scholarly journals Global Stability for a Three-Species Food Chain Model in a Patchy Environment

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Hongli Li ◽  
Yaolin Jiang ◽  
Long Zhang ◽  
Zhidong Teng
Keyword(s):  
Food Chain ◽  
Sufficient Conditions ◽  
Coupled Systems ◽  
Positive Equilibrium ◽  
Chain Model ◽  
Patchy Environment ◽  
Globally Asymptotically Stable ◽  
Predator Species ◽  
Graph Theoretical Approach ◽  
Food Chain Model

We investigate a three-species food chain model in a patchy environment where prey species, mid-level predator species, and top predator species can disperse amongndifferent patches(n≥2). By using the method of constructing Lyapunov functions based on graph-theoretical approach for coupled systems, we derive sufficient conditions under which the positive equilibrium of this model is unique and globally asymptotically stable if it exists.

scholarly journals Dynamics of a Nonautonomous Leslie-Gower Type Food Chain Model with Delays

2011 ◽  
Vol 2011 ◽  
pp. 1-19 ◽  
Author(s):  
Hongying Lu ◽  
Weiguo Wang
Keyword(s):  
Periodic Solution ◽  
Asymptotic Stability ◽  
Food Chain ◽  
Time Delays ◽  
Sufficient Conditions ◽  
Positive Periodic Solution ◽  
Chain Model ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Food Chain Model

A nonautonomous Leslie-Gower type food chain model with time delays is investigated. It is proved the general nonautonomous system is permanent and globally asymptotically stable under some appropriate conditions. Furthermore, if the system is periodic one, some sufficient conditions are established, which guarantee the existence, uniqueness, and global asymptotic stability of a positive periodic solution of the system. The conditions for the permanence, global stability of system, and the existence, uniqueness of positive periodic solution depend on delays; so, time delays are profitless.

Qualitative Analysis of Three-Dimensional Single Food-Chain Model with Variable Consumption Rate in Chemostat

2014 ◽  
Vol 955-959 ◽  
pp. 463-470
Author(s):  
Jing Liu ◽  
Hong Wei Jiang ◽  
Chao Liu
Keyword(s):  
Food Chain ◽  
Consumption Rate ◽  
Three Dimensional ◽  
Quadratic Function ◽  
Positive Equilibrium ◽  
Chain Model ◽  
Equilibrium Solutions ◽  
Existence And Stability ◽  
Variable Consumption ◽  
Food Chain Model

The paper studies three-dimensional food-chain model with variable consumption rate in Chemostat. Assume the prey population's consumption rate of the nutrients is quadratic function, and the predator's consumption rate of the prey population is linear function. Use qualitative theory of ordinary differential equation to analyze the equilibrium solution of the model, especially the existence and stability of positive equilibrium solutions and Hopf bifurcation solutions. Finally,several numerical simulations illustrating the theoretical analysis are also given.

scholarly journals Dynamics of a three species ratio-dependent food chain model with diffusion and double free boundaries

2020 ◽  
Vol 15 ◽  
pp. 62
Author(s):  
Dawei Zhang ◽  
Beiping Duan ◽  
Binxiang Dai
Keyword(s):  
Food Chain ◽  
Dimensional Space ◽  
Chain Model ◽  
Time Behavior ◽  
Free Boundaries ◽  
Top Predator ◽  
Predator Species ◽  
Spreading And Vanishing ◽  
Ratio Dependent ◽  
Food Chain Model

This paper focuses on the dynamics of a three species ratio-dependent food chain model with diffusion and double free boundaries in one dimensional space, in which the free boundaries represent expanding fronts of top predator species. The existence, uniqueness and estimates of the global solution are discussed firstly. Then we prove a spreading–vanishing dichotomy, specifically, the top predator species either successfully spreads to the entire space as time t goes to infinity and survives in the new environment, or fails to establish and dies out in the long run. The long time behavior of the three species and criteria for spreading and vanishing are also obtained. Besides, our simulations illustrate the impacts of initial occupying area and expanding capability on the dynamics of top predator for free boundaries.

Stability and Hopf Bifurcation of a Fractional-Order Food Chain Model With Disease and Two Delays

2020 ◽  
Vol 15 (3) ◽  
Author(s):  
Xinhe Wang ◽  
Zhen Wang ◽  
Xiao Shen
Keyword(s):  
Fractional Order ◽  
Food Chain ◽  
Positive Equilibrium ◽  
Chain Model ◽  
Two Delays ◽  
Existence Conditions ◽  
Positive Equilibrium Point ◽  
The Stability ◽  
Food Chain Model ◽  
Bifurcation And Stability

Abstract In this study, a fractional-order food chain model with disease and two delays is proposed. The existence conditions for a positive equilibrium point are given, and the stability conditions without the effects of delays are established. The effects of a single time delay and two time delays are discussed, the bifurcation and stability criteria are obtained, and the bifurcation points are calculated. To support the theoretical analysis, numerical simulations are presented.

