Complex systems are governed by dynamic processes whose underlying causal rules

are difficult to unravel. However, chemical reactions, molecular interactions,

and many other complex systems can be usually represented as concentrations or

quantities that vary over time, which provides a framework to study these

dynamic relationships. An increasing number of tools use these quantifications

to simulate dynamically complex systems to better understand their underlying

processes. The application of such methods covers several research areas from

biology and chemistry to ecology and even social sciences.In the following

chapter, we introduce the concept of rule-based simulations based on the

Stochastic Simulation Algorithm (SSA) as well as other mathematical methods such

as Ordinary Differential Equations (ODE) models to describe agent-based systems.

Besides, we describe the mathematical framework behind Kappa (κ), a rule-based

language for the modeling of complex systems, and some extensions for spaßtial

models implemented in PISKaS (Parallel Implementation of a Spatial Kappa

Simulator). To facilitate the understanding of these methods, we include

examples of how these models can be used to describe population dynamics in a

simple predator-prey ecosystem or to simulate circadian rhythm changes.