scholarly journals Structure and Stability of Steady State Bifurcation in a Cannibalism Model with Cross-Diffusion

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Meijun Chen ◽  
Shengmao Fu
Keyword(s):  
Steady State ◽  
Asymptotic Analysis ◽  
Spatial Patterns ◽  
Local Structure ◽  
Small Amplitude ◽  
Bifurcation Theorem ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Steady State Bifurcation ◽  
The Stability

This paper deals with spatial patterns of a predator-prey crossdiffusion model with cannibalism. By applying the asymptotic analysis and Rabinowitz bifurcation theorem, we consider the local structure of steady state to the model and determine an explicit formula of the nonconstant steady state. Furthermore, the criteria of the stability/instability for the steady state with small amplitude are established.

scholarly journals Pattern Formation in Predator-Prey Model with Delay and Cross Diffusion

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Xinze Lian ◽  
Shuling Yan ◽  
Hailing Wang
Keyword(s):  
Time Delay ◽  
Pattern Formation ◽  
Turing Instability ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Prey Model ◽  
Diffusion Controlled ◽  
The Stability ◽  
Self Replication

We consider the effect of time delay and cross diffusion on the dynamics of a modified Leslie-Gower predator-prey model incorporating a prey refuge. Based on the stability analysis, we demonstrate that delayed feedback may generate Hopf and Turing instability under some conditions, resulting in spatial patterns. One of the most interesting findings is that the model exhibits complex pattern replication: the model dynamics exhibits a delay and diffusion controlled formation growth not only to spots, stripes, and holes, but also to spiral pattern self-replication. The results indicate that time delay and cross diffusion play important roles in pattern formation.

Rayleigh–Bénard convection and pattern formation in magnetohydrodynamics

1998 ◽  
Vol 60 (3) ◽  
pp. 529-539 ◽  
Author(s):  
RENU BAJAJ ◽  
S. K. MALIK
Keyword(s):  
Magnetic Field ◽  
Steady State ◽  
Thermal Instability ◽  
Electrically Conducting ◽  
Electrically Conducting Fluid ◽  
Steady State Bifurcation ◽  
The Stability ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard ◽  
The Magnetic Field

A nonlinear thermal instability in a layer of electrically conducting fluid in the presence of a magnetic field is discussed. Steady-state bifurcation results in the formation of patterns: rolls, squares and hexagons. The stability of various patterns is also investigated. It is found that in the absence of a magnetic field only rolls are stable, but when the magnetic field strength exceeds a certain finite value, squares and hexagons also become stable.

Steady-State Bifurcation for a Biological Depletion Model

2016 ◽  
Vol 26 (04) ◽  
pp. 1650066 ◽  
Author(s):  
Yan’e Wang ◽  
Jianhua Wu ◽  
Yunfeng Jia
Keyword(s):  
Steady State ◽  
Bifurcation Theory ◽  
Steady States ◽  
Two Dimensional ◽  
Implicit Function ◽  
Flux Boundary Condition ◽  
One Dimensional ◽  
Steady State Bifurcation ◽  
The Stability ◽  
Theoretical Results

A two-species biological depletion model in a bounded domain is investigated in which one species is a substrate and the other is an activator. Firstly, under the no-flux boundary condition, the asymptotic stability of constant steady-states is discussed. Secondly, by viewing the feed rate of the substrate as a parameter, the steady-state bifurcations from constant steady-states are analyzed both in one-dimensional kernel case and in two-dimensional kernel case. Finally, numerical simulations are presented to illustrate our theoretical results. The main tools adopted here include the stability theory, the bifurcation theory, the techniques of space decomposition and the implicit function theorem.

Dynamic Simulation of Mixing-Limited Pattern Formation in Homogeneous Autocatalytic Reactions

2008 ◽  
Vol 3 (2) ◽  
Author(s):  
Ankur Gupta ◽  
Saikat Chakraborty
Keyword(s):  
Steady State ◽  
Pattern Formation ◽  
Three Dimensional ◽  
Diffusion Reaction ◽  
Dynamic Simulations ◽  
Transverse Mixing ◽  
Autocatalytic Reactions ◽  
Steady State Bifurcation ◽  
The Stability ◽  
Parametric Analyses

Interaction between transport and reaction generates a variety of complex spatio-temporal patterns in chemical reactors. These patterned states, which are typically initiated by autocatalytic effects and sustained by differences in diffusion/local mixing rates, often cause undesired effects in the reactor. In this work, we analyze the dynamic evolution of mixing-limited spatial pattern formation in fast, homogeneous autocatalytic reactions occurring in isothermal tubular reactors using two-dimensional (2-D) convection-diffusion-reaction (CDR) models that are obtained through rigorous spatial averaging of the three-dimensional (3-D) CDR model using Liapunov-Schmidt technique of bifurcation theory. We use the spatially-averaged 2-D CDR model (and its "regularized" form) to perform steady-state bifurcation analysis that captures the region of multiple solutions, and we analyze the stability of these multiple steady states to transverse perturbations using linear stability analysis. Parametric analyses of the steady-state bifurcation diagrams and stability boundaries show that when transverse mixing is significantly slower than the rate of autocatalytic reaction, mixing-limited patterns emerge from the unstable middle branch that connects the ignition and extinction points of an S-shaped bifurcation curve. Our dynamic simulations show the emergence of three different types of spatial patterns namely, Band, Anti-phase and Target, depending on the nature of transverse perturbation. The temporal evolution of these patterns consists of rapid intensification of the concentration-segregation process (especially when transverse mixing is much slower than reaction) followed by slow diffusion-mediated return to symmetry that occurs at time scales much larger than the reactor residence time. Our parametric analysis of the dynamics reveals that while larger Péclet numbers (both axial and transverse) increase the stability and decay time of the patterned states, larger Damköhler numbers lead to faster ignition resulting in the opposite effect.

