Geometric criteria for the nonexistence of cycles in Gause-type predator-prey systems

Author:
Yaping Liu

Journal:
Proc. Amer. Math. Soc. **133** (2005), 3619-3626

MSC (2000):
Primary 34D23; Secondary 34C07, 92D25

DOI:
https://doi.org/10.1090/S0002-9939-05-08026-3

Published electronically:
June 8, 2005

MathSciNet review:
2163598

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Abstract | References | Similar Articles | Additional Information

Abstract: The global stability of a multi-species interacting system has apparently important biological implications. In this paper we study the global stability of Gause-type predator-prey models by providing new criteria for the nonexistence of cycles and limit cycles. Our criteria have clear geometrical interpretations and are easier to apply than other methods employed in recent studies. Using these criteria and related techniques we are able to develop new results on the existence and uniqueness of cycles in Gause-type models with various growth and response functions.

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Additional Information

**Yaping Liu**

Affiliation:
Department of Mathematics, Pittsburg State University, Pittsburg, Kansas 66762

Email:
yliu@pittstate.edu

DOI:
https://doi.org/10.1090/S0002-9939-05-08026-3

Keywords:
Geometric criterion,
global stability,
limit cycle,
predator-prey system,
Dulac's theorem,
Li\'{e}nard system.

Received by editor(s):
August 13, 2004

Published electronically:
June 8, 2005

Communicated by:
Carmen C. Chicone

Article copyright:
© Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.