scholarly journals Global stability in a diffusive Beddington-DeAngelis and Tanner predator-prey model

Filomat ◽  
2019 ◽  
Vol 33 (17) ◽  
pp. 5511-5517
Author(s):  
Demou Luo
Keyword(s):  
Boundary Condition ◽  
Global Stability ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Flux Boundary Condition ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Constant Equilibrium ◽  
Parameter Condition

Our goal is to study a diffusive Beddington-DeAngelis and Tanner predator-prey system with no-flux boundary condition. It is proved that the unique constant equilibrium is globally asymptotically stable under a new simpler parameter condition.

scholarly journals The study of global stability of a diffusive Beddington-Deangelis and tanner predator-prey model

Filomat ◽  
2019 ◽  
Vol 33 (12) ◽  
pp. 3937-3946
Author(s):  
Demou Luo
Keyword(s):  
Boundary Condition ◽  
Global Stability ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Constant Equilibrium ◽  
Prey Model ◽  
Parameter Condition

In this article, a diffusive Beddington-DeAngelis and Tanner predator-prey model with noflux boundary condition is investigated, and it is proved that the unique constant equilibrium is globally asymptotically stable under a new simpler parameter condition.

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 Periodicity and Permanence of a Discrete Impulsive Lotka-Volterra Predator-Prey Model Concerning Integrated Pest Management

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Chang Tan ◽  
Jun Cao
Keyword(s):  
Discrete System ◽  
Numerical Experiments ◽  
Critical Value ◽  
Euler Method ◽  
Impulsive Effect ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model

By piecewise Euler method, a discrete Lotka-Volterra predator-prey model with impulsive effect at fixed moment is proposed and investigated. By using Floquets theorem, we show that a globally asymptotically stable pest-eradication periodic solution exists when the impulsive period is less than some critical value. Further, we prove that the discrete system is permanence if the impulsive period is larger than some critical value. Finally, some numerical experiments are given.

Global Stability of a Non-Smooth Predator–Prey System with Holling I Functional Response and Refuge Effect

2014 ◽  
Vol 595 ◽  
pp. 283-288 ◽  
Author(s):  
Yuan Tian ◽  
Hai Ting Sun ◽  
Yu Xia He
Keyword(s):  
Global Stability ◽  
Functional Response ◽  
Tangent Point ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Refuge Effect ◽  
Prey System ◽  
Trivial Equilibrium ◽  
The Stability ◽  
Regular Equilibrium

This paper analyses the dynamics of a non-smooth predator-prey model with refuge effect, where the functional response is taken as Holling I type. To begin with, some preliminaries and the existence of regular, virtual, pseudo-equilibrium and tangent point are established. Then, the stability of trivial equilibrium and predator free equilibrium is discussed. Furthermore, it is shown that the regular equilibrium and the pseudo-equilibrium cannot coexist. Finally, the conclusion is given.

scholarly journals Pattern Formation in a Diffusive Ratio-Dependent Holling-Tanner Predator-Prey Model with Smith Growth

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Bo Yang
Keyword(s):  
Boundary Condition ◽  
Hopf Bifurcation ◽  
Turing Instability ◽  
Turing Patterns ◽  
Positive Equilibrium ◽  
Flux Boundary Condition ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Ratio Dependent

The spatiotemporal dynamics of a diffusive ratio-dependent Holling-Tanner predator-prey model with Smith growth subject to zero-flux boundary condition are investigated analytically and numerically. The asymptotic stability of the positive equilibrium and the existence of Hopf bifurcation around the positive equilibrium are shown; the conditions of Turing instability are obtained. And with the help of numerical simulations, it is found that the model exhibits complex pattern replication: stripes, spots-stripes mixtures, and spots Turing patterns.

BIFURCATIONS IN A PREDATOR-PREY MODEL WITH DIFFUSION AND MEMORY

2006 ◽  
Vol 16 (06) ◽  
pp. 1855-1863 ◽  
Author(s):  
SHABAN ALY
Keyword(s):  
Hopf Bifurcation ◽  
Allee Effect ◽  
Stationary Solutions ◽  
Asymptotically Stable ◽  
Predator Prey ◽  
Predator Prey Model ◽  
The Past ◽  
Prey System ◽  
Stable Equilibria ◽  
Per Capita

In this paper we formulate a delayed predator-prey system in two patches in which the per capita migration rate of each species is influenced only by its own density, i.e. there is no response to the density of the other and the growth rate of the predator depends on the prey that was available in the past. If the equilibrium point lies in the Allée effect zone and when the diffusion is present only, we show that at a critical value of the bifurcation parameter the system undergoes a Turing bifurcation, patterns emerge, the spatially homogeneous equilibrium loses its stability and two new spatially non-constant stable equilibria emerge which are asymptotically stable. When the delay is present only, the increase of delay destabilizes the system and causes the occurrence of periodic oscillations, Andronov–Hopf bifurcation. For the full general model (with both diffusion and delay) if the bifurcation parameters are increased through critical values of diffusion and delay the two new spatially nonconstant stationary solutions lose their stability by Hopf bifurcation.

