Steady states of Lotka-Volterra competition models with nonlinear cross-diffusion

2021 ◽  
Vol 292 ◽  
pp. 247-286
Author(s):  
Changfeng Liu ◽  
Shangjiang Guo
Keyword(s):  
Steady States ◽  
Cross Diffusion ◽  
Competition Models

scholarly journals Stability of patterns and of constant steady states for a cross-diffusion system

2016 ◽  
Vol 293 ◽  
pp. 208-216 ◽  
Author(s):  
G. Svantnerné Sebestyén ◽  
István Faragó ◽  
Róbert Horváth ◽  
R. Kersner ◽  
M. Klincsik
Keyword(s):  
Steady States ◽  
Diffusion System ◽  
Cross Diffusion

scholarly journals Positive steady states of a ratio-dependent predator-prey system with cross-diffusion

2019 ◽  
Vol 16 (6) ◽  
pp. 6753-6768
Author(s):  
Xiaoling Li ◽  
◽  
Guangping Hu ◽  
Xianpei Li ◽  
Zhaosheng Feng ◽  
...  
Keyword(s):  
Steady States ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Prey System ◽  
Positive Steady States ◽  
Ratio Dependent

scholarly journals A meeting point of entropy and bifurcations in cross-diffusion herding

2016 ◽  
Vol 28 (2) ◽  
pp. 317-356 ◽  
Author(s):  
ANSGAR JÜNGEL ◽  
CHRISTIAN KUEHN ◽  
LARA TRUSSARDI
Keyword(s):  
Gradient Flow ◽  
Stationary Solutions ◽  
Steady States ◽  
Numerical Continuation ◽  
Continuation Methods ◽  
System Modelling ◽  
Local Bifurcation ◽  
Diffusion Term ◽  
Cross Diffusion ◽  
The Cross

A cross-diffusion system modelling the information herding of individuals is analysed in a bounded domain with no-flux boundary conditions. The variables are the species' density and an influence function which modifies the information state of the individuals. The cross-diffusion term may stabilize or destabilize the system. Furthermore, it allows for a formal gradient-flow or entropy structure. Exploiting this structure, the global-in-time existence of weak solutions and the exponential decay to the constant steady state is proved in certain parameter regimes. This approach does not extend to all parameters. We investigate local bifurcations from homogeneous steady states analytically to determine whether this defines the validity boundary. This analysis shows that generically there is a gap in the parameter regime between the entropy approach validity and the first local bifurcation. Next, we use numerical continuation methods to track the bifurcating non-homogeneous steady states globally and to determine non-trivial stationary solutions related to herding behaviour. In summary, we find that the main boundaries in the parameter regime are given by the first local bifurcation point, the degeneracy of the diffusion matrix and a certain entropy decay validity condition. We study several parameter limits analytically as well as numerically, with a focus on the role of changing a linear damping parameter as well as a parameter controlling the cross-diffusion. We suggest that our paradigm of comparing bifurcation-generated obstructions to the parameter validity of global-functional methods could also be of relevance for many other models beyond the one studied here.

scholarly journals Stationary Patterns of a Cross-Diffusion Epidemic Model

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Yongli Cai ◽  
Dongxuan Chi ◽  
Wenbin Liu ◽  
Weiming Wang
Keyword(s):  
Global Stability ◽  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Complex Dynamics ◽  
Reproduction Number ◽  
Steady States ◽  
Nonnegative Constant ◽  
Cross Diffusion ◽  
Existence And Nonexistence ◽  
The Basic Reproduction Number

We investigate the complex dynamics of cross-diffusionSIepidemic model. We first give the conditions of the local and global stability of the nonnegative constant steady states, which indicates that the basic reproduction number determines whether there is an endemic outbreak or not. Furthermore, we prove the existence and nonexistence of the positive nonconstant steady states, which guarantees the existence of the stationary patterns.

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