scholarly journals Global Asymptotic Stability of 3-Species Mutualism Models with Diffusion and Delay Effects

2009 ◽  
Vol 2009 ◽  
pp. 1-20 ◽  
Author(s):  
Chang-you Wang ◽  
Shu Wang ◽  
Xiang-ping Yan
Keyword(s):  
Steady State ◽  
Asymptotic Stability ◽  
Global Asymptotic Stability ◽  
Upper And Lower Solutions ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Steady State Solution ◽  
State Problem ◽  
Delay Effects ◽  
Steady State Problem

In this paper, the Lotka-Volterra 3-species mutualism models with diffusion and delay effects is investigated. A simple and easily verifiable condition is given to ensure the global asymptotic stability of the unique positive steady-state solution of the corresponding steady-state problem in a bounded domain with Neumann boundary condition. Our approach to the problem is based on inequality skill and the method of the upper and lower solutions for a more general reaction—diffusion system. Finally, some numerical simulations are given to illustrate our results.

scholarly journals Global Asymptotic Stability in a Class of Reaction-Diffusion Equations with Time Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Yueding Yuan ◽  
Zhiming Guo
Keyword(s):  
Asymptotic Stability ◽  
Global Asymptotic Stability ◽  
Sufficient Conditions ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Careful Analysis ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Delayed Reaction ◽  
Fluctuation Method

We study a very general class of delayed reaction-diffusion equations in which the reaction term can be nonmonotone and spatially nonlocal. By using a fluctuation method, combined with the careful analysis of the corresponding characteristic equations, we obtain some sufficient conditions for the global asymptotic stability of the trivial solution and the positive steady state to the equations subject to the Neumann boundary condition.

Convergence of a homotopy finite element method for computing steady states of Burgers’ equation

2019 ◽  
Vol 53 (5) ◽  
pp. 1629-1644 ◽  
Author(s):  
Wenrui Hao ◽  
Yong Yang
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Steady State ◽  
Burgers Equation ◽  
Initial Conditions ◽  
Steady State Solution ◽  
State Problem ◽  
Steady State Problem ◽  
Steady State Solutions ◽  
Element Method

In this paper, the convergence of a homotopy method (1.1) for solving the steady state problem of Burgers’ equation is considered. When ν is fixed, we prove that the solution of (1.1) converges to the unique steady state solution as ε → 0, which is independent of the initial conditions. Numerical examples are presented to confirm this conclusion by using the continuous finite element method. In contrast, when ν = ε →, numerically we show that steady state solutions obtained by (1.1) indeed depend on initial conditions.

The asymptotic stability for an SIQS epidemic model with diffusion

2015 ◽  
Vol 09 (01) ◽  
pp. 1650015
Author(s):  
Yanling Li ◽  
Lijing Zhang ◽  
Gaihui Guo
Keyword(s):  
Asymptotic Stability ◽  
Epidemic Model ◽  
Upper And Lower Solutions ◽  
Sufficient Conditions ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Sis Model ◽  
Globally Asymptotically Stable ◽  
Monotone Iterations ◽  
Disease Free

In this paper, an SIQS epidemic model with constant recruitment and standard incidence is investigated. Quarantine is taken into consideration on the basis of SIS model. The asymptotic stability of the equilibrium to a reaction–diffusion system with homogeneous Neumann boundary conditions is considered. Sufficient conditions for the local and global asymptotic stability are given by linearization and the method of upper and lower solutions and its associated monotone iterations. The result shows that the disease-free equilibrium is globally asymptotically stable if the contact rate is small.

Global asymptotic stability for a SEI reaction–diffusion model of infectious diseases with immigration

2018 ◽  
Vol 11 (03) ◽  
pp. 1850044
Author(s):  
Salem Abdelmalek ◽  
Samir Bendoukha
Keyword(s):  
Steady State ◽  
Asymptotic Stability ◽  
Global Stability ◽  
Global Asymptotic Stability ◽  
Epidemic Model ◽  
Lyapunov Functional ◽  
Reaction Diffusion ◽  
Stability Of Solutions ◽  
Reaction Diffusion Model ◽  
Globally Stable

This paper studies the local and global stability of solutions for a spatially spread SEI epidemic model with immigration of individuals using a Lyapunov functional. It is shown that in the presence of diffusion, the unique steady state remains globally stable. Numerical results obtained through Matlab simulations are presented to confirm the findings of this study.

