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 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 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.

Periodic Solutions for a Predator–Prey Model with Periodic Harvesting Rate

2014 ◽  
Vol 24 (07) ◽  
pp. 1450096 ◽  
Author(s):  
Yilei Tang ◽  
Dongmei Xiao
Keyword(s):  
Periodic Solution ◽  
Periodic Solutions ◽  
Integral Manifold ◽  
Polar Coordinates ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Prey Model ◽  
Existence Of Periodic Solutions

In the paper, we study the existence of periodic solutions in an invariant manifold for a predator–prey model with periodic harvesting rate. Without harvesting, the predator–prey system has a positive equilibrium of center type, which undergoes Hopf bifurcation. Under the periodic harvesting, we first reduce this predator–prey system to a complex normal form. Then, we define the Poincaré mapping of this system by using the polar coordinates, and prove that there exists an integral manifold and periodic solution with the period of the harvesting rate. An example is provided to illustrate the result.

Dynamics of a non-autonomous density-dependent predator–prey model with Beddington–DeAngelis type

2016 ◽  
Vol 09 (04) ◽  
pp. 1650050 ◽  
Author(s):  
Haiyin Li ◽  
Zhikun She
Keyword(s):  
Density Dependence ◽  
Periodic Solutions ◽  
Sufficient Condition ◽  
Positive Periodic Solutions ◽  
Predator Density ◽  
Predator Prey ◽  
Density Dependent ◽  
Predator Prey Model ◽  
Prey System ◽  
Biological Environment

The goal of this paper is to investigate the dynamics of a non-autonomous density-dependent predator–prey system with Beddington–DeAngelis functional response, where not only the prey density dependence but also the predator density dependence are considered, such that the studied predator–prey system conforms to the realistically biological environment. We firstly introduce a sufficient condition for the permanence of the system and then use a specific set to obtain a weaker sufficient condition. Afterward, we provide corresponding conditions for the extinction of the system and the existence of boundary periodical solutions, respectively. Further, we get a sufficient condition for global attractiveness of the boundary periodic solution by constructing a Lyapunov function, arriving at the uniqueness of boundary periodic solutions since the uniqueness of boundary periodic solutions can be ensured by global attractiveness. Finally, based on the existence of positive periodic solutions, which can be ensured by the Brouwer fixed-point theorem, we provide a sufficient condition for the uniqueness of positive periodic solutions.

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.

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.

Stability and Bifurcation in a State-Dependent Delayed Predator–Prey System

2016 ◽  
Vol 26 (04) ◽  
pp. 1650060 ◽  
Author(s):  
Aiyu Hou ◽  
Shangjiang Guo
Keyword(s):  
Hopf Bifurcation ◽  
Numerical Simulations ◽  
Local Stability ◽  
Perturbation Methods ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Prey System ◽  
State Dependent ◽  
The Stability ◽  
Theoretical Results

In this paper, we consider a class of predator–prey equations with state-dependent delayed feedback. Firstly, we investigate the local stability of the positive equilibrium and the existence of the Hopf bifurcation. Then we use perturbation methods to determine the sub/supercriticality of Hopf bifurcation and hence the stability of Hopf bifurcating periodic solutions. Finally, numerical simulations supporting our theoretical results are also provided.

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.

scholarly journals Stability and Bifurcation in a Predator–Prey Model with the Additive Allee Effect and the Fear Effect

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1280
Author(s):  
Liyun Lai ◽  
Zhenliang Zhu ◽  
Fengde Chen
Keyword(s):  
Allee Effect ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Predator Species ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Fear Effect ◽  
Prey Model ◽  
Strong Allee Effect ◽  
Theoretical Results

We proposed and analyzed a predator–prey model with both the additive Allee effect and the fear effect in the prey. Firstly, we studied the existence and local stability of equilibria. Some sufficient conditions on the global stability of the positive equilibrium were established by applying the Dulac theorem. Those results indicate that some bifurcations occur. We then confirmed the occurrence of saddle-node bifurcation, transcritical bifurcation, and Hopf bifurcation. Those theoretical results were demonstrated with numerical simulations. In the bifurcation analysis, we only considered the effect of the strong Allee effect. Finally, we found that the stronger the fear effect, the smaller the density of predator species. However, the fear effect has no influence on the final density of the prey.

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