scholarly journals Bifurcation Analysis in Population Genetics Model with Partial Selfing

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Yingying Jiang ◽  
Wendi Wang
Keyword(s):  
Center Manifold ◽  
Evolutionary Game ◽  
Period Doubling ◽  
Period Doubling Bifurcation ◽  
Expected Payoff ◽  
Center Manifold Theorem ◽  
Population Evolution ◽  
Dynamical Behaviors ◽  
Period 2 ◽  
Dynamics Of Population

A new model which allows both the effect of partial selfing selection and an exponential function of the expected payoff is considered. This combines ideas from genetics and evolutionary game theory. The aim of this work is to study the effects of partial selfing selection on the discrete dynamics of population evolution. It is shown that the system undergoes period doubling bifurcation, saddle-node bifurcation, and Neimark-Sacker bifurcation by using center manifold theorem and bifurcation theory. Numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, such as the period-3, 6 orbits, cascade of period-doubling bifurcation in period-2, 4, 8, and the chaotic sets. These results reveal richer dynamics of the discrete model compared with the model in Tao et al., 1999. The analysis and results in this paper are interesting in mathematics and biology.

scholarly journals Bifurcations, Complex Behaviors, and Dynamic Transition in a Coupled Network of Discrete Predator-Prey System

2019 ◽  
Vol 2019 ◽  
pp. 1-22 ◽  
Author(s):  
Tousheng Huang ◽  
Huayong Zhang ◽  
Shengnan Ma ◽  
Ge Pan ◽  
Zhaodeng Wang ◽  
...  
Keyword(s):  
Center Manifold ◽  
Period Doubling ◽  
Flip Bifurcation ◽  
Closed Curves ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Prey System ◽  
Period 2 ◽  
New Perspective ◽  
Matrix Center

The nonlinear dynamics of predator-prey systems coupled into network is an important issue in recent biological advances. In this research, we consider each node of the coupled network represents a discrete predator-prey system, and the network dynamics is investigated. By applying Jacobian matrix, center manifold theorem and bifurcation theorems, stability of fixed points, flip bifurcation and Neimark-Sacker bifurcation of the discrete predator-prey system are analyzed. Via the method of Lyapunov exponents, the nonchaos-chaos transition of the coupled network along the routes to chaos induced by bifurcations is determined. Numerical simulations are performed to demonstrate the bifurcations, various attractors and dynamic transitions of the coupled network. Via comparison, we find that the coupled network exhibits far richer and more complex behaviors than single predator-prey system, including period-doubling cascades in orbits of period-2, period-4, period-8, invariant closed curves, dynamic windows for periodic orbits and invariant curves, quasiperiodic orbits, tori, and chaotic sets. Moreover, the attractors of the coupled network show more diverse and complicated structures. These results may provide a new perspective on the predator-prey dynamics in complex networks.

scholarly journals Hopf Bifurcation, Hopf-Hopf Bifurcation, and Period-Doubling Bifurcation in a Four-Species Food Web

2018 ◽  
Vol 2018 ◽  
pp. 1-21
Author(s):  
Huayong Zhang ◽  
Ju Kang ◽  
Tousheng Huang ◽  
Xuebing Cong ◽  
Shengnan Ma ◽  
...  
Keyword(s):  
Hopf Bifurcation ◽  
Food Web ◽  
Complex Dynamics ◽  
Period Doubling ◽  
Period Doubling Bifurcation ◽  
Food Web Structure ◽  
Center Manifold Theorem ◽  
Dynamical Complexity ◽  
Period 2 ◽  
The Stability

Complex dynamics of a four-species food web with two preys, one middle predator, and one top predator are investigated. Via the method of Jacobian matrix, the stability of coexisting equilibrium for all populations is determined. Based on this equilibrium, three bifurcations, i.e., Hopf bifurcation, Hopf-Hopf bifurcation, and period-doubling bifurcation, are analyzed by center manifold theorem, bifurcation theorem, and numerical simulations. We reveal that, influenced by the three bifurcations, the food web can exhibit very complex dynamical behaviors, including limit cycles, quasiperiodic behaviors, chaotic attractors, route to chaos, period-doubling cascade in orbits of period 2, 4, and 8 and period 3, 6, and 12, periodic windows, intermittent period, and chaos crisis. However, the complex dynamics may disappear with the extinction of one of the four populations, which may also lead to collapse of the food web. It suggests that the dynamical complexity and food web stability are determined by the food web structure and existing populations.

