Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

Homoclinic orbits in predator-prey systems with a nonsmooth prey growth rate


Authors: Jitsuro Sugie and Kyoko Kimoto
Journal: Quart. Appl. Math. 64 (2006), 447-461
MSC (2000): Primary 34C37, 37N25, 70K44; Secondary 34C05, 34D23, 92D25
DOI: https://doi.org/10.1090/S0033-569X-06-01031-6
Published electronically: August 15, 2006
MathSciNet review: 2259048
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper deals with Gause-type predator-prey models with a non-smooth prey growth rate. Our models have a unique positive equilibrium and are under the influence of an Allee effect. A necessary and sufficient condition is given for the existence of homoclinic orbits whose $ \alpha$- and $ \omega$-limit sets are the positive equilibrium. The argument used here is based on some results of a system of Liénard type. The relation between homoclinic orbits and the Allee effect is clarified. A simple example is included to illustrate the main result. Some global phase portraits are also attached.


References [Enhancements On Off] (What's this?)

  • 1. K.-S. Cheng, Uniqueness of a limit cycle for a predator-prey system, SIAM J. Math. Anal. 12, 541-548 (1981). MR 0617713 (82h:34035)
  • 2. K.-S. Cheng, S.-B. Hsu and S.-S. Lin, Some results on global stability of a predator-prey system, J. Math. Biol. 12, 115-126 (1981). MR 0631003 (83c:34069)
  • 3. H. I. Freedman, Deterministic Mathematical Models in Population Ecology, Marcel Dekker, New York, 1980. MR 0586941 (83h:92043)
  • 4. G. F. Gause, The Struggle for Existence, Williams and Wilkins, Baltimore, 1934.
  • 5. C. S. Holling, The components of predation as revealed by a study of small-mammal predation of the European pine sawfly, Canad. Entomol. 91, 293-320 (1959).
  • 6. S.-B. Hsu, On global stability of a predator-prey system, Math. Biosci. 39, 1-10 (1978). MR 0472126 (57:11837)
  • 7. V. S. Ivlev, Experimental Ecology of the Feeding of Fishes, Yale University Press, New Haven, CT, 1961.
  • 8. N. D. Kazarinoff and P. van den Driessche, A model predator-prey system with functional response, Math. Biosci. 39, 125-134 (1978). MR 0475989 (57:15570)
  • 9. R. E. Kooij and A. Zegeling, A predator-prey model with Ivlev's functional response, J. Math. Anal. Appl. 198, 473-489 (1996). MR 1376275 (96j:92033)
  • 10. R. E. Kooij and A. Zegeling, Qualitative properties of two-dimensional predator-prey systems, Nonlinear Anal. 29, 693-715 (1997). MR 1452753
  • 11. Y. Kuang, Nonuniqueness of limit cycles of Gause-type predator-prey systems, Appl. Anal. 29, 269-287 (1988). MR 0959804 (89h:92047)
  • 12. Y. Kuang, Global stability of Gause-type predator-prey systems, J. Math. Biol. 28, 463-474 (1990). MR 1057049 (91g:92017)
  • 13. Y. Kuang and H. I. Freedman, Uniqueness of limit cycles in Gause-type models of predator-prey systems, Math. Biosci. 88, 67-84 (1988). MR 0930003 (89g:92045)
  • 14. R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton, 1974.
  • 15. M. L. Rosenzweig and R. H. MacArthur, Graphical representation and stability conditions of predator-prey interactions, Am. Naturalist. 97, 209-223 (1963).
  • 16. J. Sugie, Two-parameter bifurcation in a predator-prey system of Ivlev type, J. Math. Anal. Appl. 217, 349-371 (1998). MR 1492094 (98m:92024)
  • 17. J. Sugie, Uniqueness of limit cycles in a predator-prey system with Holling-type functional response, Quart. Appl. Math. 58, 577-590 (2000). MR 1770656 (2002a:92020)
  • 18. J. Sugie, Homoclinic orbits in generalized Liénard systems, J. Math. Anal. Appl. 309, 211-226 (2005). MR 2154037 (2006d:34101)
  • 19. J. Sugie and T. Hara, Existence and non-existence of homoclinic trajectories of the Liénard system, Discrete Contin. Dynam. Systems 2, 237-254 (1996). MR 1382509 (97a:34127)
  • 20. J. Sugie and M. Katayama, Global asymptotic stability of a predator-prey system of Holling type, Nonlinear Anal. 38, 105-121 (1999). MR 1693000 (2000j:92026)
  • 21. J. Sugie, R. Kohno and R. Miyazaki, On a predator-prey system of Holling type, Proc. Amer. Math. Soc. 125, 2041-2050 (1997). MR 1396998 (97m:92005)
  • 22. J. Sugie, K. Miyamoto and K. Morino, Absence of limit cycles of a predator-prey system with a sigmoid functional response, Appl. Math. Lett. 9, 85-90 (1996). MR 1415457

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC (2000): 34C37, 37N25, 70K44, 34C05, 34D23, 92D25

Retrieve articles in all journals with MSC (2000): 34C37, 37N25, 70K44, 34C05, 34D23, 92D25


Additional Information

Jitsuro Sugie
Affiliation: Department of Mathematics and Computer Science, Shimane University, Matsue 690-8504, Japan
Email: jsugie@riko.shimane-u.ac.jp

Kyoko Kimoto
Affiliation: Department of Mathematics and Computer Science, Shimane University, Matsue 690-8504, Japan

DOI: https://doi.org/10.1090/S0033-569X-06-01031-6
Keywords: Gause-type predator-prey system, Allee effect, homoclinic orbits, global asymptotic stability, Li\'enard system
Received by editor(s): July 20, 2005
Published electronically: August 15, 2006
Additional Notes: The first author was supported in part by Grant-in-Aid for Scientific Research 16540152
Dedicated: Dedicated to Professor Tadayuki Hara on the occasion of his 60th birthday
Article copyright: © Copyright 2006 Brown University
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society