Bifurcation analysis and complex dynamics of a Kopel triopoly model

Conventional, neoclassical economics assumes perfectly rational agents (firms, consumers, investors) who face well-defined problems and arrive at optimal behaviour consistent with — in equilibrium with — the overall outcome caused by this behaviour. This rational, equilibrium system produces an elegant economics, but is restrictive and often unrealistic. Complexity economics relaxes these assumptions. It assumes that agents differ, that they have imperfect information about other agents and must, therefore, try to make sense of the situation they face. Agents explore, react and constantly change their actions and strategies in response to the outcome they mutually create. The resulting outcome may not be in equilibrium and may display patterns and emergent phenomena not visible to equilibrium analysis. The economy becomes something not given and existing but constantly forming from a developing set of actions, strategies and beliefs — something not mechanistic, static, timeless and perfect but organic, always creating itself, alive and full of messy vitality.

In this work, a fractional predator-prey model with the harvesting rate is considered. Besides the existence and uniqueness of the solution to the model, local stability and global stability are experienced. A novel discretization depending on the numerical discretization of the Riemann–Liouville integral was introduced and the corresponding numerical discretization of the predator–prey fractional model was obtained. The net reproduction number R 0 was obtained for the prediction and persistence of the disease. The dynamical behavior of the equilibria was examined by using the stability criteria. Furthermore, numerical simulations of the model were performed and their graphical representations are shown to support the numerical discretizations, to visualize the effectiveness of our theoretical results and to monitor the effect of arbitrary order derivative. In our investigations, the fractional operator is understood in the Caputo sense.