scholarly journals Dynamic Behaviors of a Leslie-Gower Ecoepidemiological Model

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Aihua Kang ◽  
Yakui Xue ◽  
Jianping Fu
Keyword(s):  
Hopf Bifurcation ◽  
Asymptotic Stability ◽  
Global Stability ◽  
Lyapunov Functions ◽  
Total Population ◽  
Sufficient Conditions ◽  
Prey Population ◽  
Dynamic Behaviors ◽  
Interior Equilibrium ◽  
Local Asymptotic Stability

A Leslie-Gower ecoepidemic model with disease in the predators is constructed and analyzed. The total population is subdivided into three subclasses, namely, susceptible predator, infected predator, and prey population. The positivity, boundness of solutions, and the existence of the equilibria are studied, and the sufficient conditions of local asymptotic stability of the equilibria are obtained by the Routh-Hurwitz criterion. We analyze the global stability of the interior equilibria by using Lyapunov functions. It is observed that a Hopf bifurcation may occur around the interior equilibrium. At last, numeric simulations are performed in support of the feasibility of the main result.

Discontinuous Lyapunov functions for a class of piecewise affine systems

2018 ◽  
Vol 41 (3) ◽  
pp. 729-736 ◽  
Author(s):  
Farideh Cheraghi-Shami ◽  
Ali-Akbar Gharaveisi ◽  
Malihe M Farsangi ◽  
Mohsen Mohammadian
Keyword(s):  
Asymptotic Stability ◽  
Lyapunov Function ◽  
Lyapunov Functions ◽  
Convex Polytopes ◽  
Sufficient Conditions ◽  
Monotonicity Condition ◽  
Piecewise Affine Systems ◽  
Affine Systems ◽  
Piecewise Affine ◽  
Local Asymptotic Stability

In this paper, a Lyapunov-based method is provided to study the local asymptotic stability of planar piecewise affine systems with continuous vector fields. For such systems, the state space is supposed to be partitioned into several bounded convex polytopes. A piecewise affine function, not necessarily continuous on the boundaries of the polytopic partitions, is proposed as a candidate Lyapunov function. Then, sufficient conditions for the local asymptotic stability of the system, including a monotonicity condition at switching instants, are formulated as a linear programming problem. In addition, when the problem does not have a feasible solution based on the original partitions of the system, a new partition refinement algorithm is presented. In this way, more flexibility can be provided in searching for the Lyapunov function. Owing to relaxation of the continuity condition imposed on the system boundaries, the proposed method reaches to less conservative results, compared with the previous methods based on continuous piecewise affine Lyapunov functions. Simulation results illustrate the effectiveness of the proposed method.

GLOBAL ASYMPTOTIC STABILITY FOR A TWO-SPECIES DISCRETE RATIO-DEPENDENT PREDATOR–PREY SYSTEM

2013 ◽  
Vol 06 (01) ◽  
pp. 1250064 ◽  
Author(s):  
XIANGLAI ZHUO
Keyword(s):  
Asymptotic Stability ◽  
Global Stability ◽  
Global Asymptotic Stability ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Prey System ◽  
Ratio Dependent ◽  
Analysis Of Stability ◽  
Appl Math

The dynamical behaviors of a two-species discrete ratio-dependent predator–prey system are considered. Some sufficient conditions for the local stability of the equilibria is obtained by using the linearization method. Further, we also obtain a new sufficient condition to ensure that the positive equilibrium is globally asymptotically stable by using an iteration scheme and the comparison principle of difference equations, which generalizes what paper [G. Chen, Z. Teng and Z. Hu, Analysis of stability for a discrete ratio-dependent predator–prey system, Indian J. Pure Appl. Math.42(1) (2011) 1–26] has done. The method given in this paper is new and very resultful comparing with papers [H. F. Huo and W. T. Li, Existence and global stability of periodic solutions of a discrete predator–prey system with delays, Appl. Math. Comput.153 (2004) 337–351; X. Liao, S. Zhou and Y. Chen, On permanence and global stability in a general Gilpin–Ayala competition predator–prey discrete system, Appl. Math. Comput.190 (2007) 500–509] and it can also be applied to study the global asymptotic stability for general multiple species discrete population systems. At the end of this paper, we present an open question.

scholarly journals Global Stability of Positive Discrete-time Time-varying Nonlinear Feedback Systems

2021 ◽  
Vol 3 (1) ◽  
pp. 17-20
Author(s):  
Tadeusz Kaczorek ◽  
Łukasz Sajewski
Keyword(s):  
Nonlinear Systems ◽  
Asymptotic Stability ◽  
Global Stability ◽  
Linear Systems ◽  
Discrete Time ◽  
Sufficient Conditions ◽  
Time Varying ◽  
Nonlinear Feedback ◽  
Feedback Systems ◽  
Positive Time

The global stability of positive  discrete-time time-varying nonlinear systems with time-varying scalar feedbacks is investigated. Sufficient conditions for the asymptotic stability of discrete-time positive time-varying linear systems are given. The new conditions are applied to discrete-time positive time-varying nonlinear systems with time-varying feedbacks. Sufficient conditions are established for the global stability of the discrete-time positive time-varying nonlinear systems with feedbacks.

