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.

Dynamical behaviors of a diffusive predator–prey model with Beddington–DeAngelis functional response and disease in the prey

2017 ◽  
Vol 10 (08) ◽  
pp. 1750119 ◽  
Author(s):  
Wensheng Yang
Keyword(s):  
Lyapunov Function ◽  
Iterative Method ◽  
Functional Response ◽  
Local Stability ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Dynamical Behaviors ◽  
Prey Model

The dynamical behaviors of a diffusive predator–prey model with Beddington–DeAngelis functional response and disease in the prey is considered in this work. By applying the comparison principle, linearized method, Lyapunov function and iterative method, we are able to achieve sufficient conditions of the permanence, the local stability and global stability of the boundary equilibria and the positive equilibrium, respectively. Our result complements and supplements some known ones.

Analysis of a Predator–Prey Model with Sparssing Effect

2014 ◽  
Vol 971-973 ◽  
pp. 2234-2237
Author(s):  
Yong Po Zhang ◽  
Ming Juan Ma ◽  
Yue Shuang ◽  
Jia Hui Sun
Keyword(s):  
Sufficient Conditions ◽  
Liapunov Function ◽  
Latent Root ◽  
Asymptotical Stability ◽  
Positive Equilibrium ◽  
Effect Analysis ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Boundary Equilibrium

In this paper we formulated and analyzed a predator-prey model with sparssing effect, analysis of the existing conditions of equilibrium point, and the sufficient condition of the local asymptotical stability of the equilibrium was studied with the method of latent root, and furthermore, by constructing a Liapunov function to get the boundary equilibrium and the positive equilibrium sufficient conditions for the globally asymptotical stability.

scholarly journals Global Stability of a Predator-Prey Model with Ivlev-Type Functional Response

2020 ◽  
Vol 99 (99) ◽  
pp. 1-12
Author(s):  
Yinshu Wu ◽  
Wenzhang Huang
Keyword(s):  
Global Stability ◽  
Functional Response ◽  
Local Stability ◽  
Positive Equilibrium ◽  
Functional Responses ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Predator Prey Models ◽  
Global And Local

A predator-prey model with Ivlev-Type functional response is studied. The main purpose is to investigate the global stability of a positive (co-existence) equilibrium, whenever it exists. A recently developed approach shows that for certain classes of models, there is an implicitly defined function which plays an important rule in determining the global stability of the positive equilibrium. By performing a detailed analytic analysis we demonstrate that a crucial property of this implicitly defined function is governed by the local stability of the positive equilibrium, which enable us to show that the global and local stability of the positive equilibrium, whenever it exists, is equivalent. We believe that our approach can be extended to study the global stability of the positive equilibrium for predator-prey models with some other types of functional responses.

Dynamics of a diffusive predator–prey system with ratio-dependent functional response and time delay

2020 ◽  
Vol 13 (06) ◽  
pp. 2050036
Author(s):  
Xin Jiang ◽  
Ran Zhang ◽  
Zhikun She
Keyword(s):  
Time Delay ◽  
Asymptotical Stability ◽  
Positive Equilibrium ◽  
Global Asymptotical Stability ◽  
Diffusion Effect ◽  
Predator Prey ◽  
Prey System ◽  
Boundary Equilibrium ◽  
Ratio Dependent ◽  
And Diffusion

In this paper, we investigate the qualitative behaviors of a predator–prey system with ratio-dependent function. The system accommodates the diffusion effect to model the migration of individuals and the time delay induced by reproduction. We start with some basic properties of the system. Then the sufficient condition independent of time delay and diffusion effect for global asymptotical stability of the boundary equilibrium is obtained by using the comparison principle. Afterwards, based on the LaSalle’s invariance principle and Lyapunov functional, we investigate the global attractiveness of the positive equilibrium, arriving at its global asymptotical stability. Further, Hopf bifurcation induced by time delay around the positive equilibrium is explored. Finally, numerical examples are listed to verify the corresponding analytical results.

scholarly journals Stability and Hopf Bifurcation of a Predator-Prey Model with Distributed Delays and Competition Term

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Lv-Zhou Zheng
Keyword(s):  
Hopf Bifurcation ◽  
Local Stability ◽  
Sufficient Conditions ◽  
Distributed Delays ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Positive Equilibrium Point ◽  
The Stability

A class of predator-prey system with distributed delays and competition term is considered. By considering the time delay as bifurcation parameter, we analyze the stability and the Hopf bifurcation of the predator-prey system. According to the theorem of Hopf bifurcation, some sufficient conditions are obtained for the local stability of the positive equilibrium point.

scholarly journals Dynamic Analysis of a Delayed Reaction-Diffusion Predator-Prey System with Modified Holling-Tanner Functional Response

2015 ◽  
Vol 2015 ◽  
pp. 1-12
Author(s):  
Xinhong Pan ◽  
Min Zhao ◽  
Chuanjun Dai ◽  
Yapei Wang
Keyword(s):  
Hopf Bifurcation ◽  
Functional Response ◽  
Reaction Diffusion ◽  
Positive Equilibrium ◽  
Delayed Reaction ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Predator Prey Model ◽  
The Stability

A predator-prey model with modified Holling-Tanner functional response and time delays is considered. By regarding the delays as bifurcation parameters, the local and global asymptotic stability of the positive equilibrium are investigated. The system has been found to undergo a Hopf bifurcation at the positive equilibrium when the delays cross through a sequence of critical values. In addition, the direction of the Hopf bifurcation and the stability of bifurcated periodic solutions are also studied, and an explicit algorithm is obtained by applying normal form theory and the center manifold theorem. The main results are illustrated by numerical simulations.

Stability Analysis in a Delayed Predator-Prey Model

2012 ◽  
Vol 472-475 ◽  
pp. 2940-2943
Author(s):  
Zhi Chao Jiang ◽  
Hui Chen
Keyword(s):  
Stability Analysis ◽  
Hopf Bifurcation ◽  
Time Delay ◽  
Local Stability ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Boundary Equilibrium ◽  
System With Time Delay

A stage-structured predator-prey system with time delay is considered. By analyzing the characteristic equations, the local stability of a positive equilibrium and a boundary equilibrium is discussed, respectively. Furthermore, it is proved that the system undergoes a Hopf bifurcation at the positive equilibrium when . The estimation of the length of delay to preserve stability has also been calculated.

Persistence, Stability and Hopf Bifurcation in a Diffusive Ratio-Dependent Predator–Prey Model with Delay

2014 ◽  
Vol 24 (07) ◽  
pp. 1450093 ◽  
Author(s):  
Yongli Song ◽  
Yahong Peng ◽  
Xingfu Zou
Keyword(s):  
Hopf Bifurcation ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Boundary Equilibrium ◽  
Ratio Dependent ◽  
The Stability ◽  
Theoretical Results

In this paper, we study the persistence, stability and Hopf bifurcation in a ratio-dependent predator–prey model with diffusion and delay. Sufficient conditions independent of diffusion and delay are obtained for the persistence of the system and global stability of the boundary equilibrium. The local stability of the positive constant equilibrium and delay-induced Hopf bifurcation are investigated by analyzing the corresponding characteristic equation. We show that delay can destabilize the positive equilibrium and induce spatially homogeneous and inhomogeneous periodic solutions. By calculating the normal form on the center manifold, the formulae determining the direction and the stability of Hopf bifurcations are explicitly derived. The numerical simulations are carried out to illustrate and extend our theoretical results.

Sign in / Sign up

Export Citation Format

Share Document