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| Uniqueness of Limit Cycles in a Predator-Prey Model with Sigmoid Functional Response |
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| KeyWord:Limit cycle, predator-prey system, Li$\acute{\rm e}$nard equation, Sigmoid functional response |
| Author Name | Affiliation | | Andr$\acute{\rm e}$ Zegeling | College of Mathematics and Statistics, Guangxi Normal University, Guilin, Guangxi 541004, China | | Hailing Wang | College of Mathematics and Statistics, Guangxi Normal University, Guilin, Guangxi 541004, China | | Guangzheng Zhu | College of Physical Science and Technology, Guangxi Normal University, Guilin, Guangxi 541004, China |
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| Abstract: |
| In this paper, we prove that a predator-prey model with sigmoid functional response and logistic growth for the prey has a unique stable limit cycle, if the equilibrium point is locally unstable. This extends the results of the literature where it was proved that the equilibrium point is globally asymptotically stable, if it is locally stable. For the proof, we use a combination of three versions of Zhang Zhifen's uniqueness theorem for limit cycles in Li$\acute{\rm e}$nard systems to cover all possible limit cycle configurations. This technique can be applied to a wide range of differential equations where at most one limit cycle occurs. |
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