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 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 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 Existence and Local Stability of Stationary Solutions for Nonlinear Gilpin Ayala Competition Model with Dirichlet Boundary Value

2020 ◽  
Author(s):  
Ruofeng Rao
Keyword(s):  
Stability Criterion ◽  
Local Stability ◽  
Dirichlet Boundary ◽  
Stationary Solutions ◽  
Competition Model ◽  
Mountain Pass Lemma ◽  
Species Competition ◽  
Diffusion Behavior ◽  
Population Extinction ◽  
Lyapunov Function Method

In this paper, the existence of two nontrivial stationary solutions for the nonlinear Gilpin Ayala two species competition model is given by using the mountain pass lemma, and the local stability criterion of the trivial solution is given by using Lyapunov function method. Based on the local stability criterion, we give some suggestions on how to avoid the population extinction. This is, when the population is on the verge of extinction, we should try our best to avoid the diffusion behavior and reduce the diffusion coefficient, otherwise the species are easy to go extinct. Numerical example shows the effectiveness of the proposed method.

scholarly journals Stability and Stabilization of Ecosystem for Epidemic Virus Transmission Under Neumann Boundary Value Via Impulse Control

2021 ◽  
Author(s):  
Ruofeng Rao
Keyword(s):  
Stability Criterion ◽  
Impulse Control ◽  
Boundary Value ◽  
Virus Transmission ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Prevention Measures ◽  
Unique Existence ◽  
Globally Exponential Stability ◽  
The Stability

In this paper, by using the variational method, a sufficient condition for the unique existence of the stationary solution of the reaction-diffusion ecosystem is obtained, which directly leads to the global asymptotic stability of the unique equilibrium point. Moreover, delayed feedback ecosystem with reaction-diffusion item is considered, and utilizing impulse control results in the globally exponential stability criterion of the delayed ecosystem. It is worth mentioning that the Neumann zero-boundary value that the infected and the susceptible people or animals should be controlled in the epidemic prevention area and not allowed to cross the border, which is a good simulation of the actual situation of epidemic prevention. And numerical examples illuminate the effectiveness of impulse control, which has a certain enlightening effect on the actual epidemic prevention work . That is, in the face of the epidemic situation, taking a certain frequency of positive and effective epidemic prevention measures is conducive to the stability and control of the epidemic situation. Particularly, the newly-obtained theorems quantifies this feasible step. Besides, utilizing Laplacian semigroup derives the $p$th moment stability criterion for the impulsive ecosystem.

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.

scholarly journals The influence of density in population dynamics with strong and weak Allee effect

2021 ◽  
Vol 29 (1) ◽  
Author(s):  
Kamrun Nahar Keya ◽  
Md. Kamrujjaman ◽  
Md. Shafiqul Islam
Keyword(s):  
Population Dynamics ◽  
Allee Effect ◽  
Global Bifurcation ◽  
Reaction Diffusion ◽  
Logistic Growth ◽  
Allee Effects ◽  
Reaction Diffusion Model ◽  
Positive Steady States ◽  
The Stability ◽  
The Impact

AbstractIn this paper, we consider a reaction–diffusion model in population dynamics and study the impact of different types of Allee effects with logistic growth in the heterogeneous closed region. For strong Allee effects, usually, species unconditionally die out and an extinction-survival situation occurs when the effect is weak according to the resource and sparse functions. In particular, we study the impact of the multiplicative Allee effect in classical diffusion when the sparsity is either positive or negative. Negative sparsity implies a weak Allee effect, and the population survives in some domain and diverges otherwise. Positive sparsity gives a strong Allee effect, and the population extinct without any condition. The influence of Allee effects on the existence and persistence of positive steady states as well as global bifurcation diagrams is presented. The method of sub-super solutions is used for analyzing equations. The stability conditions and the region of positive solutions (multiple solutions may exist) are presented. When the diffusion is absent, we consider the model with and without harvesting, which are initial value problems (IVPs) and study the local stability analysis and present bifurcation analysis. We present a number of numerical examples to verify analytical results.

scholarly journals Bifurcation Analysis in a Diffusive Mussel-Algae Model with Delay

2019 ◽  
Vol 29 (11) ◽  
pp. 1950144 ◽  
Author(s):  
Zuolin Shen ◽  
Junjie Wei
Keyword(s):  
Hopf Bifurcation ◽  
Functional Differential Equations ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Delayed Reaction ◽  
Center Manifold Reduction ◽  
Existence Of Positive Solutions ◽  
Boundary Equilibrium ◽  
Functional Differential ◽  
The Stability

In this paper, we consider the dynamics of a delayed reaction–diffusion mussel-algae system subject to Neumann boundary conditions. When the delay is zero, we show the existence of positive solutions and the global stability of the boundary equilibrium. When the delay is not zero, we obtain the stability of the positive constant steady state and the existence of Hopf bifurcation by analyzing the distribution of characteristic values. By using the theory of normal form and center manifold reduction for partial functional differential equations, we derive an algorithm that determines the direction of Hopf bifurcation and the stability of bifurcating periodic solutions. Finally, some numerical simulations are carried out to support our theoretical results.

scholarly journals The Concepts of Well-Posedness and Stability in Different Function Spaces for the 1D Linearized Euler Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Stefan Balint ◽  
Agneta M. Balint
Keyword(s):  
Euler Equation ◽  
Function Space ◽  
Euler Equations ◽  
Small Perturbation ◽  
Function Spaces ◽  
Well Posedness ◽  
Small Solution ◽  
Null Solution ◽  
Linearized Euler Equations ◽  
The Stability

This paper considers the stability of constant solutions to the 1D Euler equation. The idea is to investigate the effect of different function spaces on the well-posedness and stability of the null solution of the 1D linearized Euler equations. It is shown that the mathematical tools and results depend on the meaning of the concepts “perturbation,” “small perturbation,” “solution of the propagation problem,” and “small solution, that is, solution close to zero,” which are specific for each function space.

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