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 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 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.

GLOBAL EXPONENTIAL STABILITY OF CHUA'S REACTION–DIFFUSION CNN

2009 ◽  
Vol 19 (10) ◽  
pp. 3397-3406
Author(s):  
YUNQUAN KE ◽  
CHUNFANG MIAO
Keyword(s):  
Exponential Stability ◽  
Equilibrium Point ◽  
Global Exponential Stability ◽  
Sufficient Conditions ◽  
Analytical Techniques ◽  
Reaction Diffusion ◽  
Coefficient Matrix ◽  
Homeomorphism Mapping

In this paper, the global exponential stability of Chua's reaction–diffusion CNN system is investigated. For this system, some sufficient conditions ensuring the existence and global exponential stability of the equilibrium point is derived by using homeomorphism mapping, the property of coefficient matrix and analytical techniques. Finally, three illustrative examples are given to show the effectiveness of our results.

scholarly journals Characteristic phenomena in combustion

1997 ◽  
Vol 1 (2) ◽  
pp. 147-159
Author(s):  
Dirk Meinköhn
Keyword(s):  
State Space ◽  
Reaction Diffusion ◽  
Stationary Solutions ◽  
The State ◽  
State Variable ◽  
Finite Dimensional ◽  
Dimensional State Space ◽  
Singularity Order ◽  
The Relationship ◽  
Stable Stationary Solutions

For the case of a reaction–diffusion system, the stationary states may be represented by means of a state surface in a finite-dimensional state space. In the simplest example of a single semi-linear model equation given. in terms of a Fredholm operator, and under the assumption of a centre of symmetry, the state space is spanned by a single state variable and a number of independent control parameters, whereby the singularities in the set of stationary solutions are necessarily of the cuspoid type. Certain singularities among them represent critical states in that they form the boundaries of sheets of regular stable stationary solutions. Critical solutions provide ignition and extinction criteria, and thus are of particular physical interest. It is shown how a surface may be derived which is below the state surface at any location in state space. Its contours comprise singularities which correspond to similar singularities in the contours of the state surface, i.e., which are of the same singularity order. The relationship between corresponding singularities is in terms of lower bounds with respect to a certain distinguished control parameter associated with the name of Frank-Kamenetzkii.

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.

Nonlinear stability and numerical simulations for a reaction–diffusion system modelling Allee effect on predators

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Florinda Capone ◽  
Maria Francesca Carfora ◽  
Roberta De Luca ◽  
Isabella Torcicollo
Keyword(s):  
Numerical Simulations ◽  
Nonlinear Stability ◽  
Allee Effect ◽  
Complex Dynamics ◽  
Stability Result ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
System Modelling ◽  
Oscillatory Pattern

Abstract A reaction–diffusion system governing the prey–predator interaction with Allee effect on the predators, already introduced by the authors in a previous work is reconsidered with the aim of showing destabilization mechanisms of the biologically meaning equilibrium and detecting some aspects for the eventual oscillatory pattern formation. Extensive numerical simulations, depicting such complex dynamics, are shown. In order to complete the stability analysis of the coexistence equilibrium, a nonlinear stability result is shown.

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.

Sign in / Sign up

Export Citation Format

Share Document