DYNAMIC COMPLEXITIES IN A LOTKA–VOLTERRA PREDATOR–PREY MODEL CONCERNING IMPULSIVE CONTROL STRATEGY

2005 ◽  
Vol 15 (02) ◽  
pp. 517-531 ◽  
Author(s):  
BING LIU ◽  
YUJUAN ZHANG ◽  
LANSUN CHEN
Keyword(s):  
Periodic Solution ◽  
Control Strategy ◽  
Impulsive Control ◽  
Period Doubling ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Pest Eradication ◽  
Impulsive Control Strategy

Based on the classical Lotka–Volterra predator–prey system, an impulsive differential equation to model the process of periodically releasing natural enemies and spraying pesticides at different fixed times for pest control is proposed and investigated. It is proved that there exists a globally asymptotically stable pest-eradication periodic solution when the impulsive period is less than some critical value. Otherwise, the system can be permanent. We observe that our impulsive control strategy is more effective than the classical one if we take chemical control efficiently. Numerical results show that the system we considered has more complex dynamics including period-doubling bifurcation, symmetry-breaking bifurcation, period-halving bifurcation, quasi-periodic oscillation, chaos and nonunique dynamics, meaning that several attractors coexist. Finally, a pest–predator stage-structured model for the pest concerning this kind of impulsive control strategy is proposed, and we also show that there exists a globally asymptotically stable pest-eradication periodic solution when the impulsive period is less than some threshold.

scholarly journals Global Stability and Hopf Bifurcation for Gause-Type Predator-Prey System

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Shuang Guo ◽  
Weihua Jiang
Keyword(s):  
Global Stability ◽  
Periodic Solutions ◽  
Three Dimensional ◽  
Growth Time ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Existence Of Periodic Solutions ◽  
The Stability ◽  
System With Time Delay

A class of three-dimensional Gause-type predator-prey model is considered. Firstly, local stability of equilibrium indicating the extinction of top-predator is obtained. Meanwhile, we construct a Lyapunov function, which is an extension of the Lyapunov functions constructed by Hsu for predator-prey system (2005), to give the global stability of the equilibrium. Secondly, we analyze the stability of coexisting equilibrium of predator-prey system with time delay when the predator catches the prey of pregnancy or with growth time. The delay can lead to periodic solutions, which is consistent with the law of growth for birds and some mammals. Further, an explicit formula is given which determines the stability of the bifurcating periodic solutions theoretically and the existence of periodic solutions is displayed by numerical simulations.

TURING PATTERNS OF A PREDATOR–PREY MODEL WITH A MODIFIED LESLIE–GOWER TERM AND CROSS-DIFFUSIONS

2012 ◽  
Vol 05 (06) ◽  
pp. 1250060 ◽  
Author(s):  
GUANG-PING HU ◽  
XIAO-LING LI
Keyword(s):  
Turing Instability ◽  
Turing Patterns ◽  
Positive Equilibrium ◽  
Asymptotically Stable ◽  
Strongly Coupled ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Prey System ◽  
Positive Steady States

In this paper, a strongly coupled diffusive predator–prey system with a modified Leslie–Gower term is considered. We will show that under certain hypotheses, even though the unique positive equilibrium is asymptotically stable for the dynamics with diffusion, Turing instability can produce due to the presence of the cross-diffusion. In particular, we establish the existence of non-constant positive steady states of this system. The results indicate that cross-diffusion can create stationary patterns.

Stationary patterns for a Leslie–Gower type three species model with diffusion

2014 ◽  
Vol 07 (06) ◽  
pp. 1450069 ◽  
Author(s):  
Guangping Hu ◽  
Xiaoling Li ◽  
Shiping Lu
Keyword(s):  
Global Stability ◽  
Positive Constant ◽  
Sufficient Conditions ◽  
Reaction Diffusion ◽  
Steady States ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Constant Equilibrium ◽  
Positive Steady States ◽  
The Stability

In this paper, a diffusive three species predator–prey model with two Leslie–Gower terms is considered. The stability of the unique positive constant equilibrium for the reaction–diffusion system is obtained. Sufficient conditions are derived for the global stability of the positive constant equilibrium. In particular, we establish the existence and non-existence of non-constant positive steady states of this system. The results indicate that the large diffusivity is helpful for the appearance of the non-constant positive steady states (stationary patterns).

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