scholarly journals Global Stability for a Delayed Predator-Prey System with Stage Structure for the Predator

2009 ◽  
Vol 2009 ◽  
pp. 1-24
Author(s):  
Xiao Zhang ◽  
Rui Xu ◽  
Qintao Gan
Keyword(s):  
Global Stability ◽  
Local Stability ◽  
Sufficient Conditions ◽  
Stage Structure ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Prey System ◽  
Comparison Argument ◽  
Theoretical Results ◽  
Iteration Technique

A delayed predator-prey system with stage structure for the predator is investigated. By analyzing the corresponding characteristic equations, the local stability of equilibria of the system is discussed. The existence of Hopf bifurcation at the positive equilibrium is established. By using an iteration technique and comparison argument, respectively, sufficient conditions are derived for the global stability of the positive equilibrium and two boundary equilibria of the system. Numerical simulations are carried out to illustrate the theoretical results.

scholarly journals Hopf Bifurcation of a Predator-Prey System with Delays and Stage Structure for the Prey

2012 ◽  
Vol 2012 ◽  
pp. 1-28 ◽  
Author(s):  
Zizhen Zhang ◽  
Huizhong Yang
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Local Stability ◽  
Sufficient Conditions ◽  
Stage Structure ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Prey System ◽  
The Stability

This paper is concerned with a Holling type III predator-prey system with stage structure for the prey population and two time delays. The main result is given in terms of local stability and bifurcation. By choosing the time delay as a bifurcation parameter, sufficient conditions for the local stability of the positive equilibrium and the existence of periodic solutions via Hopf bifurcation with respect to both delays are obtained. In particular, explicit formulas that can determine the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are established by using the normal form method and center manifold theorem. Finally, numerical simulations supporting the theoretical analysis are also included.

scholarly journals A Stochastic Predator-Prey System with Stage Structure for Predator

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Shufen Zhao ◽  
Minghui Song
Keyword(s):  
Global Existence ◽  
Global Stability ◽  
Numerical Simulations ◽  
Functional Response ◽  
Sufficient Conditions ◽  
Stage Structure ◽  
Predator Prey ◽  
Stability In Probability ◽  
Prey System ◽  
Theoretical Results

The authors introduce stochasticity into a predator-prey system with Beddington-DeAngelis functional response and stage structure for predator. We present the global existence and positivity of the solution and give sufficient conditions for the global stability in probability of the system. Numerical simulations are introduced to support the main theoretical results.

scholarly journals Stability and Bifurcation Analysis on an Ecoepidemiological Model with Stage Structure and Time Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Lingshu Wang ◽  
Guanghui Feng
Keyword(s):  
Time Delay ◽  
Lyapunov Functions ◽  
Local Stability ◽  
Sufficient Conditions ◽  
Stage Structure ◽  
Positive Equilibrium ◽  
Prey Refuge ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Iteration Technique

An ecoepidemiological predator-prey model with stage structure for the predator and time delay due to the gestation of the predator is investigated. The effects of a prey refuge with disease in the prey population are concerned. By analyzing the corresponding characteristic equations, the local stability of each of the feasible equilibria of the model is discussed. Further, it is proved that the model undergoes a Hopf bifurcation at the positive equilibrium. By means of appropriate Lyapunov functions and LaSalle’s invariance principle, sufficient conditions are obtained for the global stability of the semitrivial boundary equilibria. By using an iteration technique, sufficient conditions are derived for the global attractiveness of the positive equilibrium.

