Zero singularities of codimension two in a delayed predator-prey diffusion system

2017 ◽  
Vol 227 ◽  
pp. 10-17 ◽  
Author(s):  
Jinling Wang ◽  
Jinling Liang ◽  
Yurong Liu ◽  
Jin-Liang Wang
Keyword(s):  
Diffusion System ◽  
Predator Prey ◽  
Codimension Two

scholarly journals Stabilization for a Periodic Predator-Prey System

2007 ◽  
Vol 2007 ◽  
pp. 1-17
Author(s):  
Sebastian Aniţa ◽  
Carmen Oana Tarniceriu
Keyword(s):  
Internal Control ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
System Modelling ◽  
Predator Prey ◽  
Periodic Environment ◽  
Prey System

A reaction-diffusion system modelling a predator-prey system in a periodic environment is considered. We are concerned in stabilization to zero of one of the components of the solution, via an internal control acting on a small subdomain, and in the preservation of the nonnegativity of both components.

Coexistence states in a cross-diffusion system of a predator–prey model with predator satiation term

2018 ◽  
Vol 28 (11) ◽  
pp. 2131-2159 ◽  
Author(s):  
Willian Cintra ◽  
Cristian Morales-Rodrigo ◽  
Antonio Suárez
Keyword(s):  
A Priori ◽  
Diffusion System ◽  
Predator Satiation ◽  
Linear Diffusion ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Prey Model ◽  
Coexistence States ◽  
Predator Model

In this paper, we study the existence and non-existence of coexistence states for a cross-diffusion system arising from a prey–predator model with a predator satiation term. We use mainly bifurcation methods and a priori bounds to obtain our results. This leads us to study the coexistence region and compare our results with the classical linear diffusion predator–prey model. Our results suggest that when there is no abundance of prey, the predator needs to be a good hunter to survive.

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.

Bifurcation Analysis and Spatiotemporal Patterns in a Diffusive Predator–Prey Model

2014 ◽  
Vol 24 (06) ◽  
pp. 1450081 ◽  
Author(s):  
Guangping Hu ◽  
Xiaoling Li ◽  
Shiping Lu ◽  
Yuepeng Wang
Keyword(s):  
Pattern Formation ◽  
Spiral Wave ◽  
Reaction Diffusion ◽  
Spatiotemporal Chaos ◽  
Diffusion System ◽  
Target Pattern ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
The Impact

In this paper, we consider a species predator–prey model given a reaction–diffusion system. It incorporates the Holling type II functional response and a quadratic intra-predator interaction term. We focus on the qualitative analysis, bifurcation mechanisms and pattern formation. We present the results of numerical experiments in two space dimensions and illustrate the impact of the diffusion on the Turing pattern formation. For this diffusion system, we also observe non-Turing structures such as spiral wave, target pattern and spatiotemporal chaos resulting from the time evolution of these structures.

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.

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