PapersNo Access

Periodic Solutions for a Predator–Prey Model with Periodic Harvesting Rate

Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, P. R. China

and

Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, P. R. China

Research supported by the National Natural Science Foundations of China (No. 11371248) and the RFDP of Higher Education of China grant (No. 20130073110074).

Search for more papers by this author

https://doi.org/10.1142/S0218127414500965Cited by:3 (Source: Crossref)

Abstract

In the paper, we study the existence of periodic solutions in an invariant manifold for a predator–prey model with periodic harvesting rate. Without harvesting, the predator–prey system has a positive equilibrium of center type, which undergoes Hopf bifurcation. Under the periodic harvesting, we first reduce this predator–prey system to a complex normal form. Then, we define the Poincaré mapping of this system by using the polar coordinates, and prove that there exists an integral manifold and periodic solution with the period of the harvesting rate. An example is provided to illustrate the result.

Keywords:

References

P. A. Abrams and L. R. Ginzburg , Trends in Ecol. Evol. 15 , 337 ( 2000 ) . Crossref, Web of Science, Google Scholar
R. Arditi and L. R. Ginzburg , J. Theoret. Biol. 139 , 311 ( 1989 ) . Crossref, Web of Science, Google Scholar
F. Berezovskaya , G. Karev and R. Arditi , J. Math. Biol. 43 , 221 ( 2001 ) . Crossref, Web of Science, Google Scholar
F. Brauer and D. A. Sánchez , Nat. Res. Model. 16 , 233 ( 2003 ) . Crossref, Google Scholar
J. Chen et al. , SIAM J. Appl. Math. 73 , 1876 ( 2013 ) . Crossref, Web of Science, Google Scholar
S.-N. Chow , C. Li and D. Wang , Normal Forms and Bifurcation of Planar Vector Fields ( Cambridge University Press , Cambridge , 1994 ) . Crossref, Google Scholar
J. Guckenheimer and P. Holmes , Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields ( Springer-Verlag , NY , 1983 ) . Crossref, Google Scholar
A. P. Gutierrez , Ecology 73 , 1552 ( 1992 ) . Crossref, Web of Science, Google Scholar
M. Hirsch , S. Smale and R. Devaney , Differential Equations, Dynamical Systems and An Introduction to Chaos ( Elsevier , NY , 2004 ) . Google Scholar
S. Hsu , T. Hwang and Y. Kuang , J. Math. Biol. 42 , 489 ( 2001 ) . Crossref, Web of Science, Google Scholar
C. Jost , O. Arino and R. Arditi , Bull. Math. Biol. 61 , 19 ( 1999 ) . Crossref, Web of Science, Google Scholar
Y. Kuang , Fields Instit. Commun. 21 , 325 ( 1999 ) . Google Scholar
M. Qian , H. Hu and Q. Zhou , Acta Math. Sin. 26 , 37 ( 1983 ) . Google Scholar
C. J. Waiters and P. J. Bandy , J. Wildl. Manage. 36 , 128 ( 1972 ) . Crossref, Web of Science, Google Scholar
D. Xiao and S. Ruan , J. Math. Biol. 43 , 268 ( 2001 ) . Crossref, Web of Science, Google Scholar
D. Xiao and L. Jennings , SIAM J. Appl. Math. 65 , 737 ( 2005 ) . Crossref, Web of Science, Google Scholar
D. Xiao , W. Li and M. Han , J. Math. Anal. Appl. 324 , 14 ( 2006 ) . Crossref, Web of Science, Google Scholar