Advances in Complex Systems

We discuss the cellular automata approach and its extensions, the lattice Boltzmann and multiparticle methods. The potential of these techniques is demonstrated in the case of modeling complex systems. In particular, we consider applications taken from various fields of physics, such as reaction-diffusion systems, pattern formation phenomena, fluid flows, fracture processes and road traffic models.

Cancer development can be viewed as an example of spatio-temporal pattern formation. Several attempts have been made to model and predict malignant tumor behavior and also to account for immune system response and the impact of possible clinical treatments. Modeling started from a macroscopic perspective and developed towards cell-based approaches, from which cellular automaton (CA) models are an example.

In this article, we first introduce the general concept of CA systems. Then, we review CA models of tumor development, focusing on avascular and vascular growth, tumor invasion and angiogenesis. Finally, a comparative analysis of the models as well as criteria for designing new CA models are provided and future perspectives are outlined.

We investigate the spatial distribution and the global frequency of agents who can either cooperate or defect. The agent interaction is described by a deterministic, non-iterated prisoner's dilemma game, further each agent only locally interacts with his neighbors. Based on a detailed analysis of the local payoff structures we derive critical conditions for the invasion or the spatial coexistence of cooperators and defectors. These results are concluded in a phase diagram that allows us to identify five regimes, each characterized by a distinct spatiotemporal dynamics and a corresponding final spatial structure. In addition to the complete invasion of defectors, we find coexistence regimes with either a majority of cooperators in large spatial domains, or a minority of cooperators organized in small non-stationary domains or in small clusters. The analysis further allowed a verification of computer simulation results by Nowak and May (1993). Eventually, we present simulation results of a true 5-person game on a lattice. This modification leads to non-uniform spatial interactions that may even enhance the effect of cooperation.

We perform a rigorous stochastic analysis of both deterministic and stochastic cellular automata. The theory uses a mesoscopic view, i.e. it works with probabilities instead of individual configurations used in micro-simulations. An exact stochastic analysis can be done using the theory of Markov processes. But this analysis is restricted to small problems only. For larger problems we compute the distribution using a factorization into marginals. These marginals are then approximated by the given marginals of low order with iterative proportional fitting using the maximum entropy principle. This method has been developed in probabilistic logic. Our method leads to a set of difference equations which have to be iterated numerically. We use the exact methods as well as our approximations to investigate the popular nonlinear voter model (NLVM). We show that the "phase transitions" regarded in recent papers are artifacts of the mean-field approximation. They do not show up in the real automata. There exist many mathematical peculiarities of the NLVM which raise doubts concerning the suitability of the model. As an alternative we propose the Exponential Voter Model which depends on a single parameter only, the inverse "temperature" β. Our proposed method to perform a stochastic analysis is not restricted to cellular automata, but can be applied to more general discrete stochastic systems.