AM  Vol.5 No.8 , May 2014
Positive Periodic Solution for a Two-Species Predator-Prey System
ABSTRACT

A two-species predator-prey system with time delay in a two-patch environment is investigated. By using a continuation theorem based on coincidence degree theory, we obtain some sufficient conditions for the existence of periodic solution for the system.



Cite this paper
Cao, M. , Li, X. and Dai, X. (2014) Positive Periodic Solution for a Two-Species Predator-Prey System. Applied Mathematics, 5, 1099-1107. doi: 10.4236/am.2014.58103.
References
[1]   Freedman, H.I. (1980) Mathematical Models in Population Ecology. Marcel Dekker, New York.

[2]   Sugie, J. (1998) Two-Parameter Bifurcation System of Ivlev Type. Journal of Mathematical Analysis and Applications, 217, 349-371.
http://dx.doi.org/10.1006/jmaa.1997.5700

[3]   Ardito, A. and Ricciardi, P. (1995) Lyapunov Functions for a Generalized Gause-Type Model. Journal of Mathematical Biology, 33, 816-828.
http://dx.doi.org/10.1007/BF00187283

[4]   Hassel, M.P. (1978) The Dynamics of Arthropod Predator-Prey Systems. Princeton University Press, Princeton.

[5]   Hwang, T.W. (1999) Predator-Prey System. Journal of Mathematical Analysis and Applications, 238, 179-195.
http://dx.doi.org/10.1006/jmaa.1999.6520

[6]   Hsu, S.B., Hwang, T.W. and Kuang, Y. (2001) Global Analysis of the Michaelis-Menten Type Ratio-Dependent Predator-Prey. Journal of Mathematical Biology, 42, 489-506.
http://dx.doi.org/10.1007/s002850100079

[7]   Kot, M. (2001) Elements of Mathematical Biology. Cambridge University Press, Cambridge.

[8]   Kuang, Y. and Freedman, H.I. (1988) Uniqueness of Limit Cycles in Gause-Type Predator-Prey Systems. Mathematical Biosciences, 88, 67-84.
http://dx.doi.org/10.1016/0025-5564(88)90049-1

[9]   Kooij, R.E. and Zegeling, A. (1996) A Predator-Prey Model with Ivlev’s Functional Response. Journal of Mathematical Analysis and Applications, 198, 473-489.
http://dx.doi.org/10.1006/jmaa.1996.0093

[10]   Liu, X.X. and Lou, Y.J. (2010) Global Dynamics of a Predator-Prey Model. Journal of Mathematical Analysis and Applications, 371, 323-340.
http://dx.doi.org/10.1016/j.jmaa.2010.05.037

[11]   Xiao, D.M., Li, W.X. and Han, M.A. (2006) Dynamics in Ratio-Dependent Predator-Prey Model with Predator Harvesting. Journal of Mathematical Analysis and Applications, 324, 14-29.
http://dx.doi.org/10.1016/j.jmaa.2005.11.048

[12]   Xiao, D. and Zhang, Z.D. (2003) On the Uniqueness and Nonexistence of Limit Cycles for Predator-Prey Systems. Nonlinearity, 16, 1185-1201.
http://dx.doi.org/10.1088/0951-7715/16/3/321

[13]   Beddington, J.R. (1975) Mutual Interference between Parasites or Predators and Its Effect on Searching Efficiency. Journal of Animal Ecology, 3, 331-340.
http://dx.doi.org/10.2307/3866

[14]   DeAngelis, D.L., Goldstein, R.A. and O’Neil, R.V. (1975) A Model for Trophic Interaction. Ecology, 4, 881-892.
http://dx.doi.org/10.2307/1936298

[15]   Cantrell, R.S. and Cosner, C. (2001) On the Dynamics of Predator-Prey Models with the Beddington-DeAngelis Functional Response. Journal of Mathematical Analysis and Applications, 257, 206-222.
http://dx.doi.org/10.1006/jmaa.2000.7343

[16]   Chen, F., Chen, Y. and Shi, J. (2008) Stability of the Boundary Solution of a Nonautonomous Predator-Prey System with the Beddington-DeAngelis Functional Response. Journal of Mathematical Analysis and Applications, 344, 10571067.
http://dx.doi.org/10.1016/j.jmaa.2008.03.050

[17]   Cui, J. and Takeuchi, Y. (2006) Permanence, Extinction and Periodic Solution of Predator-Prey System with the Beddington-DeAngelis Functional Response. Journal of Mathematical Analysis and Applications, 317, 464-474.
http://dx.doi.org/10.1016/j.jmaa.2005.10.011

[18]   Dimitrov, D.T. and Kojouharov, H.V. (2005) Complete Mathematical Analysis of Predator-Prey System with Linear Prey Growth and Beddington-DeAngelis Functional Response. Applied Mathematics and Computation, 162, 523-538.
http://dx.doi.org/10.1016/j.amc.2003.12.106

[19]   Fan, M. and Kuang, Y. (2004) Dynamics of a Nonautonomous Predator-Prey System with the Beddington-DeAngelis Functional Response. Journal of Mathematical Analysis and Applications, 295, 15-39.
http://dx.doi.org/10.1016/j.jmaa.2004.02.038

[20]   Hwang, T.W. (2003) Global Analysis of the Predator-Prey System with the Beddington-DeAngelis Functional Response. Journal of Mathematical Analysis and Applications, 281, 395-401.
http://dx.doi.org/10.1016/S0022-247X(02)00395-5

[21]   Liu, S. and Beretta, E. (2006) A Stage-Structured Predator-Prey Model of Beddington-DeAngelis Type. SIAM Journal on Applied Mathematics, 66, 1101-1129.
http://dx.doi.org/10.1137/050630003

[22]   Li, H.Y. and Takeuchi, Y. (2011) Dynamics of the Density Dependent Predator-Prey System with Beddington-DeAngelis Functional Response. Journal of Mathematical Analysis and Applications, 374, 644-654.
http://dx.doi.org/10.1016/j.jmaa.2010.08.029

[23]   Gaines, R.E. and Mawhin, J.L. (1977) Coincidence Degree and Nonlinear Differential Equations. Springer, Berlin.

 
 
Top