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

Sugie, Jitsuro; Kimoto, Kyoko

doi:10.1090/S0033-569X-06-01031-6

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
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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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