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 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 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 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.

Analysis on stochastic predator-prey model with distributed delay

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
C. Gokila ◽  
M. Sambath
Keyword(s):  
Lyapunov Function ◽  
Numerical Simulations ◽  
Stationary Distribution ◽  
Sufficient Conditions ◽  
Distributed Delay ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Disease In Prey ◽  
Stochastic Lyapunov Function

Abstract In the present work, we consider a stochastic predator-prey model with disease in prey and distributed delay. Firstly, we establish sufficient conditions for the extinction of the disease and also permanence of healthy prey and predator. Besides, we obtain the condition for the existence of an ergodic stationary distribution through the stochastic Lyapunov function. Finally, we provide some numerical simulations to validate our theoretical findings.

scholarly journals Global stability of solutions in a reaction-diffusion system of predator-prey model

Filomat ◽  
2018 ◽  
Vol 32 (13) ◽  
pp. 4665-4672
Author(s):  
Demou Luo ◽  
Hailin Liu
Keyword(s):  
Asymptotic Stability ◽  
Global Asymptotic Stability ◽  
Equilibrium Solution ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model

In this article, we investigate the global asymptotic stability of a reaction-diffusion system of predator-prey model. By applying the comparison principle and iteration method, we prove the global asymptotic stability of the unique positive equilibrium solution of (1.1).

Globally asymptotic stability of a predator–prey model with stage structure incorporating prey refuge

2016 ◽  
Vol 09 (04) ◽  
pp. 1650058 ◽  
Author(s):  
Fengying Wei ◽  
Qiuyue Fu
Keyword(s):  
Asymptotic Stability ◽  
Sufficient Conditions ◽  
Stage Structure ◽  
Characteristic Functions ◽  
Prey Refuge ◽  
Predator Species ◽  
Predator Prey ◽  
Globally Asymptotic Stability ◽  
Predator Prey Model ◽  
Prey Model

This paper focuses on the stabilities of the equilibria to a predator–prey model with stage structure incorporating prey refuge. By analyzing the characteristic functions, we obtain that the equilibria of the model are locally stable when some suitable conditions are being satisfied. According to the comparison theorem and iteration technique, the globally asymptotic stability of the positive equilibrium is discussed. And, the sufficient conditions of the global stability to the trivial equilibrium and the boundary equilibrium are derived. The study shows that the prey refuge will enhance the density of the prey species, and it will decrease the density of predator species. Finally, some numerical simulations are carried out to show the efficiency of our main results.

A Markov-switching predator–prey model with Allee effect for preys

2020 ◽  
Vol 13 (03) ◽  
pp. 2050018
Author(s):  
Xiaoxia Guo ◽  
Zhiming Guo
Keyword(s):  
Numerical Simulations ◽  
Allee Effect ◽  
Global Attractivity ◽  
Markov Switching ◽  
Sufficient Conditions ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Theoretical Results ◽  
Prey Populations

This paper concerns with a Markov-switching predator–prey model with Allee effect for preys. The conditions under which extinction of predator and prey populations occur have been established. Sufficient conditions are also given for persistence and global attractivity in mean. In addition, stability in the distribution of the system under consideration is derived under some assumptions. Finally, numerical simulations are carried out to illustrate theoretical results.

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