Multiple Positive Periodic Solutions of a Delayed Predatory-Prey System with Holling Type II Functional Response
In this letter, we considers a delayed predatory-prey system with Holling type II functional response. Under some sufficient conditions, the existence of multiple positive periodic solutions is obtained by using Mawhin’s continuation theorem of coincidence degree theory. An example is given to illustrate the effectiveness of our results.
<p>[1] S.G. Ruan, D.M. Xiao, Global analysis in a predator-prey system with
nonmonotonic functional response, SIAM J. Appl. Math. 61 (2001) 1445-
1472.
[2] 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.
[3] A.D. Bazykin, Structural and Dynamic Stability of Model Predator-Prey
Systems, Int. Inst. Appl. Syst. Anal., Laxenburg. Res. Rep., IIASA,
Laxenburg, 1976.
[4] Wonlyul Ko, Kimun Ryu, Qualitative analysis of a predator-prey model
with Holling type II functional response incorporating a prey refuge,
Journal of Differential Equations, 231 (2006) 534-550.
[5] L.J. Chen, F.D. Chen, L.J. Chen, Qualitative analysis of a predator-prey
model with Holling type II functional response incorporating a constant
prey refuge, Nonlinear Analysis: Real World Applications, 11 (2010) 246-
252.
[6] L.F. Nie, Z.D. Teng, L. Hu, J.G. Peng, Qualitative analysis of a modified
Leslie-Gower and Holling-type II predator-prey model with state dependent
impulsive effects, Nonlinear Analysis: Real World Applications, 11
(201) 1364-1373.
[7] W. Liu, C.J. Fu, B.S. Chen, Hopf bifurcation for a predator-prey biological
economic system with Holling type II functional response, Journal of the
Franklin Institute, 348 (2011) 1114-1127
[8] R. Gaines, J. Mawhin, Coincidence Degree and Nonlinear Differetial
Equitions, Springer Verlag, Berlin, 1977.
[9] Y. Chen, Multiple periodic solutions of delayed predator-prey systems
with type IV functional responses, Nonlinear Anal. Real World Appl.
5(2004) 45-53.
[10] Q. Wang, B. Dai, Y. Chen, Multiple periodic solutions of an impulsive
predator-prey model with Holling-type IV functional response, Math.
Comput. Modelling 49 (2009) 1829-1836.
[11] D.W. Hu, Z.Q. Zhang, Four positive periodic solutions to a Lotka-
Volterra cooperative system with harvesting terms, Nonlinear Anal. Real
World Appl. 11 (2010) 1115-1121.
[12] K.H. Zhao, Y. Ye, Four positive periodic solutions to a periodic Lotka-
Volterra predatory-prey system with harvesting terms, Nonlinear Anal.
Real World Appl. 11 (2010) 2448-2455.
[13] K.H. Zhao, Y.K. Li, Four positive periodic solutions to two species
parasitical system with harvesting terms, Computers and Mathematics
with Applications. 59 (2010) 2703-2710.
[14] Y.K. Li, K.H. Zhao, Y. Ye, Multiple positive periodic solutions of n
species delay competition systems with harvesting terms, Nonlinear Anal.
Real World Appl. 12 (2011) 1013-1022.
[15] Y. Li, Y. Kuang, Periodic solutions of periodic delay Lotka-Volterra
equations and systems, J. Math. Anal. Appl. 255 (2001) 260-280.</p>
<p>[1] S.G. Ruan, D.M. Xiao, Global analysis in a predator-prey system with
nonmonotonic functional response, SIAM J. Appl. Math. 61 (2001) 1445-
1472.
[2] 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.
[3] A.D. Bazykin, Structural and Dynamic Stability of Model Predator-Prey
Systems, Int. Inst. Appl. Syst. Anal., Laxenburg. Res. Rep., IIASA,
Laxenburg, 1976.
[4] Wonlyul Ko, Kimun Ryu, Qualitative analysis of a predator-prey model
with Holling type II functional response incorporating a prey refuge,
Journal of Differential Equations, 231 (2006) 534-550.
[5] L.J. Chen, F.D. Chen, L.J. Chen, Qualitative analysis of a predator-prey
model with Holling type II functional response incorporating a constant
prey refuge, Nonlinear Analysis: Real World Applications, 11 (2010) 246-
252.
[6] L.F. Nie, Z.D. Teng, L. Hu, J.G. Peng, Qualitative analysis of a modified
Leslie-Gower and Holling-type II predator-prey model with state dependent
impulsive effects, Nonlinear Analysis: Real World Applications, 11
(201) 1364-1373.
[7] W. Liu, C.J. Fu, B.S. Chen, Hopf bifurcation for a predator-prey biological
economic system with Holling type II functional response, Journal of the
Franklin Institute, 348 (2011) 1114-1127
[8] R. Gaines, J. Mawhin, Coincidence Degree and Nonlinear Differetial
Equitions, Springer Verlag, Berlin, 1977.
[9] Y. Chen, Multiple periodic solutions of delayed predator-prey systems
with type IV functional responses, Nonlinear Anal. Real World Appl.
5(2004) 45-53.
[10] Q. Wang, B. Dai, Y. Chen, Multiple periodic solutions of an impulsive
predator-prey model with Holling-type IV functional response, Math.
Comput. Modelling 49 (2009) 1829-1836.
[11] D.W. Hu, Z.Q. Zhang, Four positive periodic solutions to a Lotka-
Volterra cooperative system with harvesting terms, Nonlinear Anal. Real
World Appl. 11 (2010) 1115-1121.
[12] K.H. Zhao, Y. Ye, Four positive periodic solutions to a periodic Lotka-
Volterra predatory-prey system with harvesting terms, Nonlinear Anal.
Real World Appl. 11 (2010) 2448-2455.
[13] K.H. Zhao, Y.K. Li, Four positive periodic solutions to two species
parasitical system with harvesting terms, Computers and Mathematics
with Applications. 59 (2010) 2703-2710.
[14] Y.K. Li, K.H. Zhao, Y. Ye, Multiple positive periodic solutions of n
species delay competition systems with harvesting terms, Nonlinear Anal.
Real World Appl. 12 (2011) 1013-1022.
[15] Y. Li, Y. Kuang, Periodic solutions of periodic delay Lotka-Volterra
equations and systems, J. Math. Anal. Appl. 255 (2001) 260-280.</p>
@article{"International Journal of Engineering, Mathematical and Physical Sciences:65296", author = "Kaihong Zhao and Jiuqing Liu", title = "Multiple Positive Periodic Solutions of a Delayed Predatory-Prey System with Holling Type II Functional Response", abstract = "In this letter, we considers a delayed predatory-prey system with Holling type II functional response. Under some sufficient conditions, the existence of multiple positive periodic solutions is obtained by using Mawhin’s continuation theorem of coincidence degree theory. An example is given to illustrate the effectiveness of our results.
", keywords = "Multiple positive periodic solutions, Predatory-prey
system, Coincidence degree, Holling type II functional response.", volume = "7", number = "3", pages = "453-5", }