Dynamics of a Food Chain Model with Two Infected Predators

2021 ◽  
Vol 31 (02) ◽  
pp. 2150019
Author(s):  
Xin-You Meng ◽  
Ni-Ni Qin ◽  
Hai-Feng Huo
Keyword(s):  
Food Chain ◽  
Disease Transmission ◽  
Sufficient Conditions ◽  
Geometric Approach ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Chain Model ◽  
Number Formula ◽  
Manifold Theory ◽  
Food Chain Model

In this paper, the dynamics of a three-species food chain model with two predators infected by an infectious disease is investigated. The positivity and boundedness of the system, the existence of the equilibria and the basic reproductive number are given. Sufficient conditions for the local stability of all equilibria are obtained by analyzing the corresponding characteristic equations. By constructing suitable Lyapunov functions and taking the geometric approach, the global stability of all equilibria is proved. According to the center manifold theory, this model undergoes the phenomenon of backward and forward bifurcations in a certain range of the basic reproductive number [Formula: see text]. By taking the disease transmission coefficient of predator as bifurcation parameter, Hopf bifurcation emerges in the neighborhood of the endemic equilibrium. Furthermore, the optimal control of the disease is discussed by the Pontryagin’s maximum principle. Various simulations are given to support the analytical results.

scholarly journals GLOBAL STABILITY OF A THREE-SPECIES FOOD-CHAIN MODEL WITH DIFFUSION AND NONLOCAL DELAYS

2011 ◽  
Vol 16 (3) ◽  
pp. 376-389 ◽  
Author(s):  
Xiao Zhang ◽  
Rui Xu ◽  
Zhe Li
Keyword(s):  
Steady State ◽  
Global Stability ◽  
Food Chain ◽  
Energy Function ◽  
Function Method ◽  
Sufficient Conditions ◽  
Reaction Diffusion ◽  
Chain Model ◽  
Food Chain Model ◽  
Theoretical Results

In this paper, a three species reaction-diffusion food-chain system with nonlocal delays is investigated. Sufficient conditions are derived for the global stability of a positive steady state and boundary steady states of the system by using the energy function method. Numerical simulations are carried out to illustrate the theoretical results.

scholarly journals Global Stability for a Predator-Prey Model with Dispersal among Patches

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Yang Gao ◽  
Shengqiang Liu
Keyword(s):  
Global Stability ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Coupled Systems ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Graph Theoretical Approach

We investigate a predator-prey model with dispersal for both predator and prey amongnpatches; our main purpose is to extend the global stability criteria by Li and Shuai (2010) on a predator-prey model with dispersal for prey amongnpatches. By using the method of constructing Lyapunov functions based on graph-theoretical approach for coupled systems, we derive sufficient conditions under which the positive coexistence equilibrium of this model is unique and globally asymptotically stable if it exists.

scholarly journals Asymptotic behavior of a three-species food chain model with time-varying delays

2021 ◽  
Author(s):  
Yuxiao Zhao ◽  
Linshan Wang ◽  
Yangfan Wang
Keyword(s):  
Asymptotic Behavior ◽  
Exponential Stability ◽  
Positive Solutions ◽  
Food Chain ◽  
Lyapunov Method ◽  
Sufficient Conditions ◽  
Chain Model ◽  
Time Varying ◽  
Razumikhin Technique ◽  
Food Chain Model

In this paper, a stochastic three-species food chain model with time-varying delays is focussed. The existence and the asymptotic behavior of global positive solutions to the model are discussed, and the sufficient conditions for the 1th moment practical exponential stability and the extinction of the model are given by using the Razumikhin technique and Lyapunov method.

scholarly journals Density Dependent Predator Death Prevalence Chaos in a Tri-trophic Food Chain Model

2008 ◽  
Vol 13 (3) ◽  
pp. 305-324 ◽  
Author(s):  
M. Bandyopadhyay ◽  
S. Chatterjee ◽  
S. Chakraborty ◽  
J. Chattopadhyay
Keyword(s):  
Food Chain ◽  
Death Rate ◽  
Chaotic Dynamics ◽  
Equilibrium Points ◽  
Model System ◽  
Chain Model ◽  
Predator Population ◽  
Predator Species ◽  
Density Dependent ◽  
Food Chain Model

Ecological systems have all the properties to produce chaotic dynamics. To predict the chaotic behavior in an ecological system and its possible control mechanism is interesting. Aziz-Alaoui [1] considered a tri-trophic food-chain model with modified Leslie-Gower type growth rate for top-predator population and established the chaotic dynamics exhibited by the model system for a certain choice of parameter values. We have modified the said model by incorporating density dependent death rate for predator population. Our mathematical findings reveal the fact that there are two coexisting equilibrium points one of which is a source and the other one is a sink. The positive equilibrium point which is sink is actually globally asymptotically stable under certain parametric conditions. Numerical experiment analysis shows that the model system are capable to produce chaotic dynamics when the rate of intra specific completion is very low and chaotic dynamics disappears for a certain value of the rate of intra specific completion for predator species. Our results suggest that the consideration of density dependent death rate for predator species have the ability to control the chaotic dynamics.

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