Existence of limit cycles in a predator-prey system with a functional response

We consider the existence of limit cycles for a predator-prey system with a functional response. The system has two or more parameters that represent the intrinsic rate of the predator population. A necessary and sufficient condition for the uniqueness of limit cycles in this system is presented. Such result will usually lead to a bifurcation curve.

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Uniqueness of limit cycles of the predator–prey system with Beddington–DeAngelis functional response

Journal of Mathematical Analysis and Applications ◽

10.1016/j.jmaa.2003.09.073 ◽

2004 ◽

Vol 290 (1) ◽

pp. 113-122 ◽

Cited By ~ 87

Author(s):

Tzy-Wei Hwang

Keyword(s):

Functional Response ◽

Limit Cycles ◽

Predator Prey ◽

Prey System ◽

Uniqueness Of Limit Cycles

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Uniqueness of limit cycles in a predator-prey system with Holling-type functional response

Quarterly of Applied Mathematics ◽

10.1090/qam/1770656 ◽

2000 ◽

Vol 58 (3) ◽

pp. 577-590 ◽

Cited By ~ 7

Author(s):

Jitsuro Sugie

Keyword(s):

Functional Response ◽

Limit Cycles ◽

Predator Prey ◽

Prey System ◽

Uniqueness Of Limit Cycles ◽

Holling Type Functional Response

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Necessary and sufficient condition for existence of periodic solutions of predator-prey dynamic systems with Beddington-DeAngelis-type functional response

Advances in Difference Equations ◽

10.1186/s13662-016-0747-0 ◽

2016 ◽

Vol 2016 (1) ◽

Cited By ~ 2

Author(s):

Neslihan Nesliye Pelen ◽

A. Feza Güvenilir ◽

Billur Kaymakçalan

Keyword(s):

Periodic Solutions ◽

Functional Response ◽

Dynamic Systems ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Predator Prey ◽

Existence Of Periodic Solutions ◽

Necessary And Sufficient

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Canard limit cycles and global dynamics in a singularly perturbed predator–prey system with non-monotonic functional response

Nonlinear Analysis Real World Applications ◽

10.1016/j.nonrwa.2016.01.013 ◽

2016 ◽

Vol 31 ◽

pp. 146-165 ◽

Cited By ~ 12

Author(s):

Jianhe Shen

Keyword(s):

Functional Response ◽

Limit Cycles ◽

Singularly Perturbed ◽

Global Dynamics ◽

Predator Prey ◽

Prey System

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Four limit cycles in a predator–prey system of Leslie type with generalized Holling type III functional response

Nonlinear Analysis Real World Applications ◽

10.1016/j.nonrwa.2019.04.003 ◽

2019 ◽

Vol 50 ◽

pp. 218-239 ◽

Cited By ~ 6

Author(s):

Yanfei Dai ◽

Yulin Zhao ◽

Bo Sang

Keyword(s):

Functional Response ◽

Limit Cycles ◽

Predator Prey ◽

Type Iii ◽

Prey System

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Absence of limit cycles of a predator-prey system with a sigmoid functional response

Applied Mathematics Letters ◽

10.1016/0893-9659(96)00056-0 ◽

1996 ◽

Vol 9 (4) ◽

pp. 85-90 ◽

Cited By ~ 12

Author(s):

J. Sugie ◽

K. Miyamoto ◽

K. Morino

Keyword(s):

Functional Response ◽

Limit Cycles ◽

Predator Prey ◽

Prey System ◽

Sigmoid Functional Response

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Uniqueness of limit cycles in predator–prey system: the role of weight functions

Journal of Mathematical Analysis and Applications ◽

10.1016/s0022-247x(02)00515-2 ◽

2003 ◽

Vol 277 (1) ◽

pp. 130-141 ◽

Cited By ~ 3

Author(s):

Karel Hasík

Keyword(s):

Limit Cycles ◽

Weight Functions ◽

Predator Prey ◽

Prey System ◽

Uniqueness Of Limit Cycles

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NECESSARY AND SUFFICIENT CONDITIONS OF EXISTENCE AND UNIQUENESS OF LIMIT CYCLES FOR A CLASS OF POLYNOMIAL SYSTEM

Acta Mathematica Scientia ◽

10.1016/s0252-9602(18)30430-2 ◽

1991 ◽

Vol 11 (1) ◽

pp. 65-71 ◽

Cited By ~ 1

Author(s):

Deming Liu

Keyword(s):

Limit Cycles ◽

Existence And Uniqueness ◽

Polynomial System ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Uniqueness Of Limit Cycles ◽

Necessary And Sufficient

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Stochastic bifurcations, a necessary and sufficient condition for a stochastic Beddington–DeAngelis predator–prey model

Applied Mathematics Letters ◽

10.1016/j.aml.2021.107069 ◽

2021 ◽

Vol 117 ◽

pp. 107069

Author(s):

Xiaoling Zou ◽

Qingwei Li ◽

Jingliang Lv

Keyword(s):

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Predator Prey ◽

Predator Prey Model ◽

Prey Model ◽

Necessary And Sufficient ◽

Stochastic Bifurcations

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Investigation on dynamics of an impulsive predator–prey system with generalized Holling type IV functional response and anti-predator behavior

Advances in Difference Equations ◽

10.1186/s13662-021-03324-w ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Sekson Sirisubtawee ◽

Nattawut Khansai ◽

Akapak Charoenloedmongkhon

Keyword(s):

Periodic Solution ◽

Functional Response ◽

Impulsive Control ◽

Positive Periodic Solution ◽

Existence Condition ◽

Predator Prey ◽

Type Iv ◽

Prey System ◽

Existence And Stability ◽

Predator Behavior

AbstractIn the present article, we propose and analyze a new mathematical model for a predator–prey system including the following terms: a Monod–Haldane functional response (a generalized Holling type IV), a term describing the anti-predator behavior of prey populations and one for an impulsive control strategy. In particular, we establish the existence condition under which the system has a locally asymptotically stable prey-eradication periodic solution. Violating such a condition, the system turns out to be permanent. Employing bifurcation theory, some conditions, under which the existence and stability of a positive periodic solution of the system occur but its prey-eradication periodic solution becomes unstable, are provided. Furthermore, numerical simulations for the proposed model are given to confirm the obtained theoretical results.

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