Local and Global Dynamics of a Constraint Profit Maximization for Bischi–Naimzada Competition Duopoly Game

The Bischi–Naimzada game is a market competition between two firms with the objective of maximizing profits under limited information. In this article, we study a more generalized and realistic situation that takes into account the sales constraints. we generalize the economic model suggested by Bischi–Naimzada by introducing and studying the maximization of profits based on sales constraints. Our motivation in this paper is the studying of profit and sales constraints maximization and their influences on the game’s dynamics. The local stability of the equilibrium points of the proposed model is discussed. It examines how the dynamics of the proposed two-dimensional competition game model focusing on changes in both the speed of the adjustment and the sales constraint parameters. The map describing the game is proven to be noninvertible and yields many multi-stable, complex dynamics and the coexistence chaotic attractors may arise. The global behavior of the map is achieved by studying the critical curves. The numerical simulations demonstrate the coexistence of two attractors and complex structures of the attraction basins. Several examples are discussed in order to confirm all the analytical results obtained.

Download Full-text

The Influence of Information Acquisition on the Complex Dynamics of Market Competition

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127416500085 ◽

2016 ◽

Vol 26 (01) ◽

pp. 1650008 ◽

Cited By ~ 11

Author(s):

Zhanbing Guo ◽

Junhai Ma

Keyword(s):

Information Sharing ◽

Information Acquisition ◽

Complex Dynamics ◽

Market Competition ◽

Estimation Accuracy ◽

Global Dynamics ◽

Game Model ◽

Amount Of Information ◽

Dynamical Game ◽

Rational Players

In this paper, we build a dynamical game model with three bounded rational players (firms) to study the influence of information on the complex dynamics of market competition, where useful information is about rival’s real decision. In this dynamical game model, one information-sharing team is composed of two firms, they acquire and share the information about their common competitor, however, they make their own decisions separately, where the amount of information acquired by this information-sharing team will determine the estimation accuracy about the rival’s real decision. Based on this dynamical game model and some creative 3D diagrams, the influence of the amount of information on the complex dynamics of market competition such as local dynamics, global dynamics and profits is studied. These results have significant theoretical and practical values to realize the influence of information.

Download Full-text

Complex Dynamics in a Model of Common Fishery Resource Harvested by Multiagents with Heterogeneous Strategy

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127415501539 ◽

2015 ◽

Vol 25 (11) ◽

pp. 1550153 ◽

Cited By ~ 2

Author(s):

En-Guo Gu

Keyword(s):

Complex Dynamics ◽

Sustainable Use ◽

Fish Stock ◽

Global Dynamics ◽

Coexisting Attractors ◽

Fishery Resource ◽

Local Bifurcations ◽

Local And Global Dynamics ◽

Strategic Behaviors ◽

Special Case

In this paper, we formulate a dynamical model of common fishery resource harvested by multiagents with heterogeneous strategy: profit maximizers and gradient learners. Special attention is paid to the problem of heterogeneity of strategic behaviors. We mainly study the existence and the local stability of non-negative equilibria for the model through mathematical analysis. We analyze local bifurcations and complex dynamics such as coexisting attractors by numerical simulations. We also study the local and global dynamics of the exclusive gradient learners as a special case of the model. We discover that when adjusting the speed to be slightly high, the increasing ratio of gradient learners may lead to instability of the fixed point and makes the system sink into complicated dynamics such as quasiperiodic or chaotic attractor. The results reveal that gradient learners with high adjusting speed may ultimately be more harmful to the sustainable use of fish stock than the profit maximizers.

Download Full-text

Dynamic Analysis and Circuit Implementation of a New 4D Lorenz-Type Hyperchaotic System

Mathematical Problems in Engineering ◽

10.1155/2019/6581586 ◽

2019 ◽

Vol 2019 ◽

pp. 1-17 ◽

Cited By ~ 1

Author(s):

A. Al-khedhairi ◽

A. Elsonbaty ◽

A. H. Abdel Kader ◽

A. A. Elsadany

Keyword(s):

Initial Conditions ◽

Equilibrium Points ◽

Dynamical Analysis ◽

Global Dynamics ◽

Hyperchaotic System ◽

Circuit Implementation ◽

Local And Global Dynamics ◽

Invariant Surfaces ◽

The Rich ◽

Global Invariant

This paper attempts to further extend the results of dynamical analysis carried out on a recent 4D Lorenz-type hyperchaotic system while exploring new analytical results concerns its local and global dynamics. In particular, the equilibrium points of the system along with solution’s continuous dependence on initial conditions are examined. Then, a detailed Z2 symmetrical Bogdanov-Takens bifurcation analysis of the hyperchaotic system is performed. Also, the possible first integrals and global invariant surfaces which exist in system’s phase space are analytically found. Theoretical results reveal the rich dynamics and the complexity of system behavior. Finally, numerical simulations and a proposed circuit implementation of the hyperchaotic system are provided to validate the present analytical study of the system.

