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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Global asymptotic stability for predator-prey systems whose prey receives time-variation of the environment


Authors: Jitsuro Sugie, Yasuhisa Saito and Meng Fan
Journal: Proc. Amer. Math. Soc. 139 (2011), 3475-3483
MSC (2010): Primary 34D23, 92D25; Secondary 34D05, 37B25
Published electronically: June 6, 2011
MathSciNet review: 2813379
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Abstract: A predator-prey model with prey receiving time-variation of the environment is considered. Such a system is shown to have a unique interior equilibrium that is globally asymptotically stable if the time-variation is bounded and weakly integrally positive. In particular, the result tells us that the equilibrium point can be stabilized even by nonnegative functions that make the limiting system structurally unstable. The method that is used to obtain the result is an analysis of asymptotic behavior of the solutions of an equivalent system to the predator-prey model.


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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

Yasuhisa Saito
Affiliation: Department of Mathematics, Pusan National University, Busan 609-735, Republic of Korea
Address at time of publication: Department of Mathematics, Chonnam National University, Gwangju 500-757, Republic of Korea
Email: saito.yasuhisa@gmail.com

Meng Fan
Affiliation: School of Mathematics and Statistics, Northeast Normal University, Changchun, 130024 Jilin, People’s Republic of China
Email: mfan@nenu.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-11124-9
PII: S 0002-9939(2011)11124-9
Keywords: Global asymptotic stability, predator-prey systems, weakly integrally positive, time-variation
Received by editor(s): June 12, 2010
Published electronically: June 6, 2011
Additional Notes: The first author was supported in part by Grant-in-Aid for Scientific Research No.22540190 from the Japan Society for the Promotion of Science
The third author was supported in part by the NSFC, NCET-08-0755 and FRFCU
Communicated by: Yingfei Yi
Article copyright: © Copyright 2011 American Mathematical Society