Positive Periodic Solutions for a Predator-prey Model with Modified Leslie-Gower Holling-type II Schemes and a Deviating Argument
In this paper, by utilizing the coincidence degree theorem a predator-prey model with modified Leslie-Gower Hollingtype II schemes and a deviating argument is studied. Some sufficient conditions are obtained for the existence of positive periodic solutions of the model.
[1] P.H. Leslie, Some further notes on the use of matrices in population
mathematics, Biometrica 35 (1948) 213-245.
[2] P.H. Leslie, J.C. Gower, The properties of a stochastic model for the
predatorCprey type of interaction between two species, Biometrica 47
(1960) 219-234.
[3] E.C. Pielou, An Introduction to Mathematical Ecology, Wiley-
Interscience, New York, 1969.
[4] M.A. Aziz-Alaoui, Study of a Leslie-Gower-type tritrophic population,
Chaos Solitons & Fractals 14 (2002) 1275-1293.
[5] Z.D. Teng, L.S. Chen, Necessary and sufficient conditions for existence of
positive periodic solutions of periodic predator-prey systems, Acta. Math.
Scientia 18 (4) (1998) 402-406(in Chinese).
[6] C. Chen, F.D. Chen, Conditions for global attractivity of multispecise
ecological Competition-Predator system with Holling III type functional
response, J. Biomathematics 19 (2) (2004) 136-140.
[7] F.D. Chen, The permanence and global attractivity of Lotka-Volterra
competition system with feedback controls, Nonlinear Anal. Real World
Appl. 7 (2006) 133-143.
[8] K. Wang, Y.L. Zhu, Global attractivity of positive periodic solution for
a Volterra model. Appl. Math. Comput. 203 (2) (2008) 493-501.
[9] K. Wang, Existence and global asymptotic stability of positive periodic
solution for a Predator-Prey system with mutual interference, Nonlinear
Anal. Real World Appl. 10 (5) (2009) 2774-2783.
[10] X.Y. Song, Y.F. Li, Dynamic behaviors of the periodic predator-prey
model with modified Leslie-Gower Holling-type II schemes and impulsive
effect, Nonlinear Anal. Real World Appl. 9 (2008) 64-79.
[11] A.F. Nindjin, M.A. Aziz-Alaoui, M. Cadivel, Analysis of a predatorprey
model with modified Leslie-Gower and Holling-type II schemes with
time delay, Nonlinear Anal. Real World Appl. 7 (2006) 1104-1118.
[12] R.E. Gaines, J.L. Mawhin, Cioncidence Degree and Nonlinear Differential
Equations, Springer-Verlag, Berlin, 1977.
[1] P.H. Leslie, Some further notes on the use of matrices in population
mathematics, Biometrica 35 (1948) 213-245.
[2] P.H. Leslie, J.C. Gower, The properties of a stochastic model for the
predatorCprey type of interaction between two species, Biometrica 47
(1960) 219-234.
[3] E.C. Pielou, An Introduction to Mathematical Ecology, Wiley-
Interscience, New York, 1969.
[4] M.A. Aziz-Alaoui, Study of a Leslie-Gower-type tritrophic population,
Chaos Solitons & Fractals 14 (2002) 1275-1293.
[5] Z.D. Teng, L.S. Chen, Necessary and sufficient conditions for existence of
positive periodic solutions of periodic predator-prey systems, Acta. Math.
Scientia 18 (4) (1998) 402-406(in Chinese).
[6] C. Chen, F.D. Chen, Conditions for global attractivity of multispecise
ecological Competition-Predator system with Holling III type functional
response, J. Biomathematics 19 (2) (2004) 136-140.
[7] F.D. Chen, The permanence and global attractivity of Lotka-Volterra
competition system with feedback controls, Nonlinear Anal. Real World
Appl. 7 (2006) 133-143.
[8] K. Wang, Y.L. Zhu, Global attractivity of positive periodic solution for
a Volterra model. Appl. Math. Comput. 203 (2) (2008) 493-501.
[9] K. Wang, Existence and global asymptotic stability of positive periodic
solution for a Predator-Prey system with mutual interference, Nonlinear
Anal. Real World Appl. 10 (5) (2009) 2774-2783.
[10] X.Y. Song, Y.F. Li, Dynamic behaviors of the periodic predator-prey
model with modified Leslie-Gower Holling-type II schemes and impulsive
effect, Nonlinear Anal. Real World Appl. 9 (2008) 64-79.
[11] A.F. Nindjin, M.A. Aziz-Alaoui, M. Cadivel, Analysis of a predatorprey
model with modified Leslie-Gower and Holling-type II schemes with
time delay, Nonlinear Anal. Real World Appl. 7 (2006) 1104-1118.
[12] R.E. Gaines, J.L. Mawhin, Cioncidence Degree and Nonlinear Differential
Equations, Springer-Verlag, Berlin, 1977.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:56058", author = "Yanling Zhu and Kai Wang", title = "Positive Periodic Solutions for a Predator-prey Model with Modified Leslie-Gower Holling-type II Schemes and a Deviating Argument", abstract = "In this paper, by utilizing the coincidence degree theorem a predator-prey model with modified Leslie-Gower Hollingtype II schemes and a deviating argument is studied. Some sufficient conditions are obtained for the existence of positive periodic solutions of the model.
", keywords = "Predator-prey model, Holling II type functional response, positive periodic solution, coincidence degree theorem.", volume = "6", number = "8", pages = "988-4", }
