Time-delay networks
The main project I'm working on deals with the analysis of time-delay networks. In these networks, the nodes represent dynamical systems, and the links represent interactions between them. Time-delays are considered in the interactions as well as in the node dynamics.
The interaction delays usually describe the travel-time of signals or material between nodes. The internal delay can be used to consider the time needed for dynamical processes on the node that are too complex to be described in the model. For instance considering an ecological model, the increase of the number of sexually mature individuals is delayed by the process of maturation. Thus the increase of sexually mature individuals depends on the number of sexually mature individuals a certain time ago. Hence, instead of describing the maturation process by an age-structured model, we may also introduce a time-delay into a model that considers only the number of mature individuals.
Time-delays turn ordinary differential equations into infinite dimensional system. Thus, even a scalar system with just a single state variable can give rise to complex dynamics as oscillations and chaos. None-surprisingly, the delays hinder the mathematical analysis of such systems. Therefore, there is a demand for efficient method to analyze time-delay networks.
Recently, it has been shown that undelayed network systems can be analyzed efficiently with the
The generalized modeling approach avoids the calculation of the steady state. This calculation can be computational demanding, so that it might set limits to the investigation of large systems. But often the steady state itself is of little interest, so that the generalized modeling approach can be used to study large systems with high numerical efficiency. The high efficiency also allows to study a large number of different networks so that the influence of topological properties can be studied statistically.
Coarse graining of evolutionary models
This project is about the analysis of complex individual-based evolutionary models. Whereas such models are usually analyzed by simulations, we try to analyze them using the ``equation-free" approach introduced by I.G. Kevrekidis [1].
Compared to equation-based models, describing a system on an emergent macroscopic level, a major drawback of microscopic individual-based models is that they can not be analyzed directly by the tools of dynamical systems theory and are therefore almost exclusively studied by simulation. In comparison to more advanced mathematical tools simulation reveals less information on the long-term dynamics of the system at a significantly higher computational cost. In particular the simulation of large individual-based evolutionary models can become prohibitively demanding as evolution manifests itself on much longer timescales than the underlying interactions between individuals.
Even though microscopic individual-based models usually contain a huge amount of degrees of freedom, most systems can be described faithfully by a small number of macroscopic variables. Additional microscopic degrees of freedom change much faster than the slow macroscopic variables. In the long term behavior they therefore equilibrate and become enslaved to the macroscopic variables. If such variables can be identified, then analyzing the dynamics directly on the emergent level becomes possible. However, the derivation of closed equations of motion on this level may be prohibitively difficult.
The ``equation-free" approach is based on the insight that coarse graining does not have to be performed analytically in order to analyze the system with the tools of dynamical systems theory. The established numerical tools do not use information on the functional form of the equations of motion, but only evaluate them in a finite number of points. If we can identify a set of slow variables and are able to evaluate the change of these variables over time for a given system, we can thus extract all information that is needed by the numerical methods. Note, that in an individual based model the change of the slow macroscopic variables can be extracted from a short burst of microscopic simulation. The numerical tools of dynamical systems theory can therefore be used to investigate the dynamics of equations of motion, which are not available but can be evaluated by running properly-initialized bursts of microscopic simulations on-demand.
[1] Kevrekidis IG, Gear CW, Hyman JM, Kevrekidis PG, Runborg O, and Theodoropoulos C. Equation-free, Coarse-Grained Multiscale Computation: Enabling Microscopic Simulatiors To Perform System-Level Analysis. Communication in Mathematical Science, 2003.