Download Full-text

On the Stability and Multistability in a Duopoly Game with R&D Spillover and Price Competition

Discrete Dynamics in Nature and Society ◽

10.1155/2019/2369898 ◽

2019 ◽

Vol 2019 ◽

pp. 1-20 ◽

Cited By ~ 10

Author(s):

Wei Zhou ◽

Xiao-Xue Wang

Keyword(s):

Price Competition ◽

Equilibrium Points ◽

Research Result ◽

Basin Of Attraction ◽

Profit Maximization ◽

Feasible Region ◽

Multiple Attractors ◽

Critical Curves ◽

Coexistence Of Multiple Attractors ◽

The Stability

In this paper, a dynamical two-stage game with R&D competition and joint profit maximization is built. The stability of all the equilibrium points is discussed through Jury condition, and the stability region of the Nash equilibrium point is then given. The influence of the parameters on the system is discussed, and we find that the firm can even benefit from chaos, when it has higher innovation efficiency and higher adjusting speed. And then the coexistence of multiple attractors is studied using basin of attraction. Our research result shows that the coexisting attractors can be observed in the two-parameter bifurcation diagram. At last, the boundary of feasible region, global bifurcations, and formation mechanism of fractal structure of attracting basin are analyzed through critical curves and noninvertible map theory.

Download Full-text

Modeling and Analysis of the Multiannual Cholera Outbreaks with Host-Pathogen Encounters

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127420501205 ◽

2020 ◽

Vol 30 (08) ◽

pp. 2050120

Author(s):

Wenjing Zhang

Keyword(s):

Complex Dynamics ◽

Backward Bifurcation ◽

Global Dynamics ◽

Modeling And Analysis ◽

Local And Global Dynamics ◽

Model Dynamics ◽

Host Pathogen ◽

Necessary And Sufficient ◽

And Control ◽

Parameter Values

Re-emergence of cholera threatens people’s health globally. However, its periodic re-emerging outbreaks are still poorly understood. In this paper, we develop a simple ordinary differential equation (ODE) model to study the cholera outbreak cycles. Our model involves both direct (i.e. human-to-human) and indirect (i.e. environment-to-human) transmission routes, due to the multiple interactions between the human host, the pathogen, and the environment. In particular, we model the pathogen searching distance as a Poisson point process, and then formulate the host-pathogen encounter (HPE) rate. A thorough mathematical analysis is performed to investigate local and global dynamics of the model. Necessary and sufficient condition under which the backward bifurcation occurs is derived. Fold, Hopf, and Bogdanov–Takens bifurcations are studied with original model parameter values to reveal their relations with model behaviors. One- and two-dimensional bifurcation diagrams are provided to categorize model dynamics with respect to its parameter values. Analytical and numerical analyses show that our simple model is sufficient to exhibit complex epidemic patterns of cholera dynamics including bistability and annual and multiannual periodic outbreaks. Our result regarding the backward bifurcation and complex dynamics of cholera epidemics highlight the challenges in the prevention and control of the disease.

Download Full-text

Order and Chaos in a Prey-Predator Model Incorporating Refuge, Disease, and Harvesting

Journal of Applied Mathematics ◽

10.1155/2020/5373817 ◽

2020 ◽

Vol 2020 ◽

pp. 1-19

Author(s):

Dahlia Khaled Bahlool ◽

Huda Abdul Satar ◽

Hiba Abdullah Ibrahim

Keyword(s):

Equilibrium Points ◽

Global Dynamics ◽

Persistence Conditions ◽

Local Bifurcation Analysis ◽

Uniqueness Of The Solution ◽

Harvesting Process ◽

Predator Model ◽

The Stability ◽

Predator System ◽

Type Ii Functional Response

In this paper, a mathematical model consisting of a prey-predator system incorporating infectious disease in the prey has been proposed and analyzed. It is assumed that the predator preys upon the nonrefugees prey only according to the modified Holling type-II functional response. There is a harvesting process from the predator. The existence and uniqueness of the solution in addition to their bounded are discussed. The stability analysis of the model around all possible equilibrium points is investigated. The persistence conditions of the system are established. Local bifurcation analysis in view of the Sotomayor theorem is carried out. Numerical simulation has been applied to investigate the global dynamics and specify the effect of varying the parameters. It is observed that the system has a chaotic dynamics.

Download Full-text

Local and Global Dynamics in a Discrete Time Growth Model with Nonconcave Production Function

Discrete Dynamics in Nature and Society ◽

10.1155/2012/536570 ◽

2012 ◽

Vol 2012 ◽

pp. 1-22 ◽

Cited By ~ 13

Author(s):

Serena Brianzoni ◽

Cristiana Mammana ◽

Elisabetta Michetti

Keyword(s):

Production Function ◽

Growth Model ◽

Discrete Time ◽

Production Functions ◽

Elasticity Of Substitution ◽

Global Dynamics ◽

Production Factors ◽

Local And Global Dynamics ◽

Complex Features

We study the dynamics shown by the discrete time neoclassical one-sector growth model with differential savings while assuming a nonconcave production function. We prove that complex features exhibited are related both to the structure of the coexixting attractors and to their basins. We also show that complexity emerges if the elasticity of substitution between production factors is low enough and shareholders save more than workers, confirming the results obtained while considering concave production functions.

Download Full-text

The thalamic low-threshold Ca 2+ potential: a key determinant of the local and global dynamics of the slow (<1 Hz) sleep oscillation in thalamocortical networks

Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rsta.2011.0126 ◽

2011 ◽

Vol 369 (1952) ◽

pp. 3820-3839 ◽

Cited By ~ 14

Author(s):

Vincenzo Crunelli ◽

Adam C. Errington ◽

Stuart W. Hughes ◽

Tibor I. Tóth

Keyword(s):

Action Potential ◽

Slow Oscillation ◽

Global Dynamics ◽

Action Potential Firing ◽

Local And Global Dynamics ◽

Up States ◽

Potential Firing ◽

Low Threshold

During non-rapid eye movement sleep and certain types of anaesthesia, neurons in the neocortex and thalamus exhibit a distinctive slow (<1 Hz) oscillation that consists of alternating UP and DOWN membrane potential states and which correlates with a pronounced slow (<1 Hz) rhythm in the electroencephalogram. While several studies have claimed that the slow oscillation is generated exclusively in neocortical networks and then transmitted to other brain areas, substantial evidence exists to suggest that the full expression of the slow oscillation in an intact thalamocortical (TC) network requires the balanced interaction of oscillator systems in both the neocortex and thalamus. Within such a scenario, we have previously argued that the powerful low-threshold Ca 2+ potential (LTCP)-mediated burst of action potentials that initiates the UP states in individual TC neurons may be a vital signal for instigating UP states in related cortical areas. To investigate these issues we constructed a computational model of the TC network which encompasses the important known aspects of the slow oscillation that have been garnered from earlier in vivo and in vitro experiments. Using this model we confirm that the overall expression of the slow oscillation is intricately reliant on intact connections between the thalamus and the cortex. In particular, we demonstrate that UP state-related LTCP-mediated bursts in TC neurons are proficient in triggering synchronous UP states in cortical networks, thereby bringing about a synchronous slow oscillation in the whole network. The importance of LTCP-mediated action potential bursts in the slow oscillation is also underlined by the observation that their associated dendritic Ca 2+ signals are the only ones that inform corticothalamic synapses of the TC neuron output, since they, but not those elicited by tonic action potential firing, reach the distal dendritic sites where these synapses are located.

Download Full-text

COMPETITION AND COOPERATION ON PREDATION: BIFURCATION THEORY OF MUTUALISM

Journal of Biological System ◽

10.1142/s0218339021500030 ◽

2021 ◽

pp. 1-25

Author(s):

SRIJANA GHIMIRE ◽

XIANG-SHENG WANG

Keyword(s):

Quantitative Information ◽

Model Systems ◽

Exclusion Principle ◽

Positive Equilibrium ◽

Global Dynamics ◽

Severe Condition ◽

Predator Species ◽

Competition And Cooperation ◽

Local And Global Dynamics ◽

Predator Prey Models

In this paper, we investigate two predator–prey models which take into consideration hunting cooperation (i.e., mutualism) between two different predators and within one predator species, respectively. Local and global dynamics are obtained for the model systems. By a detailed bifurcation analysis, we investigate the dependence of predation dynamics on mutualism (cooperative predation). From our study, we prove that mutualism may enhance the survival of mutualist predators in a severe condition and break the competitive exclusion principle. We further provide quantitative information about how the cooperative predation (mutualism) may (i) establish multiple stability switches on the positive equilibrium; (ii) generate backward bifurcation on equilibria; (iii) induce supercritical or subcritical Hopf bifurcations; and (iv) establish bi-stability phenomenon between the predator-free equilibrium and a positive equilibrium (or a limit cycle).

Download Full-text