scholarly journals Stability Property for the Predator-Free Equilibrium Point of Predator-Prey Systems with a Class of Functional Response and Prey Refuges

2012 ◽  
Vol 2012 ◽  
pp. 1-5 ◽  
Author(s):  
Fengde Chen ◽  
Yumin Wu ◽  
Zhaozhi Ma
Keyword(s):  
Equilibrium Point ◽  
Functional Response ◽  
Stability Property ◽  
Analytical Approach ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Predator Prey ◽  
The Stability ◽  
Prey Refuges

We investigate the stability property for the predator-free equilibrium point of predator-prey systems with a class of functional response and prey refuges by using the analytical approach. Under some very weakly assumption, we show that conditions that ensure the locally asymptotically stable of the predator-free equilibrium point are consistent with that of the globally asymptotically stable ones. Our result supplements the corresponding result of Ma et al., 2009.

Dynamical Analysis of Prey Refuge in a Predator-Prey System with Rosenzweig Functional Response

2011 ◽  
Vol 271-273 ◽  
pp. 577-580
Author(s):  
Zhi Hui Ma ◽  
Shu Fan Wang ◽  
Wen Ting Wang
Keyword(s):  
Functional Response ◽  
Stability Property ◽  
Dynamical Analysis ◽  
Prey Refuge ◽  
Predator Prey ◽  
Prey System ◽  
Stabilizing Effect ◽  
The Stability ◽  
Prey Refuges

In this paper, we proposed a predator-prey system incorporating Rosenzweig functional response and prey refuges. We will consider the stability property of the equilibria. Our results show that refuges using by prey have stabilizing effect on the considered system.

Harvesting Analysis of a Predator-Prey System with Functional Response

2013 ◽  
Vol 805-806 ◽  
pp. 1957-1961
Author(s):  
Ting Wu
Keyword(s):  
Numerical Simulation ◽  
Equilibrium Point ◽  
Functional Response ◽  
Sufficient Conditions ◽  
Optimal Harvesting ◽  
Predator Prey ◽  
Optimal Harvesting Policy ◽  
Prey System ◽  
The Stability ◽  
Maximal Principle

In this paper, a predator-prey system with functional response is studied,and a set of sufficient conditions are obtained for the stability of equilibrium point of the system. Moreover, optimal harvesting policy is obtained by using the maximal principle,and numerical simulation is applied to illustrate the correctness.

scholarly journals Globale Stability in a Viral Infection Model with Beddington-DeAngelis Functional Response

2015 ◽  
Vol 9 (1) ◽  
pp. 27-29
Author(s):  
Wang Zhanwei ◽  
He Xia
Keyword(s):  
Lyapunov Function ◽  
Viral Infection ◽  
Functional Response ◽  
Reproduction Number ◽  
Infection Model ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
The Stability ◽  
Viral Infection Model ◽  
The Basic Reproduction Number

The stability of a mathematical model for viral infection with Beddington-DeAngelis functional response is considered in this paper. If the basic reproduction number R ≤1, by the Routh-Hurwitz criterion and Lyapunov function, the uninfected equilibrium E is globally asymptotically stable. Then, the global stability of the infected equilibrium E is obtained by the method of Lyapunov function

scholarly journals Dynamics of Stage-Structured Predator–Prey Model with Beddington–DeAngelis Functional Response and Harvesting

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2169
Author(s):  
Haiyin Li ◽  
Xuhua Cheng
Keyword(s):  
Functional Response ◽  
Local Stability ◽  
Asymptotical Stability ◽  
Positive Equilibrium ◽  
Global Asymptotical Stability ◽  
Globally Asymptotically Stable ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Boundary Equilibrium ◽  
The Stability

In this paper, we investigate the stability of equilibrium in the stage-structured and density-dependent predator–prey system with Beddington–DeAngelis functional response. First, by checking the sign of the real part for eigenvalue, local stability of origin equilibrium and boundary equilibrium are studied. Second, we explore the local stability of the positive equilibrium for τ=0 and τ≠0 (time delay τ is the time taken from immaturity to maturity predator), which shows that local stability of the positive equilibrium is dependent on parameter τ. Third, we qualitatively analyze global asymptotical stability of the positive equilibrium. Based on stability theory of periodic solutions, global asymptotical stability of the positive equilibrium is obtained when τ=0; by constructing Lyapunov functions, we conclude that the positive equilibrium is also globally asymptotically stable when τ≠0. Finally, examples with numerical simulations are given to illustrate the obtained results.

scholarly journals Dynamics Analysis of Modified Leslie-Gower Model with Simplified Holling Type IV Functional Response

