090.

[8]

C. A. Evripidou, P. Kassotakis and P. Vanhaecke, Integrable reductions of the dressing chain, preprint, arXiv: math/1903.02876.

[9]

M. Hénon and C. Heiles, The applicability of the third integral of motion: Some numerical experiments, Astronom. J., 69 (1964), 73-79.  doi: 10.1086/109234.

[10]

B. Hernández-Bermejo and V. Fairén, Hamiltonian structure and Darboux theorem for families of generalized Lotka-Volterra systems, J. Math. Phys., 39 (1998), 6162-6174.  doi: 10.1063/1.532621.

[11]

Y. Itoh, Integrals of a Lotka-Volterra system of odd number of variables, Progr. Theoret. Phys., 78 (1987), 507-510.  doi: 10.1143/PTP.78.507.

[12]

Y. Itoh, A combinatorial method for the vanishing of the Poisson brackets of an integrable Lotka-Volterra system, J. Phys. A, 42 (2009), 11pp.  doi: 10.1088/1751-8113/42/2/025201.

[13]

P. H. van der Kamp, T. E. Kouloukas, G. R. W. Quispel, D. T. Tran and P. Vanhaecke, Integrable and superintegrable systems associated with multi-sums of products, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 470 (2014), 23pp.  doi: 10.1098/rspa.2014.0481.

[14]

T. E. Kouloukas, G. R. W. Quispel and P. Vanhaecke, Liouville integrability and superintegrability of a generalized Lotka-Volterra system and its Kahan discretization, J. Phys. A, 49 (2016), 13pp.  doi: 10.1088/1751-8113/49/22/225201.

[15]

A. J. Lotka, Analytical Theory of Biological Populations, The Plenum Series on Demographic Methods and Population Analysis, Plenum Press, New York, 1998. doi: 10.1007/978-1-4757-9176-1.

[16]

O. Ragnisco and M. Scalia, The Volterra Integrable case, preprint, arXiv: math/1903.03595.

[17]

Y. B. Suris and O. Ragnisco, What is the relativistic Volterra lattice?, Comm. Math. Phys., 200 (1999), 445-485.  doi: 10.1007/s002200050537.

[18]

J. A. Vano, J. C. Wildenberg, M. B. Anderson, J. K. Noel and J. C. Sprott, Chaos in low-dimensional Lotka-Volterra models of competition, Nonlinearity, 19 (2006), 2391-2404.  doi: 10.1088/0951-7715/19/10/006.

[19]

A. P. Veselov and A. V. Penskoï, On algebro-geometric Poisson brackets for the Volterra lattice, Regul. Chaotic Dyn., 3 (1998), 3-9.  doi: 10.1070/rd1998v003n02ABEH000066.

[20]

V. Volterra, Leçons sur la théorie mathématique de la lutte pour la vie, Les Grands Classiques Gauthier-Villars, Éditions Jacques Gabay, Sceaux, 1990.

Figure 1.  The Poincaré surface of section $ x_2 = 1,x_1>1 $ for the Lotka–Volterra system with $ a_i = 1 $ and $ E = 6 $ for various $ k_i $, $ i = 1,2,3,4 $ values: (a) $ (k_1,k_2, k_3, k_4) = (-1, -2,- 1,-1) $ (b) $ (k_1,k_2, k_3, k_4) = (-1, -2,- 1,-2) $, (c) $ (k_1,k_2, k_3, k_4) = (-1,-4,-2,-3) $, (d) $ (k_1,k_2, k_3, k_4) = (-1, -4,- 2,-1) $
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Figure 2.  3D projections on the $ x_2,x_3,x_4 $ plane for the system with $ (k_1,k_2, k_3, k_4) = (-1, -4,- 2,-1) $ and for initial conditions: (a) close to a fixed point of Fig. 1(d) ($ E = 6 $), (b) on an ellipse around the fixed point ($ E = 6 $), (c) randomly chosen from Fig. 1(d) ($ E = 6 $) and (d) randomly chosen at a higher total energy ($ E = 20 $) exhibiting chaotic behavior
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Figure 3.  The Poincaré surface of section $ x_2 = 1, x_1>1 $ for the Lotka–Volterra system with $ a_i = 1 $ and $ k_i = -1 $, $ i = 1,2,3,4 $ for the energies: (a) $ E = 4.2 $, (b) $ E = 6 $, (c) $ E = 8 $, (d) $ E = 29 $
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Figure 4.  The largest Lyapunov exponent $ \lambda $ for the Lotka–Volterra system with $ a_i = 1 $ and $ k_i = -1 $, $ i = 1,2,3,4 $ for the energies: (a) $ E = 4.2 $ and (b) $ E = 29 $
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Figure 5.  The evolution in time of the phase space variables for the integrable cases: (a) $ (k_1,k_2, k_3, k_4) = (0, 0,- 1,-1) $ and (b) $ (k_1,k_2, k_3, k_4) = (-2, -2,- 2,2) $
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Figure 6.  The trajectories projected on the 3D plane $ x_1,x_3,x_4 $ plane for the integrable systems: (a) $ (k_1,k_2, k_3, k_4) = (0, 0,- 1,-1) $ and (b) $ (k_1,k_2, k_3, k_4) = (-2, -2,- 2,2) $
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Figure 7.  The largest Lyapunov exponent $ \lambda $ for the system with $ (k_1,k_2, k_3, k_4) = (-2, -2,- 2,2) $ at (b) $ E = 10 $ and (c) $ E = 72 $
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