Introduction to the Modeling and Analysis of Complex Systems

Chapter 13Continuous Field Models I: Modeling13.1 Continuous Field Models with Partial DifferentialEquationsSpatio-temporal dynamics of complex systems can also be modeled and analyzed usingpartial differential equations (PDEs), i.e., differential equations whose independent variablesinclude not just time, but also space. As a modeling framework, PDEs are mucholder than CA. But interestingly, the applications of PDEs to describe self-organizing dynamicsof spatially extended systems began about the same time as the studies of CA. Asdiscussed in Section 11.5, Turing’s monumental work on the chemical basis of morphogenesis[44] played an important role in igniting researchers’ attention to the PDE-basedcontinuous field models as a mathematical framework to study self-organization of complexsystems.There are many different ways to formulate a PDE-based model, but here we stick tothe following simple first-order mathematical formulation:(∂f∂t = F f, ∂f)∂x , ∂2 f∂x , . . . , x, t (13.1)2Now the partial derivatives (e.g., ∂f/∂t) have begun to show up in the equations, butdon’t be afraid; they are nothing different from ordinary derivatives (e.g., df/dt). Partialderivatives simply mean that the function being differentiated has more than one independentvariable (e.g., x, t) and that the differentiation is being done while other independentvariables are kept as constants. The above formula is still about instantaneous change ofsomething over time (as seen on the left hand side), which is consistent with what we havedone so far, so you will find this formulation relatively easy to understand and simulate.227