Homoclinic bifurcation of prey-predator model with impulsive state feedback control

Abstract In this paper, homoclinic bifurcations of a prey–predator system with state impulse are investigated. Firstly, the existence of order-1 homoclinic cycle of system (1.2) with δ = 0 is investigated; Secondly, choosing q as a control parameter, the sufficient conditions of existence and stability of order-1 periodic solution of system (1.2) with δ = 0 are obtained by means of the geometry theory of semi-continuous dynamic systems; Thirdly, on the basis of the theory of rotated vector fields, homoclinic bifurcation of system (1.2) about parameter δ are also studied; Finally, some simulations are provided to prove the main results. The methods used in this paper are intuitive to prove the existence of order-1 homoclinic cycle and homoclinic bifurcations.

[1]  Zhijun Liu,et al.  Impulsive harvesting and stocking in a Monod–Haldane functional response predator–prey system , 2007 .

[2]  Benjamin Leard,et al.  Analysis of predator-prey models with continuous threshold harvesting , 2011, Appl. Math. Comput..

[3]  Bing Liu,et al.  DYNAMICS ON A HOLLING II PREDATOR–PREY MODEL WITH STATE-DEPENDENT IMPULSIVE CONTROL , 2012 .

[4]  Weiming Wang,et al.  Dynamics of a Ivlev-type predator–prey system with constant rate harvesting , 2009 .

[5]  A. C. Soudack,et al.  Stability regions and transition phenomena for harvested predator-prey systems , 1979 .

[6]  Konstantin Mischaikow,et al.  Competing Species near a Degenerate Limit , 2003, SIAM J. Math. Anal..

[7]  Colin W. Clark,et al.  Mathematical Bioeconomics: The Optimal Management of Renewable Resources. , 1993 .

[8]  Stein Ivar Steinshamn,et al.  Rescuing the Prey by Harvesting the Predator: Is It Possible? , 2010 .

[9]  H. I. Freedman.  Deterministic mathematical models in population ecology , 1982 .

[10]  Min Zhao,et al.  HOMOCLINIC BIFURCATION IN SEMI-CONTINUOUS DYNAMIC SYSTEMS , 2012 .

[11]  Dongmei Xiao,et al.  Bifurcations of a Ratio-Dependent Predator-Prey System with Constant Rate Harvesting , 2005, SIAM J. Appl. Math..

[12]  Qingling Zhang,et al.  The geometrical analysis of a predator-prey model with two state impulses. , 2012, Mathematical biosciences.

[13]  Jianjun Jiao,et al.  Harvesting policy for a delayed stage-structured Holling II predator–prey model with impulsive stocking prey , 2009 .

[14]  Zuxiong Li,et al.  Periodic solution of a chemostat model with Beddington-DeAnglis uptake function and impulsive state feedback control. , 2009, Journal of theoretical biology.

[15]  En-Guo Gu,et al.  Complex dynamics analysis for a duopoly model of common fishery resource , 2010 .

[16]  Colin W. Clark,et al.  Bioeconomic Modelling and Fisheries Management. , 1985 .

[17]  Eduardo González-Olivares,et al.  Optimal harvesting in a predator–prey model with Allee effect and sigmoid functional response , 2012 .

[18]  Malay Banerjee,et al.  Bifurcation Analysis and Control of Leslie–Gower Predator–Prey Model with Michaelis–Menten Type Prey-Harvesting , 2012 .