# A broad exploration of nonlinear dynamics in microbial systems motivated by chemostat experiments producing deterministic chaos.

Abstract: ................................................................................................................ 3 Chapter 1: Mathematical Analysis of the Becks et al. (1995) Experiments. ............. 4 1.

## Summary (1 min read)

### 1.1. Introduction.

- The 4 coupled dependent variables were concentrations of nutrient (mg/cc) and each of the 3 microbes (cells/cc).
- Each set of data constituted a time series (concentrations at discrete times), and deterministic chaos was identified by using a computerized version of the analytical procedure developed by Rosenstein et al. (1993) for calculating the largest Lyapunov exponent (TISEAN package [Hegger et al., (1999)).

### Y K m m Y K m and

- So Equations (1.2) adapted to the Becks et al. supplemental experiments, admittedly in a non-unique way, may be written as Equations (1.10) after dividing through by the microbial masses (changed parameters in red.).
- Now that the predators have been made to prefer rods over cocci with an increasing rod population, the rods are disadvantaged and would tend to die out.
- These dimensionless equations may also serve as a basis for further study of a more abstract mathematical nature.
- Further details concerning the mathematical nature of the introduced preference change are given in the “Supplemental Information” at the end of this chapter.

### 1.3. Results.

- A set of parameters that produced chaotic dynamics is listed in Table 1.1 Based on parameter values selected in Kot et al. (1992) and value ranges given in Kravchenko et al. (2004), the Table 1.1 values appear reasonable in a physiological sense.
- D was measured carefully in the experiments, so the authors decided to work only with those values.
- For a D value of 0.9/d (0.0375/hr), both simulations and experiments produced a classical steady state with one microbe dying out.

### 1.4. Discussion and Conclusions.

- Clearly, the developed model with the mix of measured and selected parameters and coupling functions is not capturing all of the experimental details, and this would be expected with a chaotic/classical model having many unmeasured parameters and microbe coupling functions.
- There were several interesting parallels between the experimental and model results.
- The authors therefore conclude that the availability of experimental results and a mathematical model, both producing classical and deterministic chaotic dynamics under similar conditions, is a useful first step that provides new insight that may lead to better understanding of complex phenomena in microbial systems and motivate further studies.
- The statistical aspects of deterministic chaotic time series also have information measures, but classical steady or periodic states do not.
- This leads one to consider biofilms (Benefield and Molz, 1985).

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2,659 citations

### "A broad exploration of nonlinear dy..." refers methods in this paper

...The method of Rosenstein et al. (1993) was used to calculate Lyapunov exponents, which fell roughly in a range from 0.05 to 0.2 d-1....

[...]

...…set of data constituted a time series (concentrations at discrete times), and deterministic chaos was identified by using a computerized version of the analytical procedure developed by Rosenstein et al. (1993) for calculating the largest Lyapunov exponent (TISEAN package [Hegger et al., (1999))....

[...]

1,315 citations

695 citations

### "A broad exploration of nonlinear dy..." refers background in this paper

...…in a biophysical/chemical sense, needs to be studied further, which is a truly interdisciplinary problem involving biochemistry, irreversible thermodynamics, nonlinear dynamics and perhaps Shannon information theory (Schneider and Kay, 1994; Ben-Naim, 2008; Feistel and Ebling, 2011; Sagawa, 2014)....

[...]

335 citations

### "A broad exploration of nonlinear dy..." refers methods in this paper

...…the most abundant microbes are weakened leads to evolutionary dynamics of specific predator-prey pairs, which oscillate in time similar to those predicted by the classical Lotka-Volterra equations, and which are also called “Kill-the-Winner” systems (Bratbak et al., 1990;; Rohwer and Barott, 2012)....

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^{1}, University of Rostock

^{2}, Leibniz Association

^{3}, Cornell University

^{4}

333 citations

### "A broad exploration of nonlinear dy..." refers background in this paper

...Beninca et al. (2008) conducted a laboratory experiment over a period of 6.3 years, which demonstrated chaotic dynamics in a plankton community in a water sample obtained from the Baltic Sea....

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...Molz and Faybishenko (2013) have recently concluded that three papers (Becks et al, 2005; Graham et al. 2007; Beninca et al., 2008), using experimental studies and relevant mathematics, may provide convincing demonstrations that deterministic chaos is present in relatively simple biochemical…...

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###### Q2. What future works have the authors mentioned in the paper "A broad exploration of coupled nonlinear dynamics in microbial systems motivated by chemostat experiments producing deterministic chaos" ?

The potential relation of chaotic dynamics to the appropriate information flows, being careful of what “ information ” actually means in an ecological context, appears very interesting, and it should be considered as a prime area for future research.