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Chemostat Theory with Several Nutrients
Today’s starting point:
dN i
 i ( R1 , R2 ,..., Rm ) N i  DN i
dt
dR
 D  S j  R j    i N i qi
dt
i
i  1,..., n
j  1,..., m
i
i  0,
 0 i, j;
R j
strict inequality for at least one j for every i
Mass balance constraints = conservation principle:
T j  R j   Ni qij
i
dNi

  qij
 D  S j  R j    i Ni qij   i Ni qij  DNi qij
dt
dt
dt
i
i
i
 DS j  DT j
dT j
dR j
As t → ∞, Tj → Sj, and asymptotically
S j  R j   qij Ni
i
Alfred Lotka first noted the significance of such
multiple constraints.
For two resources and two competitors, a 2d graphical approach is
sufficient to represent dynamics (Tilman, others in the 1970’s).
Nutritionally
Substitutable Resources
Zi
Nutrient 1 Concentration, R1
Nutrient 2 Concentration, R2
Nutrient 2 Concentration, R2
Nutritionally Essential
(Complementary) Resources
Zi
Nutrient 1 Concentration, R1
Zero-Net-Growth-Isoclines display nutrient concentrations where a single
population is at steady state (dNi / dt = 0).
Nutrient 2 Concentration, R2
Z1
Z2
Nutrient 1 Concentration, R1
Superimposing Zero-Net-Growth-Isoclines for two species determines whether
steady state coexistence is possible. There must be a tradeoff between
competitive abilities for the two resources.
Nutrient 2 Concentration, R2
M2
(S1,S2)
M1
Z1
I2
I1
Z2
Nutrient 1 Concentration, R1
Adding mass conservation constraints determines feasibility and
stability of coexistence at equilibrium, given nutrient supplies.
Experimental Test:
Competition between two diatoms, Synedra filiformis (species 1) and Asterionella
formosa (species 2), for phosphorus (resource 1) and silicon (resource 2) in
laboratory cultures (drawn from the data of Tilman, 1981).
Z1
10
Z2
(S1,S2)
40
6
4
Silicon (M)
Silicon (M)
8
I2
I1
M2
20
M1
10
2
0
0.000
30
Z1
0.002
0.004
0.006
Phosphorus (M)
0.008
0
0.0
Z2
0.1
0.2
0.3
0.4
0.5
0.6
Phosphorus (M)
Resource Ratio Hypothesis: Outcome depends on supply ratio of the resources.
Synedra filiformis (sp. 1) wins when Si:P supply ratio is high; Asterionella formosa (sp.
2) wins when Si:P supply ratio is low. They coexist for intermediate ratios.
Variable-internal-stores model of competition for several
resources:
dNi
 i (Qi1 , Qi 2 ,..., Qim ) Ni  DNi
dt
dQij
 ij ( R j , Qij )  i Qij
dt
dR j
 D  S j  R j    Ni ij
dt
i
For analysis of equilibria, set dQij / dt = 0 and obtain Q*ij. Then
derive an equivalent steady state model where growth depends on Rj.
For two species and two resources the same graphical approach then
applies. Parameterized models for algae suggest that coexistence can
turn to bistability at high dilution rates.
Commensalism – Graphical Theory
Nutrient 2 Concentration, R2
Z1
M1
(S1,S2)
I1
Z2
M2
I2
Nutrient 1 Concentration, R1
Two species compete for one nutrient resource. The inferior
competitor releases another resource that is essential to the
superior competitor.
Commensalism and Planktonic Ecosystems
Organic Carbon
Algae
Onshore / inland
Bacteria
Offshore
Inorganic Nutrient
Vertical Distribution of Light in a Well-Mixed Water Column
Two competing species (j = 1,2) absorb light in proportion to density Nj
and water and dissolved substances absorb light:

j

k j N j z


Irradiance (I)
Iin
Depth (z)
I ( z )  I in e

 kbg 


zm
Iout
Competition for light, following Huisman et al…
Light-dependent growth rate at depth z:
 j ( I ( z ))  D 
 max
I ( z)
j
KI , j  I ( z)
D
Depth-averaged growth rate:
dN j
1

dt
zm

zm
0
  max

I ( z)
j
 D N j dz


 KI , j  I ( z)

Only I(z) depends on z, so:
 K I , j  I in

ln 
K I
dt


 I , j out
 kbg   k j N j  zm
j


dN j
 max
j

 N j  DN j

Competition for light follows something like the R*-rule.
Consider each species growing alone, and let it go to equilibrium to get I*out,j
The species with lowest I*out,j wins competition, others go extinct.
Competition for light and a nutrient is similar to
competition for two essential nutrients.
Relationship between supply and outcome is a little more complicated…
For intermediate supply ratios of light-to-nutrient, bistability is experimentally
more likely than coexistence (for two nutrients, coexistence is common).
Competition for more than two resources (work is almost all
numerical):
Competition for one or two resources in a chemostat always goes
to equilibrium. Moreover, no. persisting species is always ≤ no.
resources.
For three resources oscillations can occur among three species, if
each species is the best competitor for one resource and the
worst for another, and if each species consumes the most of the
resource for which it is the intermediate competitor (Huisman &
Weissing 2001):
R11*  R12*  R13*
q12  q13  q11
*
21
R R R
q23  q21  q22
*
*
*
R33
 R31
 R32
q31  q32  q33
*
22
*
23
Oscillations can be limit
cycles or heteroclinic
cycles:
Huisman & Weissing, 2001
Oscillations can then
allow invasion and
persistence of more
species than resources
(“supersaturated
coexistence”):
Huisman & Weissing, 2001
With 5 