# Generalized modeling

Nonlinear dynamics offers many powerful tools enabling us to gain deep insights into the functioning of a wide array of systems. However, for applying these tools the system under consideration typically needs to be described by a specific mathematical model. For many real world systems and in particular for systems from biology, finding a good model can be difficult and it is often uncertain how sensitively the model's dynamics depends on certain modeling assumptions. Generalized modeling is based on the insight that at least some common analyses become actually simpler if they are not carried out on any specific model, but on a whole class of similar models. The core of generalized modeling is a procedure for parameterizing such a class of models. The parameters that are introduced in this way typically have a direct and intuitive interpretation in the context of the system and can be estimated directly from data.

Generalized modeling can be applied for instance to a class of models in which the interactions between the model variables are not restricted to specific mathematical functions. Here it reveals necessary and sufficient conditions for the stability of every possible steady state in the whole class of systems. Further it can identify the transitions in which the stability of stationary states is lost and determine the dynamics beyond these transitions.

A major advantage of generalized modeling is that the method is highly efficient both in computational and manual work. Even for large systems, the main part of the generalized analysis can be carried out analytically. Numerical calculations, if required at all, only form the last step of the analysis. In this case the required numerical step is an eigenvalue calculation, for which highly efficient algorithms are readily available.

In a recent work (Gross et al. Science 2009) we used generalized modeling to identify stabilizing topological properties in a class of ecological food web models. In this paper nonlinear dynamical systems of up to 50 dynamical variables were considered. In the general model neither the topology of the food web nor the specific functional forms of predator-prey interaction were specified. For dealing with this degree of generality while keeping the model realistic and interpretable, our rewriting procedure needed to introduce several thousand parameters. However, because of the efficiency of generalized modeling we were able to explore the parameter space by statistical sampling of approximately 100 Billion parameter sets, which can be accomplished within some weeks on moderate computational infrastructure.

## Key Publications

Dynamical analysis of evolution equations in generalized models
Christian Kuehn, Stefan Siegmund, and Thilo Gross
IMA Journal of Applied Mathematics 78(5), 1051-1077, 2012.

Generalized models as an universal approach to the analysis of nonlinear dynamical systems
Thilo Gross and Ulrike Feudel
Physical Review E 73, 016205-14, 2006.

The influence of dispersal on a predator-prey system with two habitats
Philipp Gramlich, Sebastian Plitzko, Lars Rudolf, Barbara Drossel, and Thilo Gross
Journal of Theoretical Biology 398, 150, 2016.

Impact of dispersal on the stability of metapopulations
Eric Tromeur, Lars Rudolf, and Thilo Gross
Journal of Theoretical Biology 392, 1-11, 2016.

Nonlocal generalized models of predator-prey systems
Christian Kuehn and Thilo Gross
Discrete and Continuous Dynamical Systems B 18(3), 693-720, 2013.

Amplitude death in networks of delay-coupled delay oscillators
Johannes Höfener, Gautam Sethia, and Thilo Gross
Philosophical Transactions of the Royal Society A 371, 20120462, 2013.

Mesoscale symmetries explain dynamical equivalence of food webs
Helge Aufderheide, Lars Rudolf, and Thilo Gross
New Journal of Physics 14, 105014, 2012.

Early warning signals for critical transitions: A generalized modeling approach
Steven J. Lade and Thilo Gross
PLoS Computational Biology 8, e1002360-6, 2012.

Stability in networks of delay-coupled delay oscillators
Johannes M. Hoefener, Gautam Sethia, and Thilo Gross
Europhysics Letters 95, 40002, 2011.

General analysis of mathematical models for bone remodeling
Martin Zumsande, Dirk Stiefs, Stefan Siegmund, and Thilo Gross
Bone 48(4), 910-917, 2011.

Food quality in producer-grazer models - A generalized analysis
Dirk Stiefs, George A. K. van Voorn, Bob W. Kooi, Ulrike Feudel, and Thilo Gross
The American Naturalist 176, 367–380, 2010.

Bifurcations and chaos in the MAPK signaling cascade
Martin Zumsande and Thilo Gross
Journal of Theoretical Biology 265(3), 481-491, 2010.

Local dynamical equivalence of certain food webs
Thilo Gross and Ulrike Feudel
Ocean Dynamics 59(2), 417-427, 2009.

Generalized models reveal stabilizing factors in food webs
Thilo Gross, Lars Rudolf, Simon A. Levin, and Ulf Dieckmann
Science 325, 747-750, 2009.

