Bogdanov–Takens bifurcation with codimension three of a predator–prey system suffering the additive Allee effect

2017 ◽  
Vol 10 (03) ◽  
pp. 1750044 ◽  
Author(s):  
Yanwei Liu ◽  
Zengrong Liu ◽  
Ruiqi Wang
Keyword(s):  
Allee Effect ◽  
Perturbation Parameter ◽  
Parameter Plane ◽  
Predator Prey ◽  
Numerical Examples ◽  
Codimension Two ◽  
Prey System ◽  
Existence Conditions ◽  
Parameter Dependent

In the present work, research efforts have focused on investigating codimension two and three Bogdanov–Takens bifurcations of a predator–prey system with additive Allee effect. According to the existence conditions of Bogdanov–Takens bifurcation, we give the associated generic unfolding, and derive the dynamical classification in the perturbation parameter plane using some smooth parameter-dependent transformations of coordinate. Moreover, some numerical examples and simulations are performed to complete and illustrate our results.

Allee effect makes possible patchy invasion in a predator-prey system

2002 ◽  
Vol 5 (3) ◽  
pp. 345-352 ◽  
Author(s):  
Sergei V. Petrovskii ◽  
Andrew Y. Morozov ◽  
Ezio Venturino
Keyword(s):  
Allee Effect ◽  
Predator Prey ◽  
Prey System

scholarly journals Neimark-Sacker-Turing Instability and Pattern Formation in a Spatiotemporal Discrete Predator-Prey System with Allee Effect

2018 ◽  
Vol 2018 ◽  
pp. 1-17
Author(s):  
Huayong Zhang ◽  
Xuebing Cong ◽  
Tousheng Huang ◽  
Shengnan Ma ◽  
Ge Pan
Keyword(s):  
Pattern Formation ◽  
Allee Effect ◽  
Initial Conditions ◽  
Turing Instability ◽  
Spatiotemporal Chaos ◽  
Closed Curves ◽  
Predator Prey ◽  
Prey System ◽  
Route To Chaos ◽  
The Rich

A spatiotemporal discrete predator-prey system with Allee effect is investigated to learn its Neimark-Sacker-Turing instability and pattern formation. Based on the occurrence of stable homogeneous stationary states, conditions for Neimark-Sacker bifurcation and Turing instability are determined. Numerical simulations reveal that Neimark-Sacker bifurcation triggers a route to chaos, with the emergence of invariant closed curves, periodic orbits, and chaotic attractors. The occurrence of Turing instability on these three typical dynamical behaviors leads to the formation of heterogeneous patterns. Under the effects of Neimark-Sacker-Turing instability, pattern evolution process is sensitive to tiny changes of initial conditions, suggesting the occurrence of spatiotemporal chaos. With application of deterministic initial conditions, transient symmetrical patterns are observed, demonstrating that ordered structures can exist in chaotic processes. Moreover, when local kinetics of the system goes further on the route to chaos, the speed of symmetry breaking becomes faster, leading to more fragmented and more disordered patterns at the same evolution time. The rich spatiotemporal complexity provides new comprehension on predator-prey coexistence in the ways of spatiotemporal chaos.

scholarly journals CODIMENSION TWO BIFURCATIONS IN A PREDATOR–PREY SYSTEM WITH GROUP DEFENSE

2001 ◽  
Vol 11 (08) ◽  
pp. 2123-2131 ◽  
Author(s):  
DONGMEI XIAO ◽  
SHIGUI RUAN
Keyword(s):  
Homoclinic Bifurcation ◽  
Qualitative Behavior ◽  
Homoclinic Loop ◽  
Saddle Node Bifurcation ◽  
Predator Prey ◽  
Codimension Two ◽  
Prey System ◽  
Group Defense ◽  
Nonmonotonic Functional Response ◽  
Parameter Values

In this paper we study the qualitative behavior of a predator–prey system with nonmonotonic functional response. The system undergoes a series of bifurcations including the saddle-node bifurcation, the supercritical Hopf bifurcation, and the homoclinic bifurcation. For different parameter values the system could have a limit cycle or a homoclinic loop, or exhibit the so-called "paradox of enrichment" phenomenon. In the generic case, the model has the bifurcation of cusp-type codimension two (i.e. the Bogdanov–Takens bifurcation) but no bifurcations of codimension three.

Predator–prey system with strong Allee effect in prey

2010 ◽  
Vol 62 (3) ◽  
pp. 291-331 ◽  
Author(s):  
Jinfeng Wang ◽  
Junping Shi ◽  
Junjie Wei
Keyword(s):  
Allee Effect ◽  
Predator Prey ◽  
Prey System ◽  
Strong Allee Effect
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