Mathematics Colloquia and Seminars

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Recent publications (e.g., Becks et al. 2005, Nature Letters; Graham et al. 2007, Int. Soc. Microb. Eco. J.; Beninca et al. 2008, Nature Letters; Saleh, 2011, IJBAS) have supported the breakthrough discovery that deterministic chaos may occur in relatively simple biochemical systems. This was accomplished through a series of experiments that produced time series of data (microbe populations, substrate concentrations, etc.), which were then analyzed using recently developed methods of nonlinear dynamics and chaos: for an overview, see Molz & Faybishenko, 2013, and Faybishenko & Molz, 2013, Procedia Environ. Sci. The chaotic dynamics may arise from nonlinear interactions within the system being studied, not induced by time-varying boundary conditions. Chaotic dynamics of the system may result in the formation of strange attractors that may be viewed as representation of an emergent behavior. In their mathematical analysis of a fully mixed, predator-prey system (2 microbes, 1 food source), based on Monod growth kinetics, Kot et al. (1992, Bull. Mathematical Bio.) showed that chaotic dynamics could not occur unless chemostat substrate availability was varied sinusoidally. With parameters identical to Kot et al. (1992) we are able to generate chaotic dynamics even with constant boundary conditions, which implies a fully internal self-organization of a deterministic chaotic system. Results of simulations of predator-prey models also show that an increase in the food available to the prey may cause the predator's population to destabilize (this is the “paradox of enrichment” described by Michael Rosenzweig in 1971). Using generated time series data, we calculated diagnostic parameters of chaos: embedding dimension, capacity (fractal) dimension, correlation dimension, information dimension, and a spectrum of Lyapunov exponents, which appear to be the most important diagnostic parameters of chaos. We will also discuss the self-organization and complexity of the system by means of calculations of the information entropy, and will show the perspective of using the phase-space systems attractor to represent a sustainable domain of the system. This is joint work with Fred Molz (Environmental Engineering & Earth Sciences, Clemson University)