Stabilization due to predator interference: Comparison of different analysis approaches
George A. K. van Voorn, Dirk Stiefs, Thilo Gross, Bob W. Kooi, Ulrike Feudel, and Sebastiaan A. L. M. Kooijman
Mathematical Biosciences and Engineering 5(3), 567 –583, 2008.

Computation and visualization of bifurcation surfaces
Dirk Stiefs, Thilo Gross, Ralf Steuer, and Ulrike Feudel
International Journal Bifurcation and Chaos 18(8), 2191-2206, 2008.

Instabilities in spatially extended predator-prey systems: spatio-temporal patterns in the neighborhood of Turing-Hopf bifurcations
Martin Baurmann, Thilo Gross, and Ulrike Feudel
Journal of Theoretical Biology 245(2), 220-229, 2007.

From structure to dynamics of metabolic pathways: application to the plant mitochondrial TCA cycle
Ralf Steuer, Adriano Nunes Nesi, Alistair R. Fernie, Thilo Gross, Bernd Blasius, and Joachim Selbig
Bioinformatics 23(11), 1378-1385, 2007.

Structural kinetic modeling of metabolic networks
Ralf Steuer, Thilo Gross, Joachim Selbig, and Bernd Blasius
Proceedings of the National Academy of Sciences 103(32), 11868-11873 , 2006.

Long food chains are in general chaotic
Thilo Gross, Wolfgang Ebenhöh, and Ulrike Feudel
Oikos 109(1), 135-155, 2005.

Enrichment and foodchain stability: the impact of different functional forms
Thilo Gross, Wolfgang Ebenhöh, and Ulrike Feudel
Journal of Theoretical Biology 227(3), 349-358, 2004.

## Media Coverage

Rachel Ehrenberg, ScienceNews, 2012-02-08
Math tools predict when systems are on the brink.
(more)

Tim Barribeau, io9, 2012-02-03
Predicting when a complex system will go through a sudden change is hard — either you need vast swathes of information, or you make rather bad predictions. But we want to spot critical transitions before they happen, regardless of whether we're talking about irreversible damage to coral reefs, stock market crashes, or fishery collapses.
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Sabiene Sütterlin, Science Blogs, 2010-08-26
Seit ich mit der Netzwerk-Gruppe zu tun habe, sehe ich überall Netzwerke. Die Leser dieses Blogs sind teilweise untereinander vernetzt, zum Beispiel. Und die Nervenzellen in unseren Gehirnen knüpfen stets neue Verbindungen
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Natur + Kosmos, 2009-08-20
Was hält eine Lebensgemeinschaft im Innersten zusammen? Wissenschaftler aus Deutschland, Österreich und den USA haben mithilfe von Computersimulationen fundamentale Gesetzmäßigkeiten aufgedeckt, die die Stabilität von Ökosystemen mitbestimmen
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Vielseitige Raubtiere, die sich nicht auf eine bestimmte Beute-Art spezialisiert haben, stabilisieren Ökosysteme
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ND TV, 2009-08-07
Using computer models, scientists from Germany, Austria, and the United States have discovered fundamental rules that determine the stability of ecosystems. The computations reveal that small ecosystems follow other rules than large ecosystems
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Science Daily, 2009-08-07
New findings, published in the journal Science, conclude that food-web stability is enhanced when many diverse predator-prey links connect high and intermediate trophic levels
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Natalie Bachl, Der Standard, 2009-08-07
Forscher finden Gesetzmäßigkeiten, wie Nahrungsketten stabil bleiben - Löwen gehen mit gutem Beispiel voran
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VET-Magazin, 2009-08-07
Wissenschaftler aus Deutschland, Österreich und den USA haben mithilfe von Computersimulationen fundamentale Gesetzmäßigkeiten aufgedeckt, die die Stabilität von Ökosystemen mitbestimmen. Nahrungsnetze sind demnach stabiler, wenn Raubtierarten an der Spitze der Nahrungskette sich von verschiedenen Beutetieren ernähren und Beutearten in der Mitte der Nahrungskette vielen Räubern ausgesetzt sind.
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Scinexx, 2009-08-07
Das Nahrungsnetz eines Ökosystems ist stabiler, wenn die Top-Raubtierarten keine Nahrungsspezialisten sind, sondern sich von vielen verschiedenen Beutetieren ernähren. Das haben Wissenschaftler mithilfe eines neuen Modells festgestellt. Die Berechnungen haben zudem ergeben, dass kleine Ökosysteme anderen Regeln gehorchen als große, wie die Forscher jetzt in "Science" berichten
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