Criticality, adaptability and early-warning signals in time series in a discrete quasispecies model

Abstract

Complex systems from different fields of knowledge often do not allow a mathematical description or modeling, because of their intricate structure composed of numerous interacting components. As an alternative approach, it is possible to study the way in which observables associated with the system fluctuate in time. These time series may provide valuable information about the underlying dynamics. It has been suggested that complex dynamic systems, ranging from ecosystems to financial markets and the climate, produce generic early-warning signals at the “tipping points,” where they announce a sudden shift toward a different dynamical regime, such as a population extinction, a systemic market crash, or abrupt shifts in the weather. On the other hand, the framework of Self-Organized Criticality (SOC), suggests that some complex systems, such as life itself, may spontaneously converge toward a critical point. As a particular example, the quasispecies model suggests that RNA viruses self-organize their mutation rate near the error-catastrophe threshold, where robustness and evolvability are balanced in such a way that survival is optimized. In this paper, we study the time series associated to a classical discrete quasispecies model for different mutation rates, and identify early-warning signals for critical mutation rates near the error-catastrophe threshold, such as irregularities in the kurtosis and a significant increase in the autocorrelation range, reminiscent of 1/f noise. In the present context, we find that the early-warning signals, rather than broadcasting the collapse of the system, are the fingerprint of survival optimization.

Key words： time series; complexity; early-warning signals; quasispecies; 1/f noise; optimization

Acknowledgements

We acknowledge financial support from CONACYT (grants CB-2011-01-167441, CB-2010-01-155663 and I010/266/2011/C-410-11) and PAPIIT-DGAPA (grant IN114411). This work was partly funded by the project FP7-PEOPLE-2009-IRSES-247541-MATSIQEL. The authors wish to thank Dr. R. Mansilla, Dr. I.O. Morales and Dr. Landa for fruitful discussions on aspects of non-stationarity.

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