2021 ◽  
Vol 18 (1) ◽  
pp. 12-21
Author(s):  
Nur Suci Ramadhani ◽  
Toaha Toaha ◽  
Kasbawati Kasbawati
Keyword(s):  
Population Density ◽  
Equilibrium Point ◽  
Functional Response ◽  
Prey Population ◽  
Dynamics Analysis ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Type Iv ◽  
Simulation Results ◽  
The Stability

In this paper, the modified Leslie-Gower predator-prey model with simplified Holling type IV functional response is discussed. It is assumed that the prey population is a dangerous population. The equilibrium point of the model and the stability of the coexistence equilibrium point are analyzed. The simulation results show that both prey and predator populations will not become extinct as time increases. When the prey population density increases, there is a decrease in the predatory population density because the dangerous prey population has a better ability to defend itself from predators when the number is large enough.

GLOBAL STABILITY AND HOPF BIFURCATION IN A DELAYED DIFFUSIVE LESLIE–GOWER PREDATOR–PREY SYSTEM

2012 ◽  
Vol 22 (03) ◽  
pp. 1250061 ◽  
Author(s):  
SHANSHAN CHEN ◽  
JUNPING SHI ◽  
JUNJIE WEI
Keyword(s):  
Hopf Bifurcation ◽  
Upper And Lower Solutions ◽  
Neumann Boundary ◽  
Sufficient Condition ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Predator Prey ◽  
Prey System ◽  
The Stability ◽  
Coexistence Equilibrium

In this paper, we consider a delayed diffusive Leslie–Gower predator–prey system with homogeneous Neumann boundary conditions. The stability/instability of the coexistence equilibrium and associated Hopf bifurcation are investigated by analyzing the characteristic equations. Furthermore, using the upper and lower solutions method, we give a sufficient condition on parameters so that the coexistence equilibrium is globally asymptotically stable.

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 Dynamics Analysis and Simulation of a Modified HIV Infection Model with a Saturated Infection Rate

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Qilin Sun ◽  
Lequan Min
Keyword(s):  
Drug Resistance ◽  
Antiretroviral Therapy ◽  
Hiv Infection ◽  
Equilibrium Point ◽  
Infection Rate ◽  
Reproductive Number ◽  
Infection Model ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Hiv Rna

This paper studies a modified human immunodeficiency virus (HIV) infection differential equation model with a saturated infection rate. It is proved that if the basic virus reproductive numberR0of the model is less than one, then the infection-free equilibrium point of the model is globally asymptotically stable; ifR0of the model is more than one, then the endemic infection equilibrium point of the model is globally asymptotically stable. Based on the clinical data from HIV drug resistance database of Stanford University, using the proposed model simulates the dynamics of the two groups of patients’ anti-HIV infection treatment. The numerical simulation results are in agreement with the evolutions of the patients’ HIV RNA levels. It can be assumed that if an HIV infected individual’s basic virus reproductive numberR0<1then this person will recover automatically; if an antiretroviral therapy makes an HIV infected individual’sR0<1, this person will be cured eventually; if an antiretroviral therapy fails to suppress an HIV infected individual’s HIV RNA load to be of unpredictable level, the time that the patient’s HIV RNA level has achieved the minimum value may be the starting time that drug resistance has appeared.

scholarly journals Analysis of an Ecoepidemiological Model with Prey Refuges

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Shufan Wang ◽  
Zhihui Ma
Keyword(s):  
Functional Response ◽  
Contact Process ◽  
Global Perspective ◽  
Basic System ◽  
Stability Behavior ◽  
The Stability ◽  
Holling Iii Functional Response ◽  
Disease In Prey ◽  
Prey Refuges

An ecoepidemiological system with prey refuges and disease in prey is proposed. Bilinear incidence and Holling III functional response are used to model the contact process and the predation process, respectively. We will study the stability behavior of the basic system from a local to a global perspective. Permanence of the considered system is also investigated.

scholarly journals Note on the Stability Property of a Cooperative System Incorporating Harvesting

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Xiangdong Xie ◽  
Fengde Chen ◽  
Yalong Xue
Keyword(s):  
Iterative Method ◽  
Equilibrium Point ◽  
Stability Property ◽  
Global Attractivity ◽  
Positive Equilibrium ◽  
Cooperative System ◽  
Cooperative Model ◽  
Interior Equilibrium ◽  
The Stability

The stability of a kind of cooperative model incorporating harvesting is revisited in this paper. By using an iterative method, the global attractivity of the interior equilibrium point of the system is investigated. We show that the condition which ensures the existence of a unique positive equilibrium is enough to ensure the global attractivity of the positive equilibrium. Our results significantly improve the corresponding results of Wei and Li (2013).

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