A Novel Method to Determine the Local Stability of the n-Species Lotka-Volterra System with Multiple Delays

The n-species Lotka-Volterra system with discrete delays is considered. The local asymptotic stability of positive equilibrium is investigated based on a contour integral method. The main purpose of this paper is to propose a new and general algorithm to study the local asymptotic stability of the positive equilibrium for then-dimensional Lotka-Volterra system. Some numerical experiments are carried out to show the effectiveness of the proposed method.

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Qualitative analysis of a chemostat model with inhibitory exponential substrate uptake and a time delay

International Journal of Biomathematics ◽

10.1142/s1793524514500454 ◽

2014 ◽

Vol 07 (04) ◽

pp. 1450045 ◽

Cited By ~ 2

Author(s):

Qinglai Dong ◽

Wanbiao Ma

Keyword(s):

Time Delay ◽

Asymptotic Stability ◽

Qualitative Analysis ◽

Numerical Simulations ◽

Global Asymptotic Stability ◽

Positive Equilibrium ◽

Substrate Uptake ◽

Sufficient Condition ◽

Chemostat Model ◽

Local Asymptotic Stability

In this paper, we consider a simple chemostat model with inhibitory exponential substrate uptake and a time delay. A detailed qualitative analysis about existence and boundedness of its solutions and the local asymptotic stability of its equilibria are carried out. Using Lyapunov–LaSalle invariance principle, we show that the washout equilibrium is global asymptotic stability for any time delay. Using the fluctuation lemma, the sufficient condition of the global asymptotic stability of the positive equilibrium [Formula: see text] is obtained. Numerical simulations are also performed to illustrate the results.

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GLOBAL STABILITY OF A CHEMOSTAT MODEL WITH DELAYED RESPONSE IN GROWTH AND A LETHAL EXTERNAL INHIBITOR

International Journal of Biomathematics ◽

10.1142/s1793524508000436 ◽

2008 ◽

Vol 01 (04) ◽

pp. 503-520 ◽

Cited By ~ 3

Author(s):

ZHIQI LU ◽

JINGJING WU

Keyword(s):

Global Stability ◽

Local Stability ◽

Sufficient Conditions ◽

Positive Equilibrium ◽

Delayed Response ◽

Qualitative Properties ◽

Chemostat Model ◽

Stability And Instability ◽

Discrete Delays ◽

Globally Stable

A competition model between two species with a lethal inhibitor in a chemostat is analyzed. Discrete delays are used to describe the nutrient conversion process. The proved qualitative properties of the solution are positivity, boundedness. By analyzing the local stability of equilibria, it is found that the conditions for stability and instability of the boundary equilibria are similar to those in [9]. In addition, the global asymptotic behavior of the system is discussed and the sufficient conditions for the global stability of the boundary equilibria are obtained. Moreover, by numerical simulation, it is interesting to find that the positive equilibrium may be globally stable.

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Exploring Complex Dynamics of Spatial Predator–Prey System: Role of Predator Interference and Additional Food

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127420501023 ◽

2020 ◽

Vol 30 (07) ◽

pp. 2050102

Author(s):

Vandana Tiwari ◽

Jai Prakash Tripathi ◽

Debaldev Jana ◽

Satish Kumar Tiwari ◽

Ranjit Kumar Upadhyay

Keyword(s):

Asymptotic Stability ◽

Equilibrium Solution ◽

Model System ◽

Positive Equilibrium ◽

Additional Food ◽

Predator Prey ◽

Local Asymptotic Stability ◽

Diffusive Model ◽

Predator Interference

In this paper, an attempt has been made to understand the role of predator’s interference and additional food on the dynamics of a diffusive population model. We have studied a predator–prey interaction system with mutually interfering predator by considering additional food and Crowley–Martin functional response (CMFR) for both the reaction–diffusion model and associated spatially homogeneous system. The local stability analysis ensures that as the quantity of alternative food decreases, predator-free equilibrium stabilizes. Moreover, we have also obtained a condition providing a threshold value of additional food for the global asymptotic stability of coexisting steady state. The nonspatial model system changes stability via transcritical bifurcation and switches its stability through Hopf-bifurcation with respect to certain ranges of parameter determining the quantity of additional food. Conditions obtained for local asymptotic stability of interior equilibrium solution of temporal system determines the local asymptotic stability of associated diffusive model. The global stability of positive equilibrium solution of diffusive model system has been established by constructing a suitable Lyapunov function and using Green’s first identity. Using Harnack inequality and maximum modulus principle, we have established the nonexistence of nonconstant positive equilibrium solution of the diffusive model system. A chain of patterns on increasing the strength of additional food as spots[Formula: see text][Formula: see text][Formula: see text]stripes[Formula: see text][Formula: see text][Formula: see text]spots has been obtained. Various kind of spatial-patterns have also been demonstrated via numerical simulations and the roles of predator interference and additional food are established.

