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.

scholarly journals Hopf bifurcation in a reaction-diffusive-advection two-species competition model with one delay

2021 ◽  
pp. 1-24
Author(s):  
Qiong Meng ◽  
Guirong Liu ◽  
Zhen Jin
Keyword(s):  
Steady State ◽  
Hopf Bifurcation ◽  
Competition Model ◽  
Dirichlet Boundary Conditions ◽  
Species Competition ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Existence And Multiplicity ◽  
The Stability ◽  
Steady State Solutions

In this paper, we investigate a reaction-diffusive-advection two-species competition model with one delay and Dirichlet boundary conditions. The existence and multiplicity of spatially non-homogeneous steady-state solutions are obtained. The stability of spatially nonhomogeneous steady-state solutions and the existence of Hopf bifurcation with the changes of the time delay are obtained by analyzing the distribution of eigenvalues of the infinitesimal generator associated with the linearized system. By the normal form theory and the center manifold reduction, the stability and bifurcation direction of Hopf bifurcating periodic orbits are derived. Finally, numerical simulations are given to illustrate the theoretical results.

Global higher bifurcations in coupled systems of nonlinear eigenvalue problems

1987 ◽  
Vol 106 (1-2) ◽  
pp. 113-120 ◽  
Author(s):  
Robert Stephen Cantrell
Keyword(s):  
Steady State ◽  
Eigenvalue Problems ◽  
Coupled Systems ◽  
Volterra Model ◽  
Nonlinear Eigenvalue Problems ◽  
Volterra Models ◽  
Bifurcation Phenomena ◽  
Higher Dimensional ◽  
Secondary Bifurcation ◽  
Nonlinear Eigenvalue

SynopsisCoexistent steady-state solutions to a Lotka–Volterra model for two freely-dispersing competing species have been shown by several authors to arise as global secondary bifurcation phenomena. In this paper we establish conditions for the existence of global higher dimensional n-ary bifurcation in general systems of multiparameter nonlinear eigenvalue problems which preserve the coupling structure of diffusive steady-state Lotka–Volterra models. In establishing our result, we mainly employ the recently-developed multidimensional global multiparameter theory of Alexander–Antman. Conditions for ternary steady-state bifurcation in the three species diffusive competition model are given as an application of the result.

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