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

TLDR: The Mathematical Analysis of the Becks et al. Experiments as discussed by the authors, which is the most closely related work to ours, can be found in Section 3.4.1.

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

Figures

Figure 3.1. Plots of the four dimensionless variables w (nutrient), x (rods), y (cocci) and z (predators). Chaotic dynamics is evident.

Table 4.1. Values for Shannon’s measures of uncertainty (SMU) and the resulting redundancy (RD) values for the discrete probability distributions shown in Figures 4.1 through 4.4. The maximum possible SMU = (29)1/2 = 4.392.

Figure 2.3. 3-D system space plots of Equations (4) solution with an inflowing nutrient concentration given by Dn0 = [0.15(1 - 0.5sin(0.52t))]. A strange attractor is evident.

Table 1.1. Parameter values used in Equations (1.10) that yield chaotic dynamics with the dilution rate utilized in Becks et al. (2005).

Figure 1.6. Simulated nutrient and microbe dynamics that match the Becks et al. [2005, Fig. 1] results for D = 0.9/d (0.0375/hr).

Figure 3.2. A system-space plot of dimensionless rods vs. cocci vs. predators yields the usual 3-D section of the 4-D strange attractor.

Full text

Lawrence Berkeley National Laboratory
Recent Work
Title
A broad exploration of nonlinear dynamics in microbial systems motivated by chemostat
experiments producing deterministic chaos.
Permalink
https://escholarship.org/uc/item/9wr5396s
Authors
Molz, Fred
Faybishenko, Boris
Agarwal, Deborah
Publication Date
2022-08-05
Peer reviewed
eScholarship.org Powered by the California Digital Library
University of California

1"
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A"Broad"Exploration"of"Coupled"Nonlinear"Dynamics"in"Microbial"
Systems"Motivated"by"Chemostat"Experiments"Producing"
Deterministic"Chaos."
""
Fred"Molz
1
,"Boris"Faybishenko
2
"and"Deborah"Agarwal
3"
1
Environmental"Engineering"&"Earth"Sciences"Dept.,"Clemson"University,"342"Computer"Court,"
Anderson,"SC""29625.""
2
Energy"Geosciences"Division,"Earth"and"Environmental"Sciences"Area,"Lawrence"Berkeley"
National"Laboratory,"Berkeley,"CA""94720""
3
Computer"Research"Division,"Lawrence"Berkeley"National"Laboratory,"Berkeley,"CA""94720""
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LBNL Report Number LBNL-2001172
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Acknowledgements"BF"and"DA"research"supported"by"the"U.S."DOE,"Office"of"
Science,"Office"of"Biological"and"Environmental"Research,"and"Office"of"Science,"
Office"of"Advanced"Scientific"Computing"under"the"DOE"Contract"No."DE-AC02-
05CH11231."FM"acknowledges"the"support"of"the"Clemson"University,"
Department"of"Environmental"Engineering"and"Earth"Sciences."
"