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

Nonlinear Analysis Modelling and Control ◽

10.15388/namc.2020.25.17437 ◽

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.

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Local Asymptotic Stability Conditions for the Positive Equilibrium of a System Modeling Cell Dynamics in Leukemia

Time Delay Systems: Methods, Applications and New Trends - Lecture Notes in Control and Information Sciences ◽

10.1007/978-3-642-25221-1_14 ◽

2012 ◽

pp. 187-197

Author(s):

Hitay Özbay ◽

Catherine Bonnet ◽

Houda Benjelloun ◽

Jean Clairambault

Keyword(s):

Asymptotic Stability ◽

System Modeling ◽

Positive Equilibrium ◽

Stability Conditions ◽

Cell Dynamics ◽

Local Asymptotic Stability ◽

Asymptotic Stability Conditions

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Bifurcation Analysis of a Lotka-Volterra Mutualistic System with Multiple Delays

Abstract and Applied Analysis ◽

10.1155/2014/958140 ◽

2014 ◽

Vol 2014 ◽

pp. 1-18 ◽

Cited By ~ 1

Author(s):

Xin-You Meng ◽

Hai-Feng Huo

Keyword(s):

Hopf Bifurcation ◽

Local Stability ◽

Time Delays ◽

Center Manifold ◽

Positive Equilibrium ◽

Multiple Delays ◽

Center Manifold Theorem ◽

Normal Form Method ◽

Three Delays ◽

Theoretical Results

A class of Lotka-Volterra mutualistic system with time delays of benefit and feedback delays is introduced. By analyzing the associated characteristic equation, the local stability of the positive equilibrium and existence of Hopf bifurcation are obtained under all possible combinations of two or three delays selecting from multiple delays. Not only explicit formulas to determine the properties of the Hopf bifurcation are shown by using the normal form method and center manifold theorem, but also the global continuation of Hopf bifurcation is investigated by applying a global Hopf bifurcation result due to Wu (1998). Numerical simulations are given to support the theoretical results.

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Effect of multiple delays on the dynamics of cannibalistic prey–predator system with disease in both populations

International Journal of Biomathematics ◽

10.1142/s1793524517500498 ◽

2017 ◽

Vol 10 (04) ◽

pp. 1750049 ◽

Cited By ~ 3

Author(s):

Santosh Biswas ◽

Sudip Samanta ◽

Qamar J. A. Khan ◽

Joydev Chattopadhyay

Keyword(s):

Stability Analysis ◽

Disease Transmission ◽

Local Stability ◽

Numerical Experiments ◽

Sufficient Conditions ◽

Multiple Delays ◽

Predator Population ◽

Interior Equilibrium ◽

Local Stability Analysis ◽

Predator System

In the present paper, we investigate a prey–predator system with disease in both prey and predator populations and the predator population is cannibalistic in nature. The model is extended by introducing incubation delays in disease transmission terms. Local stability analysis of the system around the biologically feasible equilibria is studied. The bifurcation analysis of the system around the interior equilibrium is also studied. The sufficient conditions for the permanence of the system are derived in the presence of delays. We observe that incubation delays have the ability to destabilize the cannibalistic prey–predator system. Finally, we perform numerical experiments to substantiate our analytical findings.

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On neutral-delay two-species Lotka-Volterra competitive systems

The Journal of the Australian Mathematical Society Series B Applied Mathematics ◽

10.1017/s0334270000006895 ◽

1991 ◽

Vol 32 (3) ◽

pp. 311-326 ◽

Cited By ~ 13

Author(s):

Y. Kuang

Keyword(s):

Steady State ◽

Asymptotic Stability ◽

Positive Solutions ◽

Sufficient Conditions ◽

Qualitative Behavior ◽

Neutral Delay ◽

Competitive System ◽

Competitive Systems ◽

Local Asymptotic Stability ◽

Discrete Delays

AbstractThe qualitative behavior of positive solutions of the neutral-delay two-species Lotka-Volterra competitive system with several discrete delays is investigated. Sufficient conditions are obtained for the local asymptotic stability of the positive steady state. In fact, some of these sufficient conditions are also necessary except at those critical values. Results on the oscillatory and non-oscillatory characteristics of the positive solutions are also included.

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