American Institute of Mathematical Sciences

Advanced
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.  Google Scholar

[2]

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

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[20]

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

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.  Google Scholar

[2]

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

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[20]

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

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) $
Figure Options

Download full-size image

Download as PowerPoint slide

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">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
Figure Options

Download full-size image

Download as PowerPoint slide

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 $
Figure Options

Download full-size image

Download as PowerPoint slide

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 $
Figure Options

Download full-size image

Download as PowerPoint slide

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) $
Figure Options

Download full-size image

Download as PowerPoint slide

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) $
Figure Options

Download full-size image

Download as PowerPoint slide

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 $
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

S. Nakaoka, Y. Saito, Y. Takeuchi. Stability, delay, and chaotic behavior in a Lotka-Volterra predator-prey system. Mathematical Biosciences & Engineering, 2006, 3 (1) : 173-187. doi: 10.3934/mbe.2006.3.173
[2]

Guo-Bao Zhang, Ruyun Ma, Xue-Shi Li. Traveling waves of a Lotka-Volterra strong competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 587-608. doi: 10.3934/dcdsb.2018035
[3]

Yuan Lou, Wei-Ming Ni, Shoji Yotsutani. On a limiting system in the Lotka--Volterra competition with cross-diffusion. Discrete & Continuous Dynamical Systems, 2004, 10 (1&2) : 435-458. doi: 10.3934/dcds.2004.10.435
[4]

Qi Wang, Yang Song, Lingjie Shao. Boundedness and persistence of populations in advective Lotka-Volterra competition system. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2245-2263. doi: 10.3934/dcdsb.2018195
[5]

Yuan Lou, Dongmei Xiao, Peng Zhou. Qualitative analysis for a Lotka-Volterra competition system in advective homogeneous environment. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 953-969. doi: 10.3934/dcds.2016.36.953
[6]

Linping Peng, Zhaosheng Feng, Changjian Liu. Quadratic perturbations of a quadratic reversible Lotka-Volterra system with two centers. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4807-4826. doi: 10.3934/dcds.2014.34.4807
[7]

Dan Wei, Shangjiang Guo. Qualitative analysis of a Lotka-Volterra competition-diffusion-advection system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2599-2623. doi: 10.3934/dcdsb.2020197
[8]

Xiaoli Liu, Dongmei Xiao. Bifurcations in a discrete time Lotka-Volterra predator-prey system. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 559-572. doi: 10.3934/dcdsb.2006.6.559
[9]

Fuke Wu, Yangzi Hu. Stochastic Lotka-Volterra system with unbounded distributed delay. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 275-288. doi: 10.3934/dcdsb.2010.14.275
[10]

Jong-Shenq Guo, Ying-Chih Lin. The sign of the wave speed for the Lotka-Volterra competition-diffusion system. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2083-2090. doi: 10.3934/cpaa.2013.12.2083
[11]

Qi Wang, Chunyi Gai, Jingda Yan. Qualitative analysis of a Lotka-Volterra competition system with advection. Discrete & Continuous Dynamical Systems, 2015, 35 (3) : 1239-1284. doi: 10.3934/dcds.2015.35.1239
[12]

Anthony W. Leung, Xiaojie Hou, Wei Feng. Traveling wave solutions for Lotka-Volterra system re-visited. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 171-196. doi: 10.3934/dcdsb.2011.15.171
[13]

Zengji Du, Shuling Yan, Kaige Zhuang. Traveling wave fronts in a diffusive and competitive Lotka-Volterra system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (9) : 3097-3111. doi: 10.3934/dcdss.2021010
[14]

Shaohua Chen, Runzhang Xu, Hongtao Yang. Global and blowup solutions for general Lotka-Volterra systems. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1757-1768. doi: 10.3934/cpaa.2016012
[15]

Suqing Lin, Zhengyi Lu. Permanence for two-species Lotka-Volterra systems with delays. Mathematical Biosciences & Engineering, 2006, 3 (1) : 137-144. doi: 10.3934/mbe.2006.3.137
[16]

Bang-Sheng Han, Zhi-Cheng Wang, Zengji Du. Traveling waves for nonlocal Lotka-Volterra competition systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1959-1983. doi: 10.3934/dcdsb.2020011
[17]

Guichen Lu, Zhengyi Lu. Permanence for two-species Lotka-Volterra cooperative systems with delays. Mathematical Biosciences & Engineering, 2008, 5 (3) : 477-484. doi: 10.3934/mbe.2008.5.477
[18]

Jian Fang, Jianhong Wu. Monotone traveling waves for delayed Lotka-Volterra competition systems. Discrete & Continuous Dynamical Systems, 2012, 32 (9) : 3043-3058. doi: 10.3934/dcds.2012.32.3043
[19]

Guo Lin, Wan-Tong Li, Shigui Ruan. Monostable wavefronts in cooperative Lotka-Volterra systems with nonlocal delays. Discrete & Continuous Dynamical Systems, 2011, 31 (1) : 1-23. doi: 10.3934/dcds.2011.31.1
[20]

Dejun Fan, Xiaoyu Yi, Ling Xia, Jingliang Lv. Dynamical behaviors of stochastic type K monotone Lotka-Volterra systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2901-2922. doi: 10.3934/dcdsb.2018291

 Impact Factor: 

Tools

Metrics

  • PDF downloads (228)
  • HTML views (330)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy