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Dynamical Behaviour in Two Prey-Predator System with Persistence
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Dynamical Behaviour in Two Prey-Predator System with Persistence

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Abstract:

In this work, the dynamical behavior of the system with two preys and one predator population is investigated. The predator exhibits a Holling type II response to one prey which is harvested and a Beddington-DeAngelis functional response to the other prey. The boundedness of the system is analyzed. We examine the occurrence of positive equilibrium points and stability of the system at those points. At trivial equilibrium E0 and axial equilibrium (E1); the system is found to be unstable. Also we obtain the necessary and sufficient conditions for existence of interior equilibrium point (E6) and local and global stability of the system at the interior equilibrium (E6): Depending upon the existence of limit cycle, the persistence condition is established for the system. The numerical simulation infer that varying the parameters such as e and λ1 it is possible to change the dynamical behavior of the system from limit cycle to stable spiral. It is also observed that the harvesting rate plays a crucial role in stabilizing the system.

Info:

Periodical:
Bulletin of Mathematical Sciences and Applications (Volume 16)
Pages:
20-37
DOI:
https://doi.org/10.18052/www.scipress.com/BMSA.16.20
Citation:
V. Madhusudanan and S. Vijaya, "Dynamical Behaviour in Two Prey-Predator System with Persistence", Bulletin of Mathematical Sciences and Applications, Vol. 16, pp. 20-37, 2016
Online since:
August 2016
Authors:
V. Madhusudanan, S. Vijaya
Keywords:
Functional Response, Global Stability, Harvesting, Local Stability, Persistence
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Distribution:
Open Access

Creative Commons License
This work is licensed under a
Creative Commons Attribution 4.0 International License

References:

[1] B. Dubey, R.K. Upadhyay, Persistence and extinction of one prey and two predator system, J. Nonlinear Analysis. Modelling and Control, Vol. 9, No. 4, (2004) 307-329.

[2] E.A. McGehee, N. Schutt, D.A. Vasquez and E. Peacock-Lopez, Bifurcations and temporal and spatial patterns of a modified Lotka-Volterra model, Int.J. Bifurcation Chaos Appl. Sci. Eng. 18 (2008) 2223-2248.

[3] S. Gakkar and R.K. Naji, Existence of chaos in two prey and one predator system. chaos, Solutions & Fractals, 17(4) (2003) 639-649.

[4] S. Gakkar and R.K. Naji, Chaos in three species ratio dependent food chain. chaos, Solutions & Fractals 14 (2002) 771-778.

[5] H.F. Huo, Z.P. Ma and C.Y. Liu, Persistence and Stability for a Generalized Leslie-Gower Model with Stage Structure and Dispersal, Hindawi Publishing Corporation Abstract and Applied Analysis, Article ID 135843 (2009), 17 pages.

[6] S.B. Hsu, T.W. Hwang and Y. Kuang, Rich dynamics of a ratio-dependent one prey-two predators model, J. Math. Biol., Vol. 43. (2001) 377-396.

[7] D. Kesh, A.K. Sarkar and A.B. Roy, Persistence of two prey-one predator system with ratiodependent predator influence, Math. Appl. Sci., Vol. 23 (2000) 347-356.

[8] S. Kumar, S.K. Srivastava and P. Chingakham, Hopf bifuracation and stability analysis in a harvested one-predator-two-prey model, Appl. Math. Comput., Vol 129, No. 1, (2002) 107-118.

[9] M. Haque, A detailed study of the Beddington-DeAngelis predator-prey model, Math. Biosci. 234 (2011) 116.

[10] P. Lenzini and J. Rebaza, Non-constant predator harvesting on ratio-dependent prey-predator models, Appl. Math. Sci. (16)(2010) 791-803.

[11] R.K. Naji and A.T. Balasim, Dynamical behavior of a three species food chain model with Beddington- DeAngelis functional response, Chaos, Solutions and Fractals, 32 (2007), 1853- 1866.

[12] S. Gakkar and Brahampal Singh, The Dynamics of food web consisting of two preys and a harvesting predator, Chaos, Solutions and Fractals 34 (2007) 1346-1356.

[13] T.K. Kar and A. Batabyal, Persistence and Stability of a Two Prey One Predator System, International Journal of Engg. Sci. and Tech., 2(2) (2010) 174-190.

[14] D. Xiao and S. Ruan, Global dynamics of a ratio dependent predator-prey system, J. Math. Biol., 43(3) (2001) 268-290.

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