Period-1 to Period-8 Motions in a Nonlinear Jeffcott Rotor System

2020 ◽  
Author(s):  
Yeyin Xu ◽  
Albert C.J. Luo
Keyword(s):  
Period Doubling ◽  
Pitchfork Bifurcation ◽  
Mapping Method ◽  
Rotor System ◽  
Period Doubling Bifurcation ◽  
Periodic Motions ◽  
Bifurcation Tree ◽  
Rotor Systems ◽  
Period 2 ◽  
Jeffcott Rotor

Abstract In this paper, a bifurcation tree of period-1 to period-8 motions in a nonlinear Jeffcott rotor system is obtained through the discrete mapping method. The bifurcations and stability of periodic motions on the bifurcation tree are discussed. The quasi-periodic motions on the bifurcation tree are caused by two (2) Neimark bifurcations of period-1 motions, one (1) Neimark bifurcation of period-2 motions and four (4) Neimark bifurcations of period-4 motions. The specific quasi-periodic motions are mainly based on the skeleton of the corresponding periodic motions. One stable and one unstable period-doubling bifurcations exist for the period-1, period-2 and period-4 motions. The unstable period-doubling bifurcation is from an unstable period-m motion to an unstable period-2m motion, and the unstable period-m motion becomes stable. Such an unstable period-doubling bifurcation is the 3rd source pitchfork bifurcation. Periodic motions on the bifurcation tree are simulated numerically, and the corresponding harmonic amplitudes and phases are presented for harmonic effects on periodic motions in the nonlinear Jeffcott rotor system. Such a study gives a complete picture of periodic and quasi-periodic motions in the nonlinear Jeffcott rotor system in the specific parameter range. One can follow the similar procedure to work out the other bifurcation trees in the nonlinear Jeffcott rotor systems.

Complex dynamic behaviors of a discrete predator–prey model with stage structure and harvesting

2016 ◽  
Vol 10 (01) ◽  
pp. 1750013 ◽  
Author(s):  
Boshan Chen ◽  
Jiejie Chen
Keyword(s):  
Numerical Simulations ◽  
Period Doubling ◽  
Stage Structure ◽  
Model Parameters ◽  
Flip Bifurcation ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Predator Prey Model ◽  
Dynamical Behaviors ◽  
Prey Model

First, a discrete stage-structured and harvested predator–prey model is established, which is based on a predator–prey model with Type III functional response. Then theoretical methods are used to investigate existence of equilibria and their local properties. Third, it is shown that the system undergoes flip bifurcation and Neimark–Sacker bifurcation in the interior of [Formula: see text], by using the normal form of discrete systems, the center manifold theorem and the bifurcation theory, as varying the model parameters in some range. In particular, the direction and the stability of the flip bifurcation and the Neimark–Sacker bifurcation are showed. Finally, numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, such as cascades of period-doubling bifurcation and chaotic sets. These results reveal far richer dynamics of the discrete model compared with the continuous model. The Lyapunov exponents are numerically computed to confirm further the complexity of the dynamical behaviors. In addition, we show also the stabilizing effect of the harvesting by using numerical simulations.

scholarly journals Dynamic complexity and bifurcation analysis of a host–parasitoid model with Allee effect and Holling type III functional response

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Hua Liu ◽  
Kai Zhang ◽  
Yong Ye ◽  
Yumei Wei ◽  
Ming Ma
Keyword(s):  
Local Stability ◽  
Bifurcation Theory ◽  
Allee Effect ◽  
Center Manifold ◽  
Discrete Time System ◽  
Time System ◽  
Dynamic Complexity ◽  
Center Manifold Theorem ◽  
Dynamical Behaviors ◽  
Local Stability Analysis

AbstractIn this paper, we focus on dynamics in a basic discrete-time system of host–parasitoid interaction. We perform local stability analysis of this system. Furthermore, both flip and Neimark–Sacker bifurcations are also analyzed in the interior of $R_{ +}^{2}$R+2 by using center manifold theorem and bifurcation theory. Finally, numerical simulations are deployed to validate our results with theoretical analysis and to exhibit the dynamical behaviors.