2"
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Table"of"Contents."
ABSTRACT."................................................................................................................"3"
Chapter"1:"Mathematical"Analysis"of"the"Becks"et"al."(1995)"Experiments."............."4"
1.1."Introduction."..................................................................................................."4"
1.2."Mathematical"Model"Development.".............................................................."7"
1.3."Results."......................................................................................................"13"
1.4."Discussion"and"Conclusions."........................................................................."21"
1.5."References."..................................................................................................."25"
S1.""Supplemental"Information"Concerning"Predator"Preference"Change."........."27"
Chapter"2:"Further"Study"of"the"Becks"et"al."Equations."........................................."30"
2.1."Introduction."................................................................................................."30"
2.2."Another"Model"Generalization."...................................................................."31"
2.3."Results."........................................................................................................."33"
2.4."Conclusions.".................................................................................................."38"
2.5."References."..................................................................................................."38"
Chapter"3.""Dimensionless"Forms"for"Equations"(1.10)."........................................."40"
3.1."Introduction."................................................................................................."40"
3.2."Dimensionless"Formulation."........................................................................."40"
3.3."Example"Solution"to"the"Dimensionless"Equations."....................................."44"
3.4."Results"and"Discussion."................................................................................"44"
3.5."References."..................................................................................................."51"
Chapter"4:""How"Might"Information"Theory"Relate"to"Chaotic"Dynamics"in"
Biological"Systems?"................................................................................................"52"
4.1."Introduction."................................................................................................."52"
4.2."Interpretation"of"Shannon’s"Measure."........................................................."53"
4.3."The"concept"of"Redundancy."........................................................................"58"
4.4."What"About"Continuous"Probability"Densities?"..........................................."62"
4.5."Calculation"of"Chaotic"Information"Measures."............................................."64"
4.6."Summary"and"Future"Research"Suggestions."..............................................."70"
4.7."References."..................................................................................................."74"
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3"
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!"#$%!&$'(
The$main$objective$of$this$report$is$to$develop$an$exploratory$mathematical$
analysis$motivated$by$the$Becks$et$al.$(2005)$experiments,$which$will$be$
presented$in$Chapter$1.$$Related$details$of$the$resulting$model$will$be$
presented$in$Chapter$2,$and$a$detailed$non-dimensionalization$will$follow$in$
Chapter$3.$$During$the$research$to$be$described,$a$potential$relationship$to$
Shannon$information$theory$was$realized,$and$this$will$be$developed$in$
Chapter$4.$$Several$avenues$for$future$research$are$suggested,$and$a$more$
mathematically-oriented$presentation$of$the$non-linear$dynamical$properties$
of$the$equations$developed$in$Chapter$1$may$be$found$in$Faybishenko$et$al.$
(2018).$$
"
"" "

4"
"
&)*+,-.(/0(1*,)-2*,34*5(!6*57838(9:(,)-("-4;8(-,(*5'(</==>?(@A+-.32-6,8'"
/'/'(B6 ,.9 C D4 ,396 ' (
$ Deterministic$chaotic$dynamics$in$biological$systems$has$not$received$as$
much$attention$as$that$in$electronic$or$fluid-mechanical$systems,$and$
mathematical$models$are$at$an$early$stage$of$development$(Faybishenko$and$
Molz$(2013).$$However,$this$has$started$to$change.$$Molz$and$Faybishenko$
(2013)$have$recently$concluded $that$three $pape rs$(Bec ks$et$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$systems$of$an$ecological$
nature.$(See$Constantino$et$al.,$1997$for$additional$support.)$$For$example,$
Graham$et$al.$(2007)$reported$experimental$results$demonstrating$the$
phenomenon$of$chaotic$instability$in$biological$nitrification$in$a$controlled$
laboratory$environment.$$In$this$study,$the$aerob ic$biorea ctors$(ae rated $
containers$of$nutrient$solution$and$microbes)$were$filled$initially$with$a$
mixture$of$wastewater$from$a$treatment$plant$and$simulated$wastewater$
involving$a$mixt ur e$o f$m a n y$m ic rob e s.$T h e$m a in $v ar ia bl es $re cor d ed $a s$a $time$
series$were$total$bacteria,$ammonia-oxidizing$bacteria$(AOB),$nitrite-oxidizing$
bacteria$(NOB),$and$protozoa,$along$with$effluent$concentrations$of$nitrate,$
nitrite$and$total$ammonia.$$The$method$of$Rosenstein$et$al.$(1993)$was$used$to$
calculate$Lyapunov$exponents,$which$fell$roughly$in$a$range$from$0.05$to$$
0.2$d
-1
.$$Graham$et$al .$(2 0 0 7) $co n cl u d ed $th a t $“n itr ifica tio n $is $p ro n e$t o$c h a ot ic$
behavior$because$of$a$fragile$AOB-NOB$mutualism,”$i.e.,$interaction.$
$ Beninca$et$al.$(2008)$conducted$a$laboratory$experiment$over$a$period$$
$
$

Frequently asked questions

Q1. What are the contributions mentioned in the paper "A broad exploration of coupled nonlinear dynamics in microbial systems motivated by chemostat experiments producing deterministic chaos" ?

Ben-Naim et al. this paper proposed Shannon 's measure of uncertainty ( SMU ) for deterministic chaotic dynamics. 

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.