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 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 Permanence and global stability of positive solutions of a nonautonomous discrete ratio-dependent predator-prey model

2005 ◽  
Vol 2005 (2) ◽  
pp. 135-144 ◽  
Author(s):  
Hai-Feng Huo ◽  
Wan-Tong Li
Keyword(s):  
Lyapunov Function ◽  
Global Stability ◽  
Positive Solution ◽  
Positive Solutions ◽  
Sufficient Conditions ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Ratio Dependent

We first give sufficient conditions for the permanence of nonautonomous discrete ratio-dependent predator-prey model. By linearization of the model at positive solutions and construction of Lyapunov function, we also obtain some conditions which ensure that a positive solution of the model is stable and attracts all positive solutions.

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.

scholarly journals Global Dynamics of a Predator-Prey Model with Stage Structure and Delayed Predator Response

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Lili Wang ◽  
Rui Xu
Keyword(s):  
Sufficient Conditions ◽  
Stage Structure ◽  
Global Dynamics ◽  
Globally Asymptotically Stable ◽  
Predator Prey ◽  
Infinite Dimensional ◽  
Infinite Dimensional Systems ◽  
Predator Prey Model ◽  
Prey Model ◽  
Coexistence Equilibrium

A Holling type II predator-prey model with time delay and stage structure for the predator is investigated. By analyzing the corresponding characteristic equations, the local stability of each of feasible equilibria of the system is discussed. The existence of Hopf bifurcations at the coexistence equilibrium is established. By means of the persistence theory on infinite dimensional systems, it is proven that the system is permanent if the coexistence equilibrium exists. By using Lyapunov functionals and LaSalle’s invariance principle, it is shown that the predator-extinction equilibrium is globally asymptotically stable when the coexistence equilibrium is not feasible, and the sufficient conditions are obtained for the global stability of the coexistence equilibrium.

scholarly journals Predator-Prey Model with Refuge, Fear and Z-Control

2021 ◽  
Vol 26 (1) ◽  
pp. 40-57
Author(s):  
Ibrahim M. Elmojtaba ◽  
Kawkab Al-Amri ◽  
Qamar J.A. Khan
Keyword(s):  
Numerical Simulations ◽  
Equilibrium Points ◽  
Predator Population ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Long Run ◽  
Constant State

In this paper, we consider a predator-prey model incorporating fear and refuge.  Our results show that the predator-free equilibrium is globally asymptotically stable if the ratio between the death rate of predators and the conversion rate of prey into predator is greater than the value of prey in refuge at equilibrium.  We also show that the co-existence equilibrium points are locally asymptotically stable if the value of the prey outside refuge is greater than half of the carrying capacity.  Numerical simulations show that when the intensity of fear increases, the fraction of the prey inside refuge increases; however, it has no effect on the fraction of the prey outside refuge, in the long run. It is shown that the intensity of fear harms predator population size. Numerical simulations show that the application of Z-control will force the system to reach any desired state within a limited time, whether the desired state is a constant state or a periodic state. Our results show that when the refuge size is taken to be a non-constant function of the prey outside refuge, the systems change their dynamics. Namely, when it is a linear function or an exponential function, the system always reaches the predator-free equilibrium.  However, when it is taken as a logistic equation, the system reaches the co-existence equilibrium after long term oscillations.

scholarly journals Global stability for a new predator–prey model with cross-dispersal among patches based on graph theory

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Yang Gao
Keyword(s):  
Graph Theory ◽  
Lyapunov Functions ◽  
Coupled Systems ◽  
Positive Equilibrium ◽  
Connected Graphs ◽  
Predator Prey ◽  
New Model ◽  
Predator Prey Model ◽  
Prey Model ◽  
Strongly Connected

AbstractIn this paper, cross-dispersal is considered in a predator–prey model with a patchy environment. A new predator–prey model with cross-dispersal among patches is constructed. A new cross-dispersal matrix is established by the coupling relationship between vertices. First, an existence theorem of the positive equilibrium for the new model is obtained. Secondly, based on the idea of constructing Lyapunov functions and a graph-theoretical approach for coupled systems, sufficient conditions that the positive equilibrium of the new model is globally asymptotically stable in $R^{2n}_{+}$ R + 2 n are derived on a network with strongly connected graphs. Thirdly, based on the theory of asymptotically autonomous systems, Lyapunov functions method and graph theory, a stability theorem for the positive equilibrium of the new model is established on a complex network without strongly connected graphs. Finally, two examples are given to illustrate main results.

Uniqueness and multiplicity of positive solutions for a diffusive predator–prey model in the heterogeneous environment

2020 ◽  
Vol 150 (6) ◽  
pp. 3321-3348
Author(s):  
Shanbing Li ◽  
Yaying Dong
Keyword(s):  
Positive Solutions ◽  
Steady State Solution ◽  
Asymptotically Stable ◽  
Heterogeneous Environment ◽  
Globally Asymptotically Stable ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
The Stability ◽  
Multiplicity Of Positive Solutions

AbstractThis is the second part of our study on the spatially heterogeneous predator–prey model where the interaction is governed by a Crowley–Martin type functional response. In part I, we have proved that when the predator competition is strong (i.e. k is large), the model has at most one positive steady-state solution for any $\mu \in \mathbb {R}$, moreover it is globally asymptotically stable for any $\mu >0$. This part is denoted to study the effect of saturation. Our result shows that the large saturation coefficient (i.e. large m) can not only lead to the uniqueness of positive solutions, but also lead to the multiplicity of positive solutions, moreover the stability of the corresponding positive solutions is also completely obtained. This work can be regarded as a supplement of Ref. [10].

scholarly journals A Predator-Prey Model in the Chemostat with Holling Type II Response Function

2020 ◽  
Vol 9999 (9999) ◽  
pp. 1-22
Author(s):  
Tedra Bolger ◽  
Brydon Eastman ◽  
Madeleine Hill ◽  
Gail Wolkowicz
Keyword(s):  
Type Ii ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Local Asymptotic Stability ◽  
Prey Model ◽  
Prey Interaction ◽  
Stable Periodic ◽  
Unstable Solutions

A model of predator-prey interaction in a chemostat with Holling Type II functional and numerical response functions ofthe Monod or Michaelis-Menten form is considered. It is proved that local asymptotic stability of the coexistence equilibriumimplies that it is globally asymptotically stable. It is also shown that when the coexistence equilibrium exists but is unstable,solutions converge to a unique, orbitally asymptotically stable periodic orbit. Thus the range of the dynamics of the chemostatpredator-prey model is the same as for the analogous classical Rosenzweig-MacArthur predator-prey model with Holling TypeII functional response. An extension that applies to other functional rsponses is also given.

Global stability of a stochastic predator–prey model with Allee effect

2015 ◽  
Vol 08 (04) ◽  
pp. 1550044 ◽  
Author(s):  
Baodan Tian ◽  
Liu Yang ◽  
Shouming Zhong
Keyword(s):  
Asymptotic Stability ◽  
Global Stability ◽  
Global Asymptotic Stability ◽  
Allee Effect ◽  
Sufficient Conditions ◽  
Initial Value ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Global Positive Solution

In this paper, we study a stochastic predator–prey model with Beddington–DeAngelis functional response and Allee effect, and show that there is a unique global positive solution to the system with the positive initial value. Sufficient conditions for global asymptotic stability are established. Some simulation figures are introduced to support the analytical findings.

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.

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