Multiple stability and uniqueness of the limit cycle in a Gause-type predator–prey model considering the Allee effect on prey

González‐Olivares, Eduardo; Meneses-Alcay, Héctor; González‐Yañez, Betsabé; Mena-Lorca, Jaime; Rojas‐Palma, Alejandro; Ramos‐Jiliberto, Rodrigo

doi:10.1016/j.nonrwa.2011.04.003

Cited by 61 publications

(32 citation statements)

References 24 publications

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“…Comparing the models here studied with those considering a sigmoid functional response but not the Allee effect, we observe that the quantity of limit cycles increases from one to two: however, this is not a general result for the Gause type predator-prey models, since the Rosenzweig-McArthur predator-prey model has a unique limit cycle when the Allee effect is considered (González-Olivares et al 2010), like the model where this effect is absent.…”

Section: Discussionmentioning

confidence: 68%

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Multiple Limit Cycles in a Gause Type Predator–Prey Model with Holling Type III Functional Response and Allee Effect on Prey

González‐Olivares

Rojas‐Palma

2010

Bull Math Biol

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This work aims to examine the global behavior of a Gause type predator-prey model considering two aspects: (i) the functional response is Holling type III and, (ii) the prey growth is affected by the Allee effect. We prove the origin of the system is an attractor equilibrium point for all parameter values. It has also been shown that it is the ω-limit of a wide set of trajectories of the system, due to the existence of a separatrix curve determined by the stable manifold of the equilibrium point (m,0), which is associated to the Allee effect on prey. When a weak Allee effect on the prey is assumed, an important result is obtained, involving the existence of two limit cycles surrounding a unique positive equilibrium point: the innermost cycle is unstable and the outermost stable. This property, not yet reported in models considering a sigmoid functional response, is an important aspect for ecologists to acknowledge as regards the kind of tristability shown here: (1) the origin; (2) an interior equilibrium; and (3) a limit cycle of large amplitude. These models have undoubtedly been rather sensitive to disturbances and require careful management in applied conservation and renewable resource contexts.

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Section: Discussionmentioning

confidence: 68%

“…However, some of these forms may produce a change in the quantity of limit cycles surrounding a positive equilibrium point in predator-prey models (Flores et al 2007;González-Olivares et al 2010, 2011.…”

Section: The Allee Effectmentioning

confidence: 99%

Multiple Limit Cycles in a Gause Type Predator–Prey Model with Holling Type III Functional Response and Allee Effect on Prey

González‐Olivares

Rojas‐Palma

2010

Bull Math Biol

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“…The comparison between the model studied here and those that consider a sigmoid functional response but the Allee effect is absent shows the quantity of limit cycle increases from one to two; however, this is not a general result for the Gause‐type predator–prey models because the Rosenzweig–McArthur predator–prey model has only a unique limit cycle when considering the Allee effect .…”

Section: Discussionmentioning

confidence: 72%

Limit cycles in a Gause‐type predator–prey model with sigmoid functional response and weak Allee effect on prey

González‐Olivares

Rojas‐Palma

2012

Math Methods in App Sciences

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The goal of this work is to examine the global behavior of a Gause‐type predator–prey model in which two aspects have been taken into account: (i) the functional response is Holling type III; and (ii) the prey growth is affected by a weak Allee effect. Here, it is proved that the origin of the system is a saddle point and the existence of two limit cycles surround a stable positive equilibrium point: the innermost unstable and the outermost stable, just like with the strong Allee effect. Then, for determined parameter constraints, the trajectories can have different ω − limit sets. The coexistence of a stable limit cycle and a stable positive equilibrium point is an important fact for ecologists to be aware of the kind of bistability shown here. So, these models are undoubtedly rather sensitive to disturbances and require careful management in applied contexts of conservation and fisheries. Copyright © 2012 John Wiley & Sons, Ltd.

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“…The third additional family that we consider appears in [11]. The model is described by the following differential system:…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Non-algebraic oscillations for predator-prey models

Ferragut¹,

Gasull²

2014

Publicacions Matemàtiques

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Multiple stability and uniqueness of the limit cycle in a Gause-type predator–prey model considering the Allee effect on prey

Cited by 61 publications

References 24 publications

Multiple Limit Cycles in a Gause Type Predator–Prey Model with Holling Type III Functional Response and Allee Effect on Prey

Multiple Limit Cycles in a Gause Type Predator–Prey Model with Holling Type III Functional Response and Allee Effect on Prey

Limit cycles in a Gause‐type predator–prey model with sigmoid functional response and weak Allee effect on prey

Non-algebraic oscillations for predator-prey models

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