High Codimension Bifurcations of a Predator–Prey System with Generalized Holling Type III Functional Response and Allee Effects

Arsie, Alessandro; Kottegoda, Chanaka; Shan, Chunhua

doi:10.1007/s10884-022-10214-6

Cited by 3 publications

(3 citation statements)

References 53 publications

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“…Case (II)(ii.1): It is obvious that E * (x * , y * ) is a cusp of codimension 3 if B 31 ̸ = 0 ( [1,2,13,20,25]).…”

Section: Case (Ii)(ii)mentioning

confidence: 99%

Bifurcations of higher codimension in a prey-predator model with generalist predator

Banerjee,

Huang,

Pan

et al. 2024

DCDS-B

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Consideration of generalist predators leads to relatively complex dynamics due to alternative food sources. Here, we propose and analyze a prey-predator model with a generalist predator. The availability of alternative food sources for the predator and a density-dependent growth rate induces not only bistability and tristability, but also more complicated dynamical behaviors. We have studied the possible number and geometric configurations of positive equilibria in detail. A systematic bifurcation analysis has revealed the existence of the degenerate Bogdanov-Takens bifurcation of codimension four and degenerate Hopf bifurcation of codimension three. We found that degenerate local bifurcations with a higher codimension are responsible for three limit cycles. Derivation of the analytical conditions for three limit cycles for a suitable range of parameters is a crucial finding of this work.

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“…Case (II)(ii.1): It is obvious that E * (x * , y * ) is a cusp of codimension 3 if B 31 ̸ = 0 ( [1,2,13,20,25]).…”

Section: Case (Ii)(ii)mentioning

confidence: 99%

Bifurcations of higher codimension in a prey-predator model with generalist predator

Banerjee,

Huang,

Pan

et al. 2024

DCDS-B

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show abstract

“…Therefore, the Bogdanov-Takens bifurcation of codimension three can occur for system (2), and the Bogdanov-Takens bifurcation set of the original system in a small neighborhood of (α, δ, γ) = (α BT , δ BT , γ BT ) is homeomorphic to that of system (18) in a small neighborhood of (ϵ 1 , ϵ 2 , ϵ 3 ) = (0, 0, 0), which is explored in [15]. See Figure 6 for details [4]. In order to have a deeper understanding of the dynamical behavior of the original system, we use the software package Matcont to give bifurcation diagrams for system (2) with respect to the original perturbation parameters in Figure 7.…”

mentioning

confidence: 99%

“…At point C, there are a stable homoclinic loop and a weak focus of order 1 whose first focus quantity is positive. Adapted from [4].…”

mentioning

confidence: 99%

High codimension bifurcations analysis in a predator-prey system with Michaelis-Menten type predator harvesting

Du,

Hu,

Zheng

et al. 2024

DCDS-B

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The main focus of this paper is to study some high codimension bifurcations of a predator-prey system with Michaelis-Menten type predator harvesting. In this paper, using the center manifold theorem, the cusp bifurcation of codimension two is discussed in detail. Furthermore, the Bautin bifurcation of codimension two and the Bogdanov-Takens bifurcation of codimension three are well studied by computing the second Lyapunov coefficient and using the normal form theory, respectively. Numerical simulations are provided to illustrate the theoretical results. The study of these bifurcations reveals the more complex dynamics of the system and may help us to better understand the complex interactions between predator and prey.

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Dynamics of the generalized Rosenzweig–MacArthur model in a changing and patchy environment

Lu,

Xiang,

Huang

et al. 2024

Physica D: Nonlinear Phenomena

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High Codimension Bifurcations of a Predator–Prey System with Generalized Holling Type III Functional Response and Allee Effects

Cited by 3 publications

References 53 publications

Bifurcations of higher codimension in a prey-predator model with generalist predator

Bifurcations of higher codimension in a prey-predator model with generalist predator

High codimension bifurcations analysis in a predator-prey system with Michaelis-Menten type predator harvesting

Dynamics of the generalized Rosenzweig–MacArthur model in a changing and patchy environment

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