Bifurcation Analysis of a Bounded Rational Duopoly Game with Consumer Surplus

2021 ◽  
Vol 31 (07) ◽  
pp. 2150097
Author(s):  
Wei Zhou ◽  
Yinxia Cao ◽  
Amr Elsonbaty ◽  
A. A. Elsadany ◽  
Tong Chu
Keyword(s):  
Normal Form ◽  
Theoretical Analysis ◽  
Center Manifold ◽  
Profit Maximization ◽  
Consumer Surplus ◽  
Normal Form Theory ◽  
Nonlinear Dynamical ◽  
Center Manifold Theorem ◽  
Dynamical Behaviors ◽  
Form Theory

The nonlinear dynamical behaviors of economic models have been extensively examined and still represented a great challenge for economists in recent and future years. A proposed boundedly rational game incorporating consumer surplus is introduced. This paper aims at studying stability and bifurcation types of the presented model. The flip and Neimark–Sacker bifurcations are analyzed via applying the normal form theory and the center manifold theorem. This study helps determine an appropriate choice of decision parameters which have significant influences on the behavior of the game. The duopoly game that is formed by considering bounded rationality and consumer surplus is more realistic than the ordinary duopoly game which only has profit maximization. And then, some numerical simulations are provided to verify the theoretical analysis. Finally, we compare the dynamical behaviors of the built model with that of Bischi–Naimzada model so as to better understand the performance of the duopoly game with consumer surplus.

scholarly journals Bifurcation and control for a discrete-time prey–predator model with Holling-IV functional response

2013 ◽  
Vol 23 (2) ◽  
pp. 247-261 ◽  
Author(s):  
Qiaoling Chen ◽  
Zhidong Teng ◽  
Zengyun Hu
Keyword(s):  
Functional Response ◽  
Discrete Time ◽  
Chaotic Dynamics ◽  
Period Doubling ◽  
Sudden Onset ◽  
Flip Bifurcation ◽  
Center Manifold Theorem ◽  
Predator Prey Model ◽  
Period 2 ◽  
Unstable Fixed Point

The dynamics of a discrete-time predator-prey model with Holling-IV functional response are investigated. It is shown that the model undergoes a flip bifurcation, a Hopf bifurcation and a saddle-node bifurcation by using the center manifold theorem and bifurcation theory. Numerical simulations not only exhibit our results with the theoretical analysis, but also show the complex dynamical behaviors, such as the period-3, 6, 9, 12, 20, 63, 70, 112 orbits, a cascade of period-doubling bifurcations in period-2, 4, 8, 16, quasi-periodic orbits, an attracting invariant circle, an inverse period-doubling bifurcation from the period-32 orbit leading to chaos and a boundary crisis, a sudden onset of chaos and a sudden disappearance of the chaotic dynamics, attracting chaotic sets and non-attracting sets. We also observe that when the prey is in chaotic dynamics the predator can tend to extinction or to a stable equilibrium. Specifically, we stabilize the chaotic orbits at an unstable fixed point by using OGY chaotic control.

scholarly journals Hopf Bifurcation of an Improved SLBS Model under the Influence of Latent Period

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Chunming Zhang ◽  
Wanping Liu ◽  
Jing Xiao ◽  
Yun Zhao
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Center Manifold ◽  
Center Manifold Theorem ◽  
Local Hopf Bifurcation ◽  
Dynamical Behaviors ◽  
Latent Time ◽  
Normal Form Method ◽  
The Stability ◽  
Theoretical Results

A model applicable to describe the propagation of computer virus is developed and studied, along with the latent time incorporated. We regard time delay as a bifurcating parameter to study the dynamical behaviors including local asymptotical stability and local Hopf bifurcation. By analyzing the associated characteristic equation, Hopf bifurcation occurs when the time delay passes through a sequence of critical values. A formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions is given by using the normal form method and center manifold theorem. Finally, illustrative examples are given to support the theoretical results.

Dynamical Behaviors in Complex-Valued Love Model With or Without Time Delays

2017 ◽  
Vol 27 (13) ◽  
pp. 1750200 ◽  
Author(s):  
Wei Deng ◽  
Xiaofeng Liao ◽  
Tao Dong
Keyword(s):  
Time Delays ◽  
Romantic Relationship ◽  
Center Manifold ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Dynamical Behaviors ◽  
Real Domain ◽  
Form Theory ◽  
Complex Valued ◽  
Love Dynamics

In this paper, a novel version of nonlinear model, i.e. a complex-valued love model with two time delays between two individuals in a love affair, has been proposed. A notable feature in this model is that we separate the emotion of one individual into real and imaginary parts to represent the variation and complexity of psychophysiological emotion in romantic relationship instead of just real domain, and make our model much closer to reality. This is because love is a complicated cognitive and social phenomenon, full of complexity, diversity and unpredictability, which refers to the coexistence of different aspects of feelings, states and attitudes ranging from joy and trust to sadness and disgust. By analyzing associated characteristic equation of linearized equations for our model, it is found that the Hopf bifurcation occurs when the sum of time delays passes through a sequence of critical value. Stability of bifurcating cyclic love dynamics is also derived by applying the normal form theory and the center manifold theorem. In addition, it is also shown that, for some appropriate chosen parameters, chaotic behaviors can appear even without time delay.

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