scholarly journals Stability and Hopf Bifurcation in Three-Dimensional Predator-Prey Models with Allee Effect

2019 ◽  
pp. 1005-1011
Author(s):  
İlknur Kuşbeyzi Aybar
Keyword(s):  
Hopf Bifurcation ◽  
Allee Effect ◽  
Three Dimensional ◽  
Predator Prey ◽  
Predator Prey Models

scholarly journals Hopf Bifurcation Analysis on General Gause-Type Predator-Prey Models with Delay

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Shuang Guo ◽  
Weihua Jiang
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Predator Prey Model ◽  
Normal Form Method ◽  
Predator Prey Models ◽  
The Stability

A class of three-dimensional Gause-type predator-prey model with delay is considered. Firstly, a group of sufficient conditions for the existence of Hopf bifurcation is obtained via employing the polynomial theorem by analyzing the distribution of the roots of the associated characteristic equation. Secondly, the direction of the Hopf bifurcation and the stability of the bifurcated periodic solutions are determined by applying the normal form method and the center manifold theorem. Finally, some numerical simulations are carried out to illustrate the obtained results.

ALLEE EFFECT, EMIGRATION AND IMMIGRATION IN A CLASS OF PREDATOR-PREY MODELS

BIOMAT 2007 ◽  
2008 ◽  
Author(s):  
EDUARDO GONZÁLEZ-OLIVARES ◽  
JAIME MENA-LORCA ◽  
HÉCTOR MENESES-ALCAY ◽  
BETSABÉ GONZÁLEZ-YAÑEZ ◽  
JOSÉ D. FLORES
Keyword(s):  
Allee Effect ◽  
Emigration And Immigration ◽  
Predator Prey ◽  
Predator Prey Models

scholarly journals Phase tipping: how cyclic ecosystems respond to contemporary climate

2021 ◽  
Vol 477 (2254) ◽  
Author(s):  
Hassan Alkhayuon ◽  
Rebecca C. Tyson ◽  
Sebastian Wieczorek
Keyword(s):  
Allee Effect ◽  
Critical Factor ◽  
Global Dynamics ◽  
Tipping Points ◽  
Time Varying ◽  
Predator Prey ◽  
Critical Transitions ◽  
External Conditions ◽  
Predator Prey Models ◽  
Contemporary Climate

We identify the phase of a cycle as a new critical factor for tipping points (critical transitions) in cyclic systems subject to time-varying external conditions. As an example, we consider how contemporary climate variability induces tipping from a predator–prey cycle to extinction in two paradigmatic predator–prey models with an Allee effect. Our analysis of these examples uncovers a counterintuitive behaviour, which we call phase tipping or P-tipping , where tipping to extinction occurs only from certain phases of the cycle. To explain this behaviour, we combine global dynamics with set theory and introduce the concept of partial basin instability for attracting limit cycles. This concept provides a general framework to analyse and identify easily testable criteria for the occurrence of phase tipping in externally forced systems, and can be extended to more complicated attractors.

BIFURCATIONS IN A PREDATOR-PREY MODEL WITH DIFFUSION AND MEMORY

2006 ◽  
Vol 16 (06) ◽  
pp. 1855-1863 ◽  
Author(s):  
SHABAN ALY
Keyword(s):  
Hopf Bifurcation ◽  
Allee Effect ◽  
Stationary Solutions ◽  
Asymptotically Stable ◽  
Predator Prey ◽  
Predator Prey Model ◽  
The Past ◽  
Prey System ◽  
Stable Equilibria ◽  
Per Capita

In this paper we formulate a delayed predator-prey system in two patches in which the per capita migration rate of each species is influenced only by its own density, i.e. there is no response to the density of the other and the growth rate of the predator depends on the prey that was available in the past. If the equilibrium point lies in the Allée effect zone and when the diffusion is present only, we show that at a critical value of the bifurcation parameter the system undergoes a Turing bifurcation, patterns emerge, the spatially homogeneous equilibrium loses its stability and two new spatially non-constant stable equilibria emerge which are asymptotically stable. When the delay is present only, the increase of delay destabilizes the system and causes the occurrence of periodic oscillations, Andronov–Hopf bifurcation. For the full general model (with both diffusion and delay) if the bifurcation parameters are increased through critical values of diffusion and delay the two new spatially nonconstant stationary solutions lose their stability by Hopf bifurcation.

scholarly journals Dynamics of a Predator-Prey System with a Mate-Finding Allee Effect