Steady-State Problem of an S-K-T Competition Model with Spatially Degenerate Coefficients

2021 ◽  
Vol 31 (11) ◽  
pp. 2150165
Author(s):  
Hao Zhou ◽  
Yu-Xia Wang
Keyword(s):  
Steady State ◽  
Global Bifurcation ◽  
Steady State Solution ◽  
Competition Model ◽  
State Solution ◽  
State Problem ◽  
Cross Diffusion ◽  
Positive Steady State ◽  
Steady State Problem ◽  
Steady State Solutions

In this paper, we study the steady-state problem of an S-K-T competition model with a spatially degenerate intraspecific competition coefficient. First, the global bifurcation continuum of positive steady-state solutions from its semitrivial steady-state solution is given, which depends on the spatial heterogeneity and cross-diffusion. Second, two limiting systems are derived as the cross-diffusion coefficient tends to infinity. Moreover, we demonstrate the existence of positive steady-state solutions near the two limiting systems, and show which one of the limiting systems characterizes the positive steady-state solution.

Conditions for the local and global asymptotic stability of the time–fractional Degn–Harrison system

2020 ◽  
Vol 21 (7-8) ◽  
pp. 749-759
Author(s):  
Rachida Mezhoud ◽  
Khaled Saoudi ◽  
Abderrahmane Zaraï ◽  
Salem Abdelmalek
Keyword(s):  
Asymptotic Stability ◽  
Global Asymptotic Stability ◽  
Lyapunov Method ◽  
Sufficient Conditions ◽  
Reaction Diffusion ◽  
Linearization Method ◽  
Direct Lyapunov Method ◽  
Fractional Systems ◽  
Reaction Diffusion Model ◽  
Theoretical Results

AbstractFractional calculus has been shown to improve the dynamics of differential system models and provide a better understanding of their dynamics. This paper considers the time–fractional version of the Degn–Harrison reaction–diffusion model. Sufficient conditions are established for the local and global asymptotic stability of the model by means of invariant rectangles, the fundamental stability theory of fractional systems, the linearization method, and the direct Lyapunov method. Numerical simulation results are used to illustrate the theoretical results.

Dynamic Stability of an Impact System Connected With Rock Drilling

1969 ◽  
Vol 36 (4) ◽  
pp. 743-749 ◽  
Author(s):  
C. C. Fu
Keyword(s):  
Computer Simulation ◽  
Exact Solution ◽  
Steady State ◽  
Asymptotic Stability ◽  
Piecewise Linear ◽  
Steady State Solution ◽  
External Excitation ◽  
Rock Drilling ◽  
Impact System ◽  
Number Of Cycles

This paper deals with asymptotic stability of an analytically derived, synchronous as well as nonsynchronous, steady-state solution of an impact system which exhibits piecewise linear characteristics connected with rock drilling. The exact solution, which assumes one impact for a given number of cycles of the external excitation, is derived, its asymptotic stability is examined, and ranges of parameters are determined for which asymptotic stability is assured. The theoretically predicted stability or instability is verified by a digital computer simulation.

scholarly journals Stability in a Simple Food Chain System with Michaelis-Menten Functional Response and Nonlocal Delays

2013 ◽  
Vol 2013 ◽  
pp. 1-14
Author(s):  
Wenzhen Gan ◽  
Canrong Tian ◽  
Qunying Zhang ◽  
Zhigui Lin
Keyword(s):  
Steady State ◽  
Functional Response ◽  
Neumann Boundary Condition ◽  
Lyapunov Functional ◽  
Sufficient Conditions ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Food Ingestion ◽  
Nonlocal Delay ◽  
Homogeneous Neumann Boundary Condition

This paper is concerned with the asymptotical behavior of solutions to the reaction-diffusion system under homogeneous Neumann boundary condition. By taking food ingestion and species' moving into account, the model is further coupled with Michaelis-Menten type functional response and nonlocal delay. Sufficient conditions are derived for the global stability of the positive steady state and the semitrivial steady state of the proposed problem by using the Lyapunov functional. Our results show that intraspecific competition benefits the coexistence of prey and predator. Furthermore, the introduction of Michaelis-Menten type functional response positively affects the coexistence of prey and predator, and the nonlocal delay is harmless for stabilities of all nonnegative steady states of the system. Numerical simulations are carried out to illustrate the main results.

Mixed steady-state problem of torsion of a sphere with an elastic core

1976 ◽  
Vol 12 (12) ◽  
pp. 1247-1251
Author(s):  
V. A. Babeshko ◽  
V. E. Veksler
Keyword(s):  
Steady State ◽  
State Problem ◽  
Elastic Core ◽  
Steady State Problem
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