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http://dx.doi.org/10.5666/KMJ.2017.57.2.265

Delayed Dynamics of Prey-Predator System with Distinct Functional Responses  

Madhusudanan, V. (Department of Mathematics, S.A. Engineering College)
Vijaya, S. (Department of Mathematics, Annamalai University)
Publication Information
Kyungpook Mathematical Journal / v.57, no.2, 2017 , pp. 265-285 More about this Journal
Abstract
In this article, a mathematical model is proposed and analyzed to study the delayed dynamics of a system having a predator and two preys with distinct growth rates and functional responses. The equilibrium points of proposed system are determined and the local stability at each of the possible equilibrium points is investigated by its corresponding characteristic equation. The boundedness of the system is established in the absence of delay and the condition for existence of persistence in the system is determined. The discrete type gestational delay of predator is also incorporated on the system. Further it is proved that the system undergoes Hopf bifurcation using delay as bifurcation parameter. This study refers that time delay may have an impact on the stability of the system. Finally Computer simulations illustrate the dynamics of the system.
Keywords
predator-prey system; growth rate; functional response; local stability; persistence; Hopf bifurcation; discrete delay;
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1 M. Agarwal and V. Singh, Rich dynamics in Ratio department III functional response, Int. J. Environ. Sci. Technol., 5 (2013), 106-123.
2 H. R. Akcakaya, R. Arditi and L. R. Ginzburg, Ratio-dependent predition: an abstraction that works, Ecology, 76 (1995), 995-1004.   DOI
3 R. Arditi and L. R. Ginzburg, Coupling in predator-prey dynamics: ratio dependence, J. Theor. Biol., 139 (1989), 311-326.   DOI
4 G. Birkoff and G. C. Rota, Ordinary differential equations, Blaisdell Publishing Co. Ginn and Co., Waltham, Mass.-Toronto, Ont.-London, 1969.
5 H. Beak, Y. Do and D. Lim, A three species food chain system with two types of functional response, Abstr. Appl. Anal., 2011 (2011), Art. ID 934569, 16 pages.
6 S. Baek, W. Ko and I. Ahn, Coexistence of a one-prey two-predators model with ratio-depenent functional responses, Appl. Math. Comput., 219 (2012), 1897-1908.
7 B. Dubey and R. K. Upadhay, Persistence and extinction of one-prey and two-predator system, Nonlinear Anal. Model. Control, 9 (2004), 307-329.
8 H. Freedman and P. Waltman, Mathematical analysis of some three species food chain models, Math. Biosci., 33 (1977), 257-276.   DOI
9 M. Fan and Y. Kuang, Dynamics of a nonautonomous predator-prey system with Beddington-DeAngelis functional response, J. Math. Anal. Appl., 295 (2004), 15-39.   DOI
10 P. Abrams, The fallacies of "ratio-dependent" predation, Ecology, 75(6) (1994), 1842-1850.   DOI
11 T. C. Gard and T. G. Hallam, Persistence in food webs-1, Lotka-volterra food chais, Bull. Math. Biol., 41 (1997), 877-891.
12 S. Gakkhar and A. Singh, Control of choas due to additional predator in the hasting-powell food chain model, J. Math, Anal, Appl, 385 (2012), 423-438.   DOI
13 S. Gakkar and R. K. Naji, Existence of chaos in two prey and one predator system, Chaos Solitons Fractals, 17(4) (2003), 639-649.   DOI
14 S. Gakkar and R. K. Naji, Chaos in three species ratio dependent food chain, Chaos Solitons Fractals, 14 (2002), 771-778.   DOI
15 S. Gakkar and Brahampal Singh, The Dynamics of food web consisting of two preys and a harvesting predator,Chaos Solitons Fractals, 34 (2007) 1346-1356.   DOI
16 K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic Publishers, Netherlands, 1992.
17 J. Hafbaucer, A general co-operation theorem for hyper cycles, Monatsh. Math., 91 (1981), 233-240.   DOI
18 J. K. Hale, Ordinary di erential equation, John Willey and Sons, New York, 1969.
19 S. B. Hsu, T. W. Hwang and Y. Kuang, Rich dynamics of a ratio-dependent one prey-two predators model, J. Math. Biol., 43 (2001), 377-396.   DOI
20 V. Hutson and G.T. Vickers, A criterion for permanent coexistence of species with an application to two prey, one predator system, Math. Biosci., 63 (1983), 253-269.   DOI
21 C. Jost and S.P. Ellner, Testing for predator dependence in predator-prey dynamics: a non-parametric Approach, Proc. R. Soc. Land B, 267 (2000), 1611-1620.   DOI
22 T.K. Kar and H. Matsuda, Global dynamics and controllability of a Harvested preypredator system with holling type III functional response,Nonlinear Anal. Hybrid Syst., 1 (2007), 59-67.   DOI
23 T.K. Kar and A. Batabyal, Persistence and Stability of a Two Prey One Predator System, Int. J. Math. Model. Simul. Appl., 2 (2009), 395-413.
24 D. Kesh, A. K. Sarkar and A. B. Roy, Persistence of two prey-one predator system with ratio-dependent predator in uence, Math. Methods Appl. Sci., 23 (2000), 347-356.   DOI
25 Y. Kuang and E. Berratta, Global qualitative analysis of a ratio-dependent predatorprey system, J. Math. Biol., 36 (1998), 389-406.   DOI
26 Y. Kuang, Delay Differential Equations: with Applications in population Dynamics, Academic Press, New York, 1993.
27 S. Kumar, S. K. Srivastava and P. Chingakham, Hopf bifuracation and stability analysis in a harvested one-predator-two-prey model, Appl. Math. Comput., 129 (2002), 107-118.
28 R. K. Naji and A. T. Balasim, Dynamical behavior of a three species food chain model with Beddington-DeAngelis functional response, Chaos Solitons Fractals, 32 (2007), 1853-1866.   DOI
29 B. Patra, A. Maiti and P. Samanta, Effect of time-delay on a Ratio-dependent food chain model, Nonlinear Anal. Model. Control 14 (2009), 199-216.
30 F. J. Richards, A exible growth function for empirical use, J. Exp. Bot., 10 (1959), 290-301.   DOI
31 G.P. Samantha and Sharma, Dynamical behavior of a two prey and one predator system, Differential Equations and Dynamical Systems App., 22 (2014) 125-145.   DOI
32 D. Xiao and S. Ruan, Global dynamics of a ratio dependent predator-prey system, J. Math. Biol., 43(3) (2001), 268-290.   DOI
33 B. Sahoo, Predator-prey model with different growth rates and different functional response: A comparative study with additional food, International Journal of Applied Mathematical Research, 2 (2012) 117-129.
34 J .P. Tripathi, S. Abbas S and M. Thauker, Stability analysis of two prey one predator model, AIP Conf. Proc., 1479 905(2012).
35 P. F. Verhulst, Notice sur la loi que la population poursuit dans son accroissement, Correspondance mathematique et physique, 10 (1838), 113-121.