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Ruiwen Wu ◽  
Xiuxiang Liu
Keyword(s):  
Hopf Bifurcation ◽  
Bifurcation Analysis ◽  
Allee Effect ◽  
Blow Up ◽  
Mate Finding ◽  
Predator Prey ◽  
Stability Properties ◽  
Prey System ◽  
The Stability ◽  
High Possibility

We consider a ratio-dependent predator-prey system with a mate-finding Allee effect on prey. The stability properties of the equilibria and a complete bifurcation analysis, including the existence of a saddle-node, a Hopf bifurcation, and, a Bogdanov-Takens bifurcations, have been proved theoretically and numerically. The blow-up method has been applied to investigate the structure of a neighborhood of the origin. Our mathematical results show the mate-finding Allee effect can reduce the complexity of system behaviors by making the complicated equilibrium less complicated, and it can be a destabilizing force as well, which makes the system has a high possibility of being threatened with extinction in ecology.

scholarly journals Double Hopf bifurcation of a diffusive predator–prey system with strong Allee effect and two delays

2021 ◽  
Vol 26 (1) ◽  
pp. 72-92
Author(s):  
Yuying Liu ◽  
Junjie Wei
Keyword(s):  
Hopf Bifurcation ◽  
Allee Effect ◽  
Bifurcation Theorem ◽  
Predator Prey ◽  
Double Hopf Bifurcation ◽  
Prey System ◽  
Two Delays ◽  
Strong Allee Effect ◽  
The Stability ◽  
Two Parameters

In this paper, we consider a diffusive predator–prey system with strong Allee effect and two delays. First, we explore the stability region of the positive constant steady state by calculating the stability switching curves. Then we derive the Hopf and double Hopf bifurcation theorem via the crossing directions of the stability switching curves. Moreover, we calculate the normal forms near the double Hopf singularities by taking two delays as parameters. We carry out some numerical simulations for illustrating the theoretical results. Both theoretical analysis and numerical simulation show that the system near double Hopf singularity has rich dynamics, including stable spatially homogeneous and inhomogeneous periodic solutions. Finally, we evaluate the influence of two parameters on the existence of double Hopf bifurcation.

scholarly journals Bifurkasi Hopf pada Model Lotka-Volterra Orde-Fraksional dengan Efek Allee Aditif pada Predator

2020 ◽  
Vol 1 (1) ◽  
pp. 16-24
Author(s):  
Hasan S. Panigoro ◽  
Dian Savitri
Keyword(s):  
Hopf Bifurcation ◽  
Allee Effect ◽  
Equilibrium Points ◽  
Biological Condition ◽  
Existence Of Equilibrium ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Caputo Fractional Order Derivative ◽  
Theoretical Results ◽  
Extinction Point

This article aims to study the dynamics of a Lotka-Volterra predator-prey model with Allee effect in predator. According to the biological condition, the Caputo fractional-order derivative is chosen as its operator. The analysis is started by identifying the existence, uniqueness, and non-negativity of the solution. Furthermore, the existence of equilibrium points and their stability is investigated. It has shown that the model has two equilibrium points namely both populations extinction point which is always a saddle point, and a conditionally stable co-existence point, both locally and globally. One of the interesting phenomena is the occurrence of Hopf bifurcation driven by the order of derivative. Finally, the numerical simulations are given to validate previous theoretical results.

A Generalist Predator and the Planar Zero-Hopf Bifurcation

2017 ◽  
Vol 27 (03) ◽  
pp. 1750034 ◽  
Author(s):  
Luis Miguel Valenzuela ◽  
Manuel Falconi ◽  
Gamaliel Blé
Keyword(s):  
Hopf Bifurcation ◽  
Dynamical Systems ◽  
Periodic Orbits ◽  
Linear Approximation ◽  
Equilibrium Points ◽  
Generalist Predator ◽  
Predator Prey ◽  
Predator Prey Models ◽  
Bifurcation Process ◽  
Planar Dynamical Systems

A typical approach for searching periodic orbits of planar dynamical systems is through the Hopf bifurcation. In this work we present a family of predator–prey models with a generalist predator which does not exhibit a Hopf bifurcation, but a planar zero-Hopf bifurcation; that means, in the whole bifurcation process the eigenvalues of the linear approximation around the equilibrium points remain as pure imaginary. Similar models with a nongeneralist predator always possess a Hopf bifurcation.

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