scholarly journals Turing Patterns in a Predator-Prey System with Self-Diffusion

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Hongwei Yin ◽  
Xiaoyong Xiao ◽  
Xiaoqing Wen
Keyword(s):  
Spatial Patterns ◽  
Nonlinear Diffusion ◽  
Turing Instability ◽  
Turing Patterns ◽  
Predator Prey ◽  
Self Diffusion ◽  
Cross Diffusion ◽  
Prey System ◽  
Diffusion Parameters ◽  
Normal Diffusion

For a predator-prey system, cross-diffusion has been confirmed to emerge Turing patterns. However, in the real world, the tendency for prey and predators moving along the direction of lower density of their own species, called self-diffusion, should be considered. For this, we investigate Turing instability for a predator-prey system with nonlinear diffusion terms including the normal diffusion, cross-diffusion, and self-diffusion. A sufficient condition of Turing instability for this system is obtained by analyzing the linear stability of spatial homogeneous equilibrium state of this model. A series of numerical simulations reveal Turing parameter regions of the interaction of diffusion parameters. According to these regions, we further demonstrate dispersion relations and spatial patterns. Our results indicate that self-diffusion plays an important role in the spatial patterns.

Bifurcations and Pattern Formation in a Predator–Prey Model

2018 ◽  
Vol 28 (11) ◽  
pp. 1850140 ◽  
Author(s):  
Yongli Cai ◽  
Zhanji Gui ◽  
Xuebing Zhang ◽  
Hongbo Shi ◽  
Weiming Wang
Keyword(s):  
Steady State ◽  
Hopf Bifurcation ◽  
Pattern Formation ◽  
Global Bifurcation ◽  
Periodic Pattern ◽  
Spatiotemporal Pattern ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Steady State Bifurcation

In this paper, we investigate the spatiotemporal dynamics of a Leslie–Gower predator–prey model incorporating a prey refuge subject to the Neumann boundary conditions. We mainly consider Hopf bifurcation and steady-state bifurcation which bifurcate from the constant positive steady-state of the model. In the case of Hopf bifurcation, by the center manifold theory and the normal form method, we establish the bifurcation direction and stability of bifurcating periodic solutions; in the case of steady-state bifurcation, by the local and global bifurcation theories, we prove the existence of the steady-state bifurcation, and find that there are two typical bifurcations, Turing bifurcation and Turing–Hopf bifurcation. Via numerical simulations, we find that the model exhibits not only stationary Turing pattern induced by diffusion which is dependent on space and independent of time, but also temporal periodic pattern induced by Hopf bifurcation which is dependent on time and independent of space, and spatiotemporal pattern induced by Turing–Hopf bifurcation which is dependent on both time and space. These results may enrich the pattern formation in the predator–prey model.

scholarly journals Global Hopf Bifurcation on Two-Delays Leslie-Gower Predator-Prey System with a Prey Refuge

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Qingsong Liu ◽  
Yiping Lin ◽  
Jingnan Cao
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Functional Differential Equations ◽  
Global Hopf Bifurcation ◽  
Bifurcation Theorem ◽  
Predator Prey ◽  
Prey System ◽  
Two Delays ◽  
Functional Differential ◽  
The Stability

A modified Leslie-Gower predator-prey system with two delays is investigated. By choosingτ1andτ2as bifurcation parameters, we show that the Hopf bifurcations occur when time delay crosses some critical values. Moreover, we derive the equation describing the flow on the center manifold; then we give the formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions. Numerical simulations are carried out to illustrate the theoretical results and chaotic behaviors are observed. Finally, using a global Hopf bifurcation theorem for functional differential equations, we show the global existence of the periodic solutions.

SPATIAL PATTERN IN A PREDATOR-PREY SYSTEM WITH BOTH SELF- AND CROSS-DIFFUSION

2009 ◽  
Vol 20 (01) ◽  
pp. 71-84 ◽  
Author(s):  
GUI-QUAN SUN ◽  
ZHEN JIN ◽  
YI-GUO ZHAO ◽  
QUAN-XING LIU ◽  
LI LI
Keyword(s):  
Spatial Patterns ◽  
Temporal Dynamics ◽  
Spatial Scales ◽  
Natural Populations ◽  
Spatial Dynamics ◽  
Density Variation ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Prey System ◽  
Dynamics Of Population

The vast majority of models for spatial dynamics of natural populations assume a homogeneous physical environment. However, in practice, dispersing organisms may encounter landscape features that significantly inhibit their movement. And spatial patterns are ubiquitous in nature, which can modify the temporal dynamics and stability properties of population densities at a range of spatial scales. Thus, in this paper, a predator-prey system with Michaelis-Menten-type functional response and self- and cross-diffusion is investigated. Based on the mathematical analysis, we obtain the condition of the emergence of spatial patterns through diffusion instability, i.e., Turing pattern. A series of numerical simulations reveal that the typical dynamics of population density variation is the formation of isolated groups, i.e., stripe-like or spotted or coexistence of both. The obtained results show that the interaction of self-diffusion and cross-diffusion plays an important role on the pattern formation of the predator-prey system.

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