Periodic solutions and homoclinic bifurcation of a predator–prey system with two types of harvesting

In this paper, a predator–prey model with both constant rate harvesting and state dependent impulsive harvesting is analyzed. By using differential equation geometry theory and the method of successor functions, the existence, uniqueness and stability of the order one periodic solution have been studied. Sufficient conditions which guarantee the nonexistence of order k (k≥2) periodic solution are given. We also present that the system exhibits the phenomenon of homoclinic bifurcation under some parametric conditions. Finally, some numerical simulations and biological explanations are given.

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

[2]  Lansun Chen,et al.  The stage-structured predator-prey model and optimal harvesting policy. , 2000, Mathematical biosciences.

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

[4]  Sunita Gakkhar,et al.  Dynamics in a Beddington–DeAngelis prey–predator system with impulsive harvesting , 2007 .

[5]  Zhidong Teng,et al.  Existence and stability of periodic solution of a Lotka-Volterra predator-prey model with state dependent impulsive effects , 2009 .

[6]  Chen Lan-sun,et al.  Continuous and impulsive harvesting strategies in a stage-structured predator-prey model with time delay , 2009 .

[7]  Weihua Jiang,et al.  Bifurcation and chaos of a delayed predator-prey model with dormancy of predators , 2012 .

[8]  Junjie Wei,et al.  Bifurcation analysis in a time-delay model for prey–predator growth with stage-structure , 2007 .

[9]  F. Brauer.  De-stabilization of predator-prey systems under enrichment† , 1976 .

[10]  Fengde Chen,et al.  GLOBAL ANALYSIS OF A HARVESTED PREDATOR–PREY MODEL INCORPORATING A CONSTANT PREY REFUGE , 2010 .

[11]  A. C. Soudack,et al.  Stabilization and de-stabilization of predator-prey systems under harvesting and nutrient enrichment , 1976 .

[12]  Guirong Jiang,et al.  Complex dynamics of a Holling type II prey–predator system with state feedback control , 2007 .

[13]  Moxun Tang,et al.  Coexistence Region and Global Dynamics of a Harvested Predator-Prey System , 1998, SIAM J. Appl. Math..

[14]  Linearly independent homoclinic bifurcations parameterized by a small function , 2007 .

[15]  A. C. Soudack,et al.  Stability regions in predator-prey systems with constant-rate prey harvesting , 1979 .

[16]  David A. Demer,et al.  Acoustical monitoring of fish density, behavior, and growth rate in a tank , 2006 .

[17]  J. Lagardère,et al.  Variabilité météorologique et hydrologique. Conséquences sur l'activité natatoire d'un poisson marin , 1998 .

[18]  J. Lagardère,et al.  Fish telemetry in aquaculture: review and perspectives , 1995, Aquaculture International.

[19]  Chen Lan-sun.  Pest Control and Geometric Theory of Semi-Continuous Dynamical System , 2011 .

[20]  Sanyi Tang,et al.  The effect of seasonal harvesting on stage-structured population models , 2004, Journal of mathematical biology.

[21]  Guangzhao Zeng,et al.  Existence of periodic solution of order one of planar impulsive autonomous system , 2006 .