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.

scholarly journals Global Stability of Delayed Ecosystem via Impulsive Differential Inequality and Minimax Principle

2021 ◽  
Author(s):  
Ruofeng Rao
Keyword(s):  
Global Stability ◽  
Stationary Solution ◽  
Differential Inequality ◽  
Impulsive Control ◽  
Stationary Solutions ◽  
Delayed Feedback ◽  
Competition Model ◽  
Minimax Principle ◽  
Impulsive Differential Inequality

This paper reports applying Minimax principle and impulsive differential inequality to derive the existence of multiple stationary solutions and the global stability of a positive stationary solution for a delayed feedback Gilpin-Ayala competition model with impulsive disturbance. The conclusion obtained in this paper reduces the conservatism of the algorithm compared with the known literature, for the impulsive disturbance is not limited to impulsive control.

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.

scholarly journals Global Stability of Delayed Ecosystem via Impulsive Differential Inequality and Minimax Principle

2021 ◽  
Author(s):  
Ruofeng Rao
Keyword(s):  
Global Stability ◽  
Stationary Solution ◽  
Differential Inequality ◽  
Impulsive Control ◽  
Stationary Solutions ◽  
Delayed Feedback ◽  
Competition Model ◽  
Minimax Principle ◽  
Impulsive Differential Inequality

This paper reports applying Minimax principle and impulsive differential inequality to derive the existence of multiple stationary solutions and the global stability of a positive stationary solution for a delayed feedback Gilpin-Ayala competition model with impulsive disturbance. The conclusion obtained in this paper reduces the conservatism of the algorithm compared with the known literature, for the impulsive disturbance is not limited to impulsive control.

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 Result ◽  
Reaction Diffusion ◽  
Weight Coefficient ◽  
Coefficient Matrix ◽  
Stationary Solutions ◽  
Competition Model ◽  
Mountain Pass Lemma ◽  
Null Solution ◽  
Harmful Species

Firstly, the author do dynamic analysis for reaction-diffusion Gilpin-Ayala competition model with Dirichlet boundary value, involved in harmful species. Existence of multiple stationary solutions is verified by way of Mountain Pass lemma, and the local stability result of the null solution is obtained by employing linear approximation principle. Secondly, the author utilize variational methods and LMI technique to deduce the LMI-based global exponential stability criterion on the null solution which becomes the unique stationary solution of the ecosystem with delayed feedback under a reasonable boundedness assumption on population densities. Particularly, LMI criterion is involved in free weight coefficient matrix, which reduces the conservatism of the algorithm. In addition, a new impulse control stabilization criterion is also derived. Finally, two numerical examples show the effectiveness of the proposed methods. It is worth mentioning that the obtained stability criteria of null solution presented some useful hints on how to eliminate pests and bacteria.

scholarly journals Global Stability of Delayed Ecosystem via Impulsive Differential Inequality and Minimax Principle

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1943
Author(s):  
Ruofeng Rao
Keyword(s):  
Global Stability ◽  
Stationary Solution ◽  
Differential Inequality ◽  
Impulsive Control ◽  
Stationary Solutions ◽  
Delayed Feedback ◽  
Competition Model ◽  
Minimax Principle ◽  
Impulsive Differential Inequality

This paper reports applying Minimax principle and impulsive differential inequality to derive the existence of multiple stationary solutions and the global stability of a positive stationary solution for a delayed feedback Gilpin–Ayala competition model with impulsive disturbance. The conclusion obtained in this paper reduces the conservatism of the algorithm compared with the known literature, for the impulsive disturbance is not limited to impulsive control.

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.

scholarly journals Asymptotically stable stationary solutions to the reaction-diffusion equations

1993 ◽  
Vol 47 (2) ◽  
pp. 273-286 ◽  
Author(s):  
Arnold Dikansky
Keyword(s):  
Stationary Solution ◽  
Galerkin Approximation ◽  
Reaction Diffusion ◽  
Stationary Solutions ◽  
Reaction Diffusion System ◽  
Reaction Diffusion Equations ◽  
Diffusion System ◽  
Asymptotically Stable ◽  
Stable Stationary Solution ◽  
Lyapunov Second Method

We assume that there exists an asymptotically stable stationary solution of a Galerkin approximation for the reaction-diffusion system. It is shown that there exists a nearby stationary solution of the full reaction-diffusion system provided the order of the Galerkin approximation is high enough. The Lyapunov second method is used to prove the asymptotic stability of the stationary solution.

scholarly journals Stability Analysis of Nontrivial Stationary Solution and Constant Equilibrium Point of Reaction-Diffusion Neural Networks With Time Delays

2020 ◽  
Author(s):  
Ruofeng Rao
Keyword(s):  
Neural Networks ◽  
Equilibrium Point ◽  
Stationary Solution ◽  
Phase Plane ◽  
Reaction Diffusion ◽  
Stationary Solutions ◽  
Diffusion Effect ◽  
Delayed Neural Networks ◽  
Constant Equilibrium ◽  
The Stability

Firstly, the existence of asymptotically stable nontrivial stationary solution is derived by the comprehensive applications of Lyapunov-Razumikhin technique, design of state-dependent switching laws, a fixed point theorem, variational methods, and construction of compact operators on a convex set. This new theorem shows that the diffusion is a double-edged sword to the stability, refuting the views in previous literature that the greater the diffusion effect, the more stable the system will be. Next, a series of new theorems are presented one by one, which illustrates that the globally asymptotical stability of ordinary differential equations model for delayed neural networks may be locally stable in actual operation due to the inevitable diffusion. Besides, the non-zero constant equilibrium point is pointed out to be not the solution of delayed reaction diffusion system so that the stability of the non-zero constant equilibrium point of reaction diffusion system must lead to a contradiction. That is, non-zero constant equilibrium points are not in the phase plane of dynamic system. In addition, new theorems are further presented to prove that the small diffusion effect will cause the essential change of the phase plane structure of the dynamic behavior of the delayed neural networks, and thereby one equilibrium solution may become several stationary solutions, even infinitely many positive stationary solutions. Finally, a numerical example illustrates the feasibility of the proposed methods.

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