Nonexistence of coexisting steady‐state solutions of Dirichlet problem for a cross‐diffusion model

2018 ◽  
Vol 42 (1) ◽  
pp. 346-353
Author(s):  
Ningning Zhu ◽  
De Tang
Keyword(s):  
Steady State ◽  
Dirichlet Problem ◽  
Diffusion Model ◽  
Cross Diffusion ◽  
Steady State Solutions

Bifurcation Phenomena in a Lotka–Volterra Model with Cross-Diffusion and Delay Effect

2017 ◽  
Vol 27 (07) ◽  
pp. 1750105 ◽  
Author(s):  
Shuling Yan ◽  
Shangjiang Guo
Keyword(s):  
Steady State ◽  
Hopf Bifurcation ◽  
Periodic Orbits ◽  
Volterra Model ◽  
Delay Effect ◽  
Cross Diffusion ◽  
Bifurcation Phenomena ◽  
Bifurcation Direction ◽  
Steady State Solutions

This paper focuses on a Lotka–Volterra model with delay and cross-diffusion. By using Lyapunov–Schmidt reduction, we investigate the existence, multiplicity, stability and Hopf bifurcation of spatially nonhomogeneous steady-state solutions. Furthermore, we obtain some criteria to determine the bifurcation direction and stability of Hopf bifurcating periodic orbits by using Lyapunov–Schmidt reduction.

Bifurcation of Reaction Cross-Diffusion Systems

2017 ◽  
Vol 27 (04) ◽  
pp. 1750049 ◽  
Author(s):  
Rong Zou ◽  
Shangjiang Guo
Keyword(s):  
Steady State ◽  
Neumann Boundary ◽  
Reaction Cross ◽  
Homogeneous Steady State ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Existence And Multiplicity ◽  
Steady State Solutions

This paper is devoted to a reaction cross-diffusion system under Neumann boundary conditions. Firstly, the existence and multiplicity of spatially nonhomogeneous/homogeneous steady-state solutions are investigated by means of Lyapunov–Schmidt reduction. Next, the linear stability and Hopf bifurcations of homogeneous steady-state solutions are described in detail. In particular, the Hopf bifurcation direction and the stability of bifurcating time-periodic solutions are determined by using center manifold reduction and normal form theory. Finally, some of the main results are illustrated by an application to a predator–prey model with Allee effect and one-dimensional spatial domain [Formula: see text].

Steady-State Problem of an S-K-T Competition Model with Spatially Degenerate Coefficients

2021 ◽  
Vol 31 (11) ◽  
pp. 2150165
Author(s):  
Hao Zhou ◽  
Yu-Xia Wang
Keyword(s):  
Steady State ◽  
Global Bifurcation ◽  
Steady State Solution ◽  
Competition Model ◽  
State Solution ◽  
State Problem ◽  
Cross Diffusion ◽  
Positive Steady State ◽  
Steady State Problem ◽  
Steady State Solutions

In this paper, we study the steady-state problem of an S-K-T competition model with a spatially degenerate intraspecific competition coefficient. First, the global bifurcation continuum of positive steady-state solutions from its semitrivial steady-state solution is given, which depends on the spatial heterogeneity and cross-diffusion. Second, two limiting systems are derived as the cross-diffusion coefficient tends to infinity. Moreover, we demonstrate the existence of positive steady-state solutions near the two limiting systems, and show which one of the limiting systems characterizes the positive steady-state solution.

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