On Solutions and Heteroclinic Orbits of Some Lotka-Volterra Systems

Solutions and Heteroclinic Orbits of Some LV Systems

Authors

  • Supriya Mandal Department of Mathematics, Visva-Bharati (A Central University), Santiniketan-731235, West Bengal, India
  • Madan Mohan Panja Department of Mathematics, Visva-Bharati (A Central University), Santiniketan-731235, West Bengal, India
  • Santanu Ray Department of Zoology, Visva-Bharati (A Central University), Santiniketan-731235, West Bengal, India

DOI:

https://doi.org/10.24297/jam.v14i2.7499

Keywords:

LV system, Exact solution, Heteroclinic orbit, Invariant.

Abstract

In this work, a principle for getting heteroclinic orbit of a dynamical system has been proposed when the solution is known in a compact form. The proposed principle has been tested through its application to a three species Lotka-Volterra system, which may appear as a mathematical model of human pathogen system. The domain in parameter  space involve in the model, and the region of initial condition  for the existence of heteroclinic orbit have been derived.

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References

F. Brauer, C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Springer, Verlag, 2000.

A. K. Misra, A simple mathematical model for the spread of the spread of two political parties, Nonlinear Analysis: Modelling and Control, 17 (2012) 343-354.

J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Field, Springer, New York, 1983.

S. Wiggins, Introduction to Applied Nonlinear Dynamical System and Chaos, New York, 1983.

T. Li, G. Chen, G. Chen, On homoclinic and heteroclinic orbit of Chen system, Int. J. Bif. Chaos, 16 (2006) 3035-3041.https://doi.org/10.1142/S021812740601663X

G. Tigan, On a method of finding homoclinic and heteroclinic orbits in multidimensional dynamical systems, Appl. Math. Inf. Sci., 4 (2010) 383-394.

X. Li, Y. Zhao, C. Sun, The heteroclinic orbits and tracking attractor in a cosmological model with a double exponential potential, Class. Quant. Grav., 22 (2005) 3759-3766.https://doi.org/10.1088/0264-9381/22/17/024

F. Beiye, The heteroclinic cycles in the model of competition between n-species and its stability, Acta Math. Appl. Sinica, 14 (1998) 404-413. https://doi.org/10.1007/BF02683825

J. Bao, Q. Yang, A new method to find homoclinic and heteroclinic orbits, Appl. Math. Comput., 217 (2011) 6526-6540.https://doi.org/10.1016/j.amc.2011.01.032

P. Gao, Direct integration method and first integrals for three-dimensional Lotka-Volterra systems, Phys. Lett. A, 255 (1999) 253-258.https://doi.org/10.1016/S0375-9601(99)00193-0

F. Gonzalez-Gascon, D. P. Salas, On the first integrals of Lotka-Volterra systems, Phys. Lett. A, 266, (2000), 336-340.https://doi.org/10.1016/S0375-9601(00)00011-6

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

J. Llibre, C.N. Valls, Polynomial, rational and analytic first integrals for a family of 3-dimensional Lotka-Volterra systems, Z. Angew. Math. Phys., 62 (2011) 761-777.https://doi.org/10.1007/s00033-011-0119-2

W. Aziz, C. Christopher, Local integrability and linearizability of three-dimensional Lotka Volterra systems, Appl. Math. Comput., 219 (2012) 4067-4081.https://doi.org/10.1016/j.amc.2012.10.051

N. A. Kudryashov, A.S. Zakharchenko, Analytical properties and exact solutions of the Lotka-Volterra competition system, Appl. Math. Comput., 254 (2015) 219-228.https://doi.org/10.1016/j.amc.2014.12.113

V. S. Varma, Exact solutions for a special prey-predator or competing species system, Bull. Math. Bio. 39 (1977) 619-622.https://doi.org/10.1016/S0092-8240(77)80064-5

M.A. Almeida, M.E. Magalhaes, I.C. Moreira, Lie symmetries and invariants of Lotka-Volterra system, J. Math. Phys., 36 (1999) 1854-1867. https://doi.org/10.1063/1.531362

B. Grammaticos, J. Moulin-Ollagnier, A. Ramani, J-M. Strelcyn, S. Wojciechowski, Integrals of quadratic ordinary differential equations in R3: the Lotka-Volterra system, Physica A, 163 (1990) 683-722.https://doi.org/10.1016/0378-4371(90)90152-I

L. Cairo, J. Llibre, Darboux integrability for 3D Lotka-Volterra systems, J. Phys. A: Math. Gen., 33 (2000) 2395-2406.https://doi.org/10.1088/0305-4470/33/12/307

R. S. Maier, The integration of three-dimensional Lotka-Volterra systems, Proc. R. Soc. A, 469 (2013) 0693.http://dx.doi.org/10.1098/rspa.2012.0693

J. M. Kim, H. C. Jung, K. Im, I. S. Song, C. Y. Kim, Synergy between Entamoeba histolytica and Escherichia coli in the induction of cytokine gene expression in human colon epithelial cells. Paras. Res, 84 (1998) 509-512.https://doi.org/10.1007/BF03356595.

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Published

2018-07-30

How to Cite

Mandal, S., Panja, M. M., & Ray, S. (2018). On Solutions and Heteroclinic Orbits of Some Lotka-Volterra Systems: Solutions and Heteroclinic Orbits of Some LV Systems. JOURNAL OF ADVANCES IN MATHEMATICS, 14(2), 7851–7859. https://doi.org/10.24297/jam.v14i2.7499