scholarly journals Analysis of a delay nonautonomous predator-prey system with disease in the prey

2010 ◽  
Vol 15 (1) ◽  
pp. 97-108 ◽  
Author(s):  
G. P. Samanta
Keyword(s):  
Time Delay ◽  
Asymptotic Stability ◽  
Global Asymptotic Stability ◽  
Sufficient Conditions ◽  
Functional Method ◽  
Policy Makers ◽  
Predator Prey ◽  
Analytical Technique ◽  
Predator Prey Model ◽  
Lyapunov Functional Method

In this paper we have considered a nonautonomous predator-prey model with time delay due to gestation, in which a disease that can be transmitted by contact spreads among the prey only. Here, we have established some sufficient conditions on the permanence of the system by using inequality analytical technique. By Lyapunov functional method, we have also obtained some sufficient conditions for global asymptotic stability of this model. We have observed that the time delay has no effect on the permanence of the system but it has an effect on the global asymptotic stability of this model. The aim of the analysis of this model is to identify the parameters of interest for further study, with a view to informing and assisting policy-makers in targeting prevention and treatment resources for maximum effectiveness.

scholarly journals Permanence and Extinction for a Nonautonomous Malaria Transmission Model with Distributed Time Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-15
Author(s):  
Xiaohong Zhang ◽  
Jianwen Jia ◽  
Xinyu Song
Keyword(s):  
Time Delay ◽  
Asymptotic Stability ◽  
Malaria Transmission ◽  
Global Asymptotic Stability ◽  
Sufficient Conditions ◽  
Analytical Techniques ◽  
Functional Method ◽  
Transmission Model ◽  
Distributed Time Delay ◽  
Lyapunov Functional Method

We study the permanence, extinction, and global asymptotic stability for a nonautonomous malaria transmission model with distributed time delay. We establish some sufficient conditions on the permanence and extinction of the disease by using inequality analytical techniques. By a Lyapunov functional method, we also obtain some sufficient conditions for global asymptotic stability of this model. A numerical analysis is given to explain the analytical findings.

PERMANENCE AND EXTINCTION OF A NONAUTONOMOUS STAGE-STRUCTURED EPIDEMIC MODEL WITH DISTRIBUTED TIME DELAY

2010 ◽  
Vol 18 (02) ◽  
pp. 377-398
Author(s):  
G. P. SAMANTA
Keyword(s):  
Time Delay ◽  
Epidemic Model ◽  
Sufficient Conditions ◽  
Functional Method ◽  
Analytical Technique ◽  
Distributed Time Delay ◽  
Lyapunov Functional Method ◽  
Varying Total Population Size ◽  
Model Computer ◽  
Two Stages

In this paper, we have considered a nonautonomous stage-structured epidemic model having two stages of the period of infection according to the progressing process of some infectious diseases (e.g. Chagas' disease, hepatitis C, etc.) with varying total population size and distributed time delay to become infectious. The infected persons in the different stages have different ability of transmitting disease. We have established some sufficient conditions on the permanence and extinction of the disease by using inequality analytical technique. We have obtained the explicit formula of the eventual lower bounds of infected persons. We have introduced some new threshold values R0 and R* and further obtained that the disease will be permanent when R0 > 1 and the disease will be going to extinct when R* < 1. By Lyapunov functional method, we have also obtained some sufficient conditions for global asymptotic stability of this model. Computer simulations are carried out to explain the analytical findings. The aim of the analysis of this model is to identify the parameters of interest for further study, with a view to informing and assisting policy-maker in targeting prevention and treatment resources for maximum effectiveness.

scholarly journals Global Asymptotic Stability of a Predator-Prey Model with Modified Leslie-Gower and Holling-Type II Schemes

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Shengbin Yu
Keyword(s):  
Asymptotic Stability ◽  
Global Asymptotic Stability ◽  
Direct Method ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Linearized Stability ◽  
Appl Math

We study the predator-prey model proposed by Aziz-Alaoui and Okiye (Appl. Math. Lett. 16 (2003) 1069–1075) First, the structure of equilibria and their linearized stability is investigated. Then, we provide two sufficient conditions on the global asymptotic stability of a positive equilibrium by employing the Fluctuation Lemma and Lyapunov direct method, respectively. The obtained results not only improve but also supplement existing ones.

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 Dynamics in a Lotka-Volterra Predator-Prey Model with Time-Varying Delays

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Changjin Xu ◽  
Yusen Wu
Keyword(s):  
Asymptotic Stability ◽  
Numerical Simulations ◽  
Global Asymptotic Stability ◽  
Differential Inequality ◽  
Sufficient Conditions ◽  
Time Varying ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Inequality Theory

A Lotka-Volterra predator-prey model with time-varying delays is investigated. By using the differential inequality theory, some sufficient conditions which ensure the permanence and global asymptotic stability of the system are established. The paper ends with some interesting numerical simulations that illustrate our analytical predictions.

scholarly journals Persistence and Stability for a Generalized Leslie-Gower Model with Stage Structure and Dispersal

