Biological SystemsNo Access

COMPLEXITY OF AN IVLEV'S PREDATOR-PREY MODEL WITH PULSE

Department of Mathematics and Computer Science, Yulin Normal University, Yulin Guangxi 537000, China

Department of Applied Mathematics, Dalian University of Technology, Dalian Liaoning 116024, China

Corresponding author. Address: Department of Mathematics and Computer Science, Yulin Normal University, Yulin Guangxi 537000, China.

Search for more papers by this author

and

LANSUN CHEN

Department of Applied Mathematics, Dalian University of Technology, Dalian Liaoning 116024, China

Minnan Science and Technology Institute Fujian Normal University, Nan'an Fujian 362332, P. R. China

Search for more papers by this author

https://doi.org/10.1142/S0219525907001021Cited by:2 (Source: Crossref)

Abstract

In this paper, we investigate the extinction, permanence and dynamic complexity of the two-prey, one-predator system with Ivlev's functional response and impulsive perturbation on the predator at fixed moments. Conditions for the extinction and permanence of the system are established via the comparison theorem. Numerical simulations are carried out to explain the conclusions we obtain. Furthermore, the resulting bifurcation diagrams clearly show that the impulsive system takes on many forms of complexity including period-doubling bifurcation, period-halving bifurcation, and chaos.

Keywords:

References

C. S. Holling, Mem. Ent. Sec. Can. 45, 1 (1965). Google Scholar
V. S. Ivlev , Experimental Ecology of the Feeding of Fishes ( Yale University Press , New Haven , 1961 ) . Google Scholar
A. Gasull and A. Guillamon , Non-Existence of Limit Cycles for Some Predator-Prey Systems ( World Scientific , Singapore , 1993 ) . Google Scholar
R. E. Kooij and A. Zegeling, J. Math. Anal. Appl. 198, 473 (1996), DOI: 10.1006/jmaa.1996.0093. Crossref, Web of Science, Google Scholar
J. Sugie, J. Math. Anal. Appl. 217, 349 (1998), DOI: 10.1006/jmaa.1997.5700. Crossref, Web of Science, Google Scholar
J. M. Cushing, J. Math. Biol. 10, 348 (1980), DOI: 10.1007/BF00276097. Google Scholar
J. Grasmanet al., Math. Biosci. 169, 207 (2001), DOI: 10.1016/S0025-5564(00)00051-1. Crossref, Web of Science, Google Scholar
M. Bardi, J. Math. Biol. 12, 127 (1981), DOI: 10.1007/BF00275208. Crossref, Web of Science, Google Scholar
J. C. Allen, Fla. Entomol. 73, 100 (1990). Crossref, Web of Science, Google Scholar
F. Doveriet al., Theor. Popul. Biol. 43, 159 (1993), DOI: 10.1006/tpbi.1993.1008. Crossref, Web of Science, Google Scholar
S. Rinaldi and S. Muratori, Ecol. Model. 69, 79 (1993), DOI: 10.1016/0304-3800(93)90050-3. Crossref, Web of Science, Google Scholar
S. Rinaldi, S. Muratori and Y. A. Kuznetsov, Bull. Math. Biol. 55, 15 (1993). Crossref, Web of Science, Google Scholar
G. C. W. Sabin and D. Summers, Math. Biosci. 113, 91 (1993), DOI: 10.1016/0025-5564(93)90010-8. Crossref, Web of Science, Google Scholar
J. Vandermeer, L. Stone and B. Blasius, Chaos Soliton. Fract. 12, 265 (2001), DOI: 10.1016/S0960-0779(00)00111-9. Crossref, Web of Science, Google Scholar
G. Zeng and L. Chen, Adv. Complex Syst. 8, 419 (2005), DOI: 10.1142/S0219525905000580. Link, Web of Science, Google Scholar
William, J. Chivers and R. D. Herbert, Adv. Complex Syst. 6, 441 (2003), DOI: 10.1142/S0219525903000980. Link, Web of Science, Google Scholar
Y. Peiet al., Adv. Complex Syst. 8, 483 (2005), DOI: 10.1142/S0219525905000579. Link, Web of Science, Google Scholar
V. Lakshmikantham , D. D. Bainov and P. C. Simeonov , Theory of Impulsive Differential Equations ( World Scientific , Singapore , 1989 ) . Link, Google Scholar
D. D. Bainov and P. S. Simeonov , Impulsive Differential Equations: Periodic Solutions and Applications ( Longman , England , 1993 ) . Google Scholar
A. Lakmeche and O. Arino, Dyn. Cont. Discr. Impulsive. Syst. 7, 265 (2000). Google Scholar
A. Venkatesan, S. Parthasarathy and M. Lakshmanan, Chaos Soliton. Fract. 18, 891 (2003), DOI: 10.1016/S0960-0779(03)00092-4. Crossref, Web of Science, Google Scholar
J. P. Eckmann , Chaotic Behavior of Deterministic Systems , eds. G. Iooss et al. ( Amsterdam Elsevier , 1983 ) . Google Scholar
R. M. May and G. F. Oster, Am. Nature 110, 573 (1976), DOI: 10.1086/283092. Crossref, Web of Science, Google Scholar
M. G. Neubert and H. Caswell, J. Math. Biol. 41, 103 (2000), DOI: 10.1007/s002850070001. Crossref, Web of Science, Google Scholar
A. Wikan, J. Math. Biol. 43, 471 (2001), DOI: 10.1007/s002850100101. Crossref, Web of Science, Google Scholar