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On the Dynamical Behavior of a Two-Prey One-Predator System with Two-Type Functional Responses
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  • Journal title : Kyungpook mathematical journal
  • Volume 53, Issue 4,  2013, pp.647-660
  • Publisher : Department of Mathematics, Kyungpook National University
  • DOI : 10.5666/KMJ.2013.53.4.647
 Title & Authors
On the Dynamical Behavior of a Two-Prey One-Predator System with Two-Type Functional Responses
Baek, Hunki;
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 Abstract
In the paper, a two-prey one-predator system with defensive ability and Holling type-II functional responses is investigated. First, the stability of equilibrium points of the system is discussed and then conditions for the persistence of the system are established according to the existence of limit cycles. Numerical examples are illustrated to attest to our mathematical results. Finally, via bifurcation diagrams, various dynamic behaviors including chaotic phenomena are demonstrated.
 Keywords
a two-prey one-predator system;two-type functional responses;Holling type-II functional responses;bifurcation diagrams;
 Language
English
 Cited by
 References
1.
R. Arditi and L. R. Ginzburg, Coupling in predator-prey dynamics:Ratio-dependence, J. Theor. Biol., 139(1989), 311-326. crossref(new window)

2.
H. Baek, Extinction and Permanence of a Holling I Type Impulsive Predator-prey Model, Kyungpook Math. Journal, 49(2009), 763-770. crossref(new window)

3.
F. Brauer and C. Castillo-Chavez, Mathematical models in population biology and epidemiology, Texts in applied mathematics 40, Springer-Verlag, New York(2001).

4.
F. Cao and L. Chen, Asymptotic behavior of nonautonomous diffusive Lotka-Volterra model, Syst. Sci. Math. Sci., 11(2)(1998), 107-111.

5.
Z. Cheng, Y. Lin and J. Cao, Dynamical behaviors of a partial-dependent predatorprey system, Chaos, Solitons and Fractals, 28(2006), 67-75. crossref(new window)

6.
J. M. Cushing, Periodic time-dependent predator-prey systems, SIAM J. Appl. Math., 32(1977), 82-95. crossref(new window)

7.
H. I. Freedman and P. Waltman, Persistence in models of three interecting predatorprey populations, Math. Biosciece, 68(1984), 213-231. crossref(new window)

8.
K. Hasik, Uniqueness of limit cycle in the predator-prey system with sysmmetric prey isocline, Math. Biosciences, 164(2000), 203-215. crossref(new window)

9.
A. Hastings and T. Powell, Chaos in a three-species food chain, Ecology, 72(3)(1991), 896-903. crossref(new window)

10.
C. S. Holling, The functional response of predator to prey density and its role in mimicry and population regulation, Mem. Entomol. Soc. Can., 45(1965), 1-60.

11.
Sze-Bi Hsu, Tzy-Wei Hwang and Yang Kuang, A ratio-dependent food chain model and its application to biological control, Math. Biosci., 181(2003), 55-83. crossref(new window)

12.
Ji-cai Huang and Dong-mei Xiao, Analyses of Bifurcations and Stability in a Predatorprey System with Holling Type-IV Functional Response, Acta Mathematicae Applicatae Sinica(English Series), 20(1)(2004), 167-178. crossref(new window)

13.
A. Klebanoff and A. Hastings, Chaos in three species food chains, J. Math. Biol., 32(1994), 427-451. crossref(new window)

14.
B. Liu, Z. Teng and L. Chen, Analsis of a predator-prey model with Holling II functional response concerning impulsive control strategy, J. of Comp. and Appl. Math., 193(1)(2006), 347-362. crossref(new window)

15.
S. Lv and M. Zhao, The dynamic complexity of a three species food chain model, Chaos Solitons and Fractals, 37(2008), 1469-1480. crossref(new window)

16.
R. K. Naji and A. T. Balasim, Dynamical behaior of a three species food chain model with Beddington-DeAngelis functional response, Chaos, Solitions and Fractals, 32(2007), 1853-1866. crossref(new window)

17.
Y. Pei, G. Zeng and L. Chen, Species extinction and permanence in a prey-predator model with two-type functional responses and impulsive biological control, Nonlinear Dyn., 52(2008), 71-81. crossref(new window)

18.
S. Ruan and D. Xiao, Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 61(4)(2001), 1445-1472. crossref(new window)

19.
G. C. W. Sabin and D. Summers, Chaos in a periodically forced predator-prey ecosystem model, Math. Bioscience, 113(1993), 91-113. crossref(new window)

20.
E, Saez and E. Gonzalez-Olivares, Dynamics of a predator-prey model, SIAM J. Appl. Math., 59(5)(1999), 1867-1878. crossref(new window)

21.
L. Segel, Modelling dynamic phenomena in molecular and cellular biology, Cambridge Universtiy Press, Cambridge, England, 1984.

22.
C. Shen, Permanence and global attractivity of the food-chain system with Holling IV type functional response, Appl. Math and Comp., 194(2007), 179-185. crossref(new window)

23.
J. Sugie, R. Kohno and R. Miyazaki, On a predator-prey system of Holling type, Proced. of the Amer. Math. Soc, 125(7)(1997), 2041-2050.

24.
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. crossref(new window)

25.
Q. Wang, M. Fan and K. Wang, Dynamics of a class of nonautonomous semi-ratiodependent predator-prey systems with functional responses, J. of Math. Analy. and Appl., 278(2003), 443-471. crossref(new window)

26.
H. Zhu, S. A. Campbell and Gail S. K. Wolkowicz, Bifurcation analysis of a predator-prey system with nonmonotonic Functional Response, SIAM J. Appl. Math., 63(2)(2002), 636-682.