2009 ◽  
Vol 2009 ◽  
pp. 1-17 ◽  
Author(s):  
Hai-Feng Huo ◽  
Zhan-Ping Ma ◽  
Chun-Ying Liu
Keyword(s):  
Asymptotic Stability ◽  
Numerical Simulations ◽  
Global Asymptotic Stability ◽  
Lyapunov Functional ◽  
Sufficient Conditions ◽  
Stage Structure ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Boundedness Of Solution

A generalized version of the Leslie-Gower predator-prey model that incorporates the prey structure and predator dispersal in two-patch environments is introduced. The focus is on the study of the boundedness of solution, permanence, and extinction of the model. Sufficient conditions for global asymptotic stability of the positive equilibrium are derived by constructing a Lyapunov functional. Numerical simulations are also presented to illustrate our main results.

scholarly journals Hopf bifurcation and stability in a Beddington-DeAngelis predator-prey model with stage structure for predator and time delay incorporating prey refuge

2019 ◽  
Vol 17 (1) ◽  
pp. 141-159 ◽  
Author(s):  
Zaowang Xiao ◽  
Zhong Li ◽  
Zhenliang Zhu ◽  
Fengde Chen
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Sufficient Conditions ◽  
Stage Structure ◽  
Prey Refuge ◽  
Predator Species ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Bifurcation And Stability

Abstract In this paper, we consider a Beddington-DeAngelis predator-prey system with stage structure for predator and time delay incorporating prey refuge. By analyzing the characteristic equations, we study the local stability of the equilibrium of the system. Using the delay as a bifurcation parameter, the model undergoes a Hopf bifurcation at the coexistence equilibrium when the delay crosses some critical values. After that, by constructing a suitable Lyapunov functional, sufficient conditions are derived for the global stability of the system. Finally, the influence of prey refuge on densities of prey species and predator species is discussed.

scholarly journals Uniformly Strong Persistence for a Delayed Predator-Prey Model

2012 ◽  
Vol 2012 ◽  
pp. 1-7 ◽  
Author(s):  
Changjin Xu ◽  
Yuanfu Shao ◽  
Peiluan Li
Keyword(s):  
Time Delay ◽  
Sufficient Conditions ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Asymptotically Periodic

An asymptotically periodic predator-prey model with time delay is investigated. Some sufficient conditions for the uniformly strong persistence of the system are obtained. Our result is an important complementarity to the earlier results.

GLOBAL ASYMPTOTIC STABILITY FOR A TWO-SPECIES DISCRETE RATIO-DEPENDENT PREDATOR–PREY SYSTEM

2013 ◽  
Vol 06 (01) ◽  
pp. 1250064 ◽  
Author(s):  
XIANGLAI ZHUO
Keyword(s):  
Asymptotic Stability ◽  
Global Stability ◽  
Global Asymptotic Stability ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Prey System ◽  
Ratio Dependent ◽  
Analysis Of Stability ◽  
Appl Math

The dynamical behaviors of a two-species discrete ratio-dependent predator–prey system are considered. Some sufficient conditions for the local stability of the equilibria is obtained by using the linearization method. Further, we also obtain a new sufficient condition to ensure that the positive equilibrium is globally asymptotically stable by using an iteration scheme and the comparison principle of difference equations, which generalizes what paper [G. Chen, Z. Teng and Z. Hu, Analysis of stability for a discrete ratio-dependent predator–prey system, Indian J. Pure Appl. Math.42(1) (2011) 1–26] has done. The method given in this paper is new and very resultful comparing with papers [H. F. Huo and W. T. Li, Existence and global stability of periodic solutions of a discrete predator–prey system with delays, Appl. Math. Comput.153 (2004) 337–351; X. Liao, S. Zhou and Y. Chen, On permanence and global stability in a general Gilpin–Ayala competition predator–prey discrete system, Appl. Math. Comput.190 (2007) 500–509] and it can also be applied to study the global asymptotic stability for general multiple species discrete population systems. At the end of this paper, we present an open question.

scholarly journals A new study for the global asymptotic stability of a general predator-prey model

2021 ◽  
Author(s):  
Manh Tuan Hoang
Keyword(s):  
Asymptotic Stability ◽  
Functional Response ◽  
Mathematical Analysis ◽  
Global Asymptotic Stability ◽  
Lyapunov Stability Theory ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Prey Model ◽  
Theoretical Results

In a previous paper [L. M. Ladino, E. I. Sabogal, Jose C. Valverde, General functional response and recruitment in a predator-prey system with capture on both species, Math. Methods Appl. Sci. 38(2015) 2876-2887], a mathematical model for a predator-prey model with general functional response and recruitment including capture on both species was formulated and analyzed. However, the global asymptotic stability (GAS) of this model was only partially resolved. In the present paper, we provide a rigorously mathematical analysis for the complete GAS of the predator-prey model. By using the Lyapunov stability theory in combination with some nonstandard techniques of mathematical analysis for dynamical systems, the GAS of equilibria of the model is determined fully. The obtained results not only provide an important improvement for the population dynamics of the predator-prey model but also can be extended to study its modified versions in the context of fractional-order derivatives. The theoretical results are supported and illustrated by a set of numerical examples.

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