scholarly journals Hybrid nonnegative and computational dynamical systems

2002 ◽  
Vol 8 (6) ◽  
pp. 493-515 ◽  
Author(s):  
Wassim M. Haddad ◽  
Vijaysekhar Chellaboina ◽  
Sergey G. Nersesov
Keyword(s):  
Dynamical Systems ◽  
Asymptotic Stability ◽  
Conservation Laws ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Hybrid Structures ◽  
Stability Criteria ◽  
General Stability ◽  
Feedback Interconnections ◽  
Flow Laws

Nonnegative and Compartmental dynamical systems are governed by conservation laws and are comprised of homogeneous compartments which exchange variable nonnegative quantities of material via intercompartmental flow laws. These systems typically possess hierarchical (and possibly hybrid) structures and are remarkably effective in capturing the phenomenological features of many biological and physiological dynamical systems. In this paper we develop several results on stability and dissipativity of hybrid nonnegative and Compartmental dynamical systems. Specifically, usinglinearLyapunov functions we develop sufficient conditions for Lyapunov and asymptotic stability for hybrid nonnegative dynamical systems. In addition, usinglinearandnonlinearstorage functions withlinearhybrid supply rates we developnewnotions of dissipativity theory for hybrid nonnegative dynamical systems. Finally, these results are used to develop general stability criteria for feedback interconnections of hybrid nonnegative dynamical systems.

scholarly journals Turing instability and spatially homogeneous Hopf bifurcation in a diffusive Brusselator system

2020 ◽  
Vol 25 (4) ◽  
Author(s):  
Xiang-Ping Yan ◽  
Pan Zhang ◽  
Cun-Huz Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Asymptotic Stability ◽  
Neumann Boundary ◽  
Turing Instability ◽  
Spatial Diffusion ◽  
Positive Equilibrium ◽  
Local Asymptotic Stability ◽  
Ode System ◽  
Homogeneous Neumann Boundary Condition ◽  
Spatially Homogeneous

The present paper deals with a reaction–diffusion Brusselator system subject to the homogeneous Neumann boundary condition. When the effect of spatial diffusion is neglected, the local asymptotic stability and the detailed Hopf bifurcation of the unique positive equilibrium of the associated ODE system are analyzed. In the stable domain of the ODE system, the effect of spatial diffusion is explored, and local asymptotic stability, Turing instability and existence of Hopf bifurcation of the constant positive equilibrium are demonstrated. In addition, the direction of spatially homogeneous Hopf bifurcation and the stability of the spatially homogeneous bifurcating periodic solutions are also investigated. Finally, numerical simulations are also provided to check the obtained theoretical results.

On the dynamics of a class of perturbed cyclic Lotka-Volterra systems

2018 ◽  
Vol 11 (08) ◽  
pp. 1850098
Author(s):  
Chayu Yang ◽  
Ashlee Edwards ◽  
Jin Wang
Keyword(s):  
Hopf Bifurcation ◽  
Special Class ◽  
Sufficient Conditions ◽  
General Setting ◽  
Interaction Matrix ◽  
Interior Equilibrium ◽  
Perturbation Term ◽  
Volterra Systems ◽  
Higher Dimensional ◽  
Existence And Stability

We consider a special class of Lotka–Volterra systems where the associated interaction matrix is cyclic, but asymmetric, with a perturbation term on each row. After some discussion of the dynamics under a general setting, we focus our attention on 3D systems for a more detailed study. We derive sufficient conditions for the existence and stability of the nontrivial interior equilibrium. We also show that Hopf bifurcation occurs when the size of the perturbation is large. Such analysis can be similarly extended to higher dimensional systems, and we mention some results in 4D case.

scholarly journals Stability of Generalized Proportional Caputo Fractional Differential Equations by Lyapunov Functions

2022 ◽  
Vol 6 (1) ◽  
pp. 34
Author(s):  
Ravi Agarwal ◽  
Snezhana Hristova ◽  
Donal O’Regan
Keyword(s):  
Differential Equation ◽  
Asymptotic Stability ◽  
Fractional Derivative ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Nonlinear Fractional Differential Equation ◽  
Comparison Results ◽  
Caputo Fractional Differential Equations ◽  
Fractional Differential ◽  
Nonautonomous Equations

In this paper, nonlinear nonautonomous equations with the generalized proportional Caputo fractional derivative (GPFD) are considered. Some stability properties are studied by the help of the Lyapunov functions and their GPFDs. A scalar nonlinear fractional differential equation with the GPFD is considered as a comparison equation, and some comparison results are proven. Sufficient conditions for stability and asymptotic stability were obtained. Examples illustrating the results and ideas in this paper are also provided.

scholarly journals Stability and Bifurcation of Two Kinds of Three-Dimensional Fractional Lotka-Volterra Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Jinglei Tian ◽  
Yongguang Yu ◽  
Hu Wang
Keyword(s):  
Hopf Bifurcation ◽  
Asymptotic Stability ◽  
Theoretical Analysis ◽  
Fractional Order ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Critical Value ◽  
The Other ◽  
Bifurcation Parameter ◽  
Volterra Systems

Two kinds of three-dimensional fractional Lotka-Volterra systems are discussed. For one system, the asymptotic stability of the equilibria is analyzed by providing some sufficient conditions. And bifurcation property is investigated by choosing the fractional order as the bifurcation parameter for the other system. In particular, the critical value of the fractional order is identified at which the Hopf bifurcation may occur. Furthermore, the numerical results are presented to verify the theoretical analysis.

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