THE PERIODIC PREDATOR-PREY LOTKA–VOLTERRA MODEL WITH IMPULSIVE EFFECT

In this paper, a classical periodic Lotka–Volterra predator-prey system with impulsive effect is investigated. We analyze the dynamics of positive solutions of such models. Among other results we show that if some trivial or semi-trivial positive solution is linearly stable, then it is globally asymptotically stable with respect to the positive solutions. By using the method of coincidence degree, a set of sufficient conditions are derived for the existence of at least one strictly positive (componentwise) periodic solution. We use bifurcation theorem to show the existence of coexistence states which arise near the sem-trivial periodic solution. As an application, we also examine some special cases of the system which can be used in the biological pest control.

[1]  K. S. Chaudhuri,et al.  A bioeconomic model of harvesting a multispecies fishery , 1986 .

[2]  B. Shulgin,et al.  Pulse vaccination strategy in the SIR epidemic model , 1998, Bulletin of mathematical biology.

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

[4]  Xinzhi Liu,et al.  Permanence of population growth models with impulsive effects , 1997 .

[5]  Rafael Ortega,et al.  The periodic predator-prey Lotka-Volterra model , 1996, Advances in Differential Equations.

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

[7]  R. Ortega,et al.  A Periodic Prey-Predator System , 1994 .

[8]  R. Anderson,et al.  Pulse mass measles vaccination across age cohorts. , 1993, Proceedings of the National Academy of Sciences of the United States of America.

[9]  M. Crandall,et al.  Bifurcation from simple eigenvalues , 1971 .

[10]  A. C. Soudack,et al.  Coexistence properties of some predator-prey systems under constant rate harvesting and stocking , 1982 .

[11]  Eric T. Funasaki,et al.  Invasion and Chaos in a Periodically Pulsed Mass-Action Chemostat , 1993 .

[12]  J. C. Lenteren,et al.  Biological and Integrated Pest control in Greenhouses , 1988 .

[13]  J. Panetta,et al.  A mathematical model of periodically pulsed chemotherapy: tumor recurrence and metastasis in a competitive environment. , 1996, Bulletin of mathematical biology.

[14]  Brian A. Croft,et al.  Arthropod biological control agents and pesticides , 1990 .

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

[16]  C. S. Holling,et al.  The functional response of predators to prey density and its role in mimicry and population regulation. , 1965 .

[17]  A. C. Soudack,et al.  Constant-rate stocking of predator-prey systems , 1981 .

[18]  R R Kao,et al.  The dynamics of an infectious disease in a population with birth pulses. , 1998, Mathematical biosciences.

[19]  Michael G. Crandall,et al.  Bifurcation, perturbation of simple eigenvalues, itand linearized stability , 1973 .

[20]  R. Gaines,et al.  Coincidence Degree and Nonlinear Differential Equations , 1977 .

[21]  I. Bajo,et al.  Periodic Boundary Value Problem for First Order Differential Equations with Impulses at Variable Times , 1996 .

[22]  F. Brauer.  Boundedness of solutions of predator-prey systems , 1979 .

[23]  Snezhana Hristova,et al.  Existence of periodic solutions of nonlinear systems of differential equations with impulse effect , 1987 .