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 Bifurcation on diffusive Holling–Tanner predator–prey model with stoichiometric density dependence

2020 ◽  
Vol 25 (2) ◽  
Author(s):  
Maruthai Selvaraj Surendar ◽  
Muniyagounder Sambath ◽  
Krishnan Balachandran
Keyword(s):  
Density Dependence ◽  
Periodic Solutions ◽  
Local Stability ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Spatially Homogeneous ◽  
Bifurcation And Stability ◽  
Theoretical Results

This paper studies a diffusive Holling–Tanner predator–prey system with stoichiometric density dependence. The local stability of positive equilibrium, the existence of Hopf bifurcation and stability of bifurcating periodic solutions have been obtained in the absence of diffusion. We also study the spatially homogeneous and nonhomogeneous periodic solutions through all parameters of the system, which are spatially homogeneous. In order to verify our theoretical results, some numerical simulations are carried out. 

scholarly journals Hopf bifurcation analysis in a stage-structure predator–prey model with two time delay

2022 ◽  
Vol 355 ◽  
pp. 03048
Author(s):  
Bochen Han ◽  
Shengming Yang ◽  
Guangping Zeng
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Local Stability ◽  
Time Delays ◽  
Stage Structure ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Predator Model

In this paper, we consider a predator-prey system with two time delays, which describes a prey–predator model with parental care for predators. The local stability of the positive equilibrium is analysed. By choosing the two time delays as the bifurcation parameter, the existence of Hopf bifurcation is studied. Numerical simulations show the positive equilibrium loses its stability via the Hopf bifurcation when the time delay increases beyond a threshold.

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 Persistence and global stability in a delayed predator-prey system with Holling-type functional response

2004 ◽  
Vol 46 (1) ◽  
pp. 121-141 ◽  
Author(s):  
Rui Xu ◽  
Lansun Chen ◽  
M. A. J. Chaplain
Keyword(s):  
Asymptotic Stability ◽  
Global Stability ◽  
Functional Response ◽  
Global Asymptotic Stability ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Lyapunov Functionals ◽  
Predator Prey ◽  
Prey System ◽  
Holling Type Functional Response

AbstractA delayed predator-prey system with Holling type III functional response is investigated. It is proved that the system is uniformly persistent under some appropriate conditions. By means of suitable Lyapunov functionals, sufficient conditions are derived for the local and global asymptotic stability of a positive equilibrium of the system. Numerical simulations are presented to illustrate the feasibility of our main results.

scholarly journals Global Stability of an Eco-Epidemiological Model with Time Delay and Saturation Incidence

2011 ◽  
Vol 2011 ◽  
pp. 1-22 ◽  
Author(s):  
Shuxue Mao ◽  
Rui Xu ◽  
Zhe Li ◽  
Yunfei Li
Keyword(s):  
Global Stability ◽  
Local Stability ◽  
Sufficient Conditions ◽  
Epidemiological Model ◽  
Positive Equilibrium ◽  
Numerical Examples ◽  
Disease In Predator ◽  
Comparison Arguments ◽  
Disease Free ◽  
Iteration Technique

We investigate a delayed eco-epidemiological model with disease in predator and saturation incidence. First, by comparison arguments, the permanence of the model is discussed. Then, we study the local stability of each equilibrium of the model by analyzing the corresponding characteristic equations and find that Hopf bifurcation occurs when the delayτpasses through a sequence of critical values. Next, by means of an iteration technique, sufficient conditions are derived for the global stability of the disease-free planar equilibrium and the positive equilibrium. Numerical examples are carried out to illustrate the analytical results.

Stability and Hopf Bifurcation in Delayed Predator-Prey System with Ratio Dependent

2014 ◽  
Vol 687-691 ◽  
pp. 655-660
Author(s):  
Yuan Yuan
Keyword(s):  
Hopf Bifurcation ◽  
Sufficient Conditions ◽  
Stage Structure ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Characteristic Roots ◽  
Comparison Arguments ◽  
Ratio Dependent ◽  
Iteration Technique

A delayed predator-prey model with stage structure for predator and ratio dependent response function is considered. By calculating characteristic equations and analyzing characteristic roots, the sufficient conditions for local stability of all the equilibria and Hopf bifurcation are obtained. Moreover, We use an iteration technique and comparison arguments to derive the sufficient conditions of the global stability of the boundary and positive equilibrium.

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