Multiple limit cycles in a Leslie–Gower-type predator–prey model considering weak Allee effect on prey

ABSTRACT In this work we present the predator-prey model with Allee effect and Hawk and Dove tactics in fighting over caught prey implemented as fast strategy evolution dynamics. We extend the work of Auger, Parra, Morand and S´anchez (2002) using the prey population embodying Allee effect and analogously to this work we get two connected submodels with polymorphic and monomorphic predator population.We get much richer dynamics, in each submodel we find local bifurcations (saddle-node, supercritical Hopf caused by Allee effect and Bogdanov- -Takens) and a global bifurcation of limit cycles caused by the strategy evolution that is not possible in any of the submodels that can lead to a bluesky extinction of both populations.

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Two limit cycles in a Leslie–Gower predator–prey model with additive Allee effect

Nonlinear Analysis Real World Applications ◽

10.1016/j.nonrwa.2008.01.022 ◽

2009 ◽

Vol 10 (3) ◽

pp. 1401-1416 ◽

Cited By ~ 62

Author(s):

Pablo Aguirre ◽

Eduardo González-Olivares ◽

Eduardo Sáez

Keyword(s):

Limit Cycles ◽

Allee Effect ◽

Predator Prey ◽

Predator Prey Model ◽

Prey Model

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Consequences of double Allee effect on the number of limit cycles in a predator–prey model

Computers & Mathematics with Applications ◽

10.1016/j.camwa.2011.08.061 ◽

2011 ◽

Vol 62 (9) ◽

pp. 3449-3463 ◽

Cited By ~ 26

Author(s):

Eduardo González-Olivares ◽

Betsabé González-Yañez ◽

Jaime Mena Lorca ◽

Alejandro Rojas-Palma ◽

José D. Flores

Keyword(s):

Limit Cycles ◽

Allee Effect ◽

Predator Prey ◽

Predator Prey Model ◽

Prey Model

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The dynamics of a Leslie type predator–prey model with fear and Allee effect

Advances in Difference Equations ◽

10.1186/s13662-021-03490-x ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

S. Vinoth ◽

R. Sivasamy ◽

K. Sathiyanathan ◽

Bundit Unyong ◽

Grienggrai Rajchakit ◽

...

Keyword(s):

Local Stability ◽

Allee Effect ◽

Jacobian Matrix ◽

Predator Prey ◽

Predator Prey Model ◽

Fear Effect ◽

Prey Model ◽

Lyapunov Coefficient ◽

Points Of View ◽

Two Parameter

AbstractIn this article, we discuss the dynamics of a Leslie–Gower ratio-dependent predator–prey model incorporating fear in the prey population. Moreover, the Allee effect in the predator growth is added into account from both biological and mathematical points of view. We explore the influence of the Allee and fear effect on the existence of all positive equilibria. Furthermore, the local stability properties and possible bifurcation behaviors of the proposed system about positive equilibria are discussed with the help of trace and determinant values of the Jacobian matrix. With the help of Sotomayor’s theorem, the conditions for existence of saddle-node bifurcation are derived. Also, we show that the proposed system admits limit cycle dynamics, and its stability is discussed with the value of first Lyapunov coefficient. Moreover, the numerical simulations including phase portrait, one- and two-parameter bifurcation diagrams are performed to validate our important findings.

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