![]() It is observed that the complex dynamical network exhibits similar dynamical behavior such as stable equilibrium point, Flip bifurcation and chaos depending on the changing the coupling strength parameter c s. Discrete Cournot duopoly game model is also considered on a Scale free network with N=10 and N=100 nodes. Phase portraits, bifurcation diagrams, Lyaponov exponents show the existence of many complex dynamical behavior in the model such as stable equilibrium point, period-2 orbit, period-4 orbit, period-8 orbit, period-16 orbit and chaos according to changing the speed of adjustment parameter v 1. ![]() By using the center manifold theorem and the bifurcation theory, it is shown that the discrete dynamical system undergoes flip bifurcation about the Nash equilibrium point. We have obtained two dimensional discrete dynamical system as a result of the discretization process is applied to the model. In this study, an Cournot duopoly model describing conformable fractional order differential equations with piecewise constant arguments is discussed. This approach can possibly be used not only to find the equilibrium state for complex and simple ecosystems but also to remove the limitations of current methods to determine the attraction domain or stable points through statistical or difference equations in regime shift studies. The application of the model to monitor the aboveground biomass of a long-term dataset of un-grazed steppe achieved the description and prediction of the regime shift. The model successfully simulated ecosystem energy transfer under equilibrium and quantified ecosystem state. This study utilizes the energy-transfer-network equilibrium model based on Nash-equilibrium theory and the maximum power principle to quantify and predict the equilibrium state of a complex ecosystem with multiple trophic levels. To resolve this dilemma, one of the most critical steps is to determine and quantify the equilibrium states reached by complex ecosystems under a given disturbance. Complex ecosystems exhibit more nonlinearity and stochasticity than the simple ones, rendering timely and accurate detection regime shifts in complex dynamic ecosystems a challenge.
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