Bifurcation and stability of a two-species reaction–diffusion–advection competition model

2021 ◽  
Vol 59 ◽  
pp. 103241
Author(s):  
Li Ma ◽  
Shangjiang Guo
Keyword(s):  
Reaction Diffusion ◽  
Competition Model ◽  
Bifurcation And Stability

scholarly journals Unique Existence of Globally Asymptotical Input-to-State Stability of Positive Stationary Solution for Impulsive Gilpin-Ayala Competition Model with Diffusion and Delayed Feedback under Dirichlet Zero Boundary Value

2020 ◽  
Author(s):  
Ruofeng Rao
Keyword(s):  
Fixed Point ◽  
Stationary Solution ◽  
Function Method ◽  
Boundary Value ◽  
Reaction Diffusion ◽  
Delayed Feedback ◽  
Competition Model ◽  
Numerical Examples ◽  
Unique Existence ◽  
Lyapunov Function Method

By partly generalizing the Lipschitz condition of existing results to the generalized Lipschitz one, the author utilizes a fixed point theorem, variational method and Lyapunov function method to derive the unique existence of globally asymptotical input-to-state stability of positive stationary solution for Gilpin-Ayala competition model with diffusion and delayed feedback under Dirichlet zero boundary value. Remarkably, it is the first paper to derive the unique existence of the stationary solution of reaction-diffusion Gilpin-Ayala competition model, which is globally asymptotical input-to-state stability. And numerical examples illuminate the effectiveness and feasibility of the proposed methods.

A reaction-diffusion-advection competition model with a free boundary

2021 ◽  
Vol 65 (3) ◽  
pp. 25-37
Keyword(s):  
Free Boundary ◽  
A Priori Estimates ◽  
A Priori ◽  
Reaction Diffusion ◽  
Competition Model ◽  
Baseline Data ◽  
The Other ◽  
Quasilinear System ◽  
Priori Estimates ◽  
Competitive Diffusion

In this paper, we study a competitive diffusion quasilinear system with a free boundary. First, we investigate the mathematical questions of the problem. A priori estimates of Schauder type are established, which are necessary for the solvability of the problem. One of two competing species is an invader, which initially exists on a certain sub-interval. The other is initially distributed throughout the area under consideration. Examining the influence of baseline data on the success or failure of the first invasion. We conclude that there is a dichotomy of spread and extinction and give precise criteria for spread and extinction in this model.

scholarly journals Local and Global Stability Analysis for Gilpin-Ayala Competition Model Involved in Harmful Species Via LMI Approach and Variational Methods

2021 ◽  
Author(s):  
Ruofeng Rao
Keyword(s):  
Variational Methods ◽  
Stability Criterion ◽  
Reaction Diffusion ◽  
Stationary Solutions ◽  
Competition Model ◽  
Mountain Pass Lemma ◽  
Lmi Approach ◽  
Null Solution ◽  
The Stability ◽  
Harmful Species

In this paper, stability of reaction-diffusion Gilpin-Ayala competition model with Dirichlet boundary value, involved in harmful species, was investigated. Employing Mountain Pass Lemma and linear approximation principle results in the local stability criterion of the null solution of the ecosystem which owns at least three stationary solutions. On the other hand, globally asymptotical stability criterion for the null solution of the ecosystem was derived by variational methods and LMI approach. It is worth mentioning that the stability criteria of null solution presented some useful hints on how to eliminate pests and bacteria. Finally, two numerical examples show the effectiveness of the proposed methods.

Numerical bifurcation and stability analysis of solitary pulses in an excitable reaction—diffusion medium

1999 ◽  
Vol 170 (3-4) ◽  
pp. 253-275 ◽  
Author(s):  
J. Krishnan ◽  
Ioannis G. Kevrekidis ◽  
Michael Or-Guil ◽  
Martin G. Zimmerman ◽  
Bär Markus
Keyword(s):  
Stability Analysis ◽  
Reaction Diffusion ◽  
Numerical Bifurcation ◽  
Bifurcation And Stability

Traveling waves and spreading properties for a reaction-diffusion competition model with seasonal succession*

Nonlinearity ◽  
2021 ◽  
Vol 35 (1) ◽  
pp. 134-169
Author(s):  
Mingxin Wang ◽  
Qianying Zhang ◽  
Xiao-Qiang Zhao
Keyword(s):  
Upper And Lower Solutions ◽  
Seasonal Succession ◽  
Traveling Wave Solutions ◽  
Reaction Diffusion ◽  
Positive Periodic Solution ◽  
Competition Model ◽  
Competition Condition ◽  
Whole Space ◽  
Comparison Arguments ◽  
Globally Stable

Abstract In this paper, we investigate the propagation dynamics of a reaction–diffusion competition model with seasonal succession in the whole space. Under the weak competition condition, the corresponding kinetic system admits a globally stable positive periodic solution ( u ^ ( t ) , v ^ ( t ) ) . By the method of upper and lower solutions and the Schauder fixed point theorem, we first obtain the existence and nonexistence of traveling wave solutions connecting (0, 0) to ( u ^ ( t ) , v ^ ( t ) ) . Then we use the comparison arguments to establish the spreading properties for a large class of solutions.

scholarly journals Multiple Stationary Solutions and Global Stabilization of Reaction-diffusion Gilpin-Ayala Competition Model under Event-triggered Impulsive Control

2021 ◽  
Author(s):  
Ruofeng Rao
Keyword(s):  
Variational Methods ◽  
Stationary Solution ◽  
Impulsive Control ◽  
Reaction Diffusion ◽  
Stationary Solutions ◽  
Competition Model ◽  
Global Stabilization ◽  
Impulsive Event ◽  
The Cost ◽  
Event Triggered

In this paper, the author utilizes Saddle Theorem and variational methods to deduce existence of at least six stationary solutions for reaction-diffusion Gilpin-Ayala competition model (RDGACM). To obtain the global stabilization of the positive stationary solution of the RDGACM, the author designs a suitable impulsive event triggered mechanism (IETM) to derive the global exponential stability of the the positive stationary solution. It is worth mentioning that the new mechanism can exclude Zeno behavior and effectively reduce the cost of impulse control through event triggering mechanism. Besides, compared with existing literature, the restrictions on the parameters of the RDGACM are relaxed so that the methods used in existing literature can not be applied to the relaxed case of this paper, and so the author makes comprehensive use of Saddle Theorem, orthogonal decomposition of Sobolev space $H_0^1(\Omega)$ and variational methods to overcome the mathematical difficulty. Numerical examples show the effectiveness of the methods proposed in this paper.

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