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  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
Published electronically: August 15, 2006
MathSciNet review: 2259048
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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.


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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: http://dx.doi.org/10.1090/S0033-569X-06-01031-6
PII: S 0033-569X(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.



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