Dynamic characteristics of a discrete-time activator-inhibitor system

Enzymes are specialized proteins that act as biological catalysts, accelerating chemical reactions essential for the functioning of all living cells. They play a critical role in metabolism, and their activity is regulated by molecules known as activators, which enhance their function, and inhibitors, which reduce it. These regulatory mechanisms ensure that metabolic processes occur efficiently and at the right time. Systems biology applies mathematical modeling to study and simulate these complex biological networks, allowing researchers to better understand and predict the outcomes of various metabolic reactions, providing valuable insights into cellular behavior. So, this work investigates the dynamical properties of a discrete activator-inhibitor system. It proves the existence of an interior equilibrium solution and analyzes its local dynamics. The study explores possible bifurcations, showing that the system undergoes Neimark-Sacker and flip bifurcations. Chaos in the system is also examined. Finally, simulations are provided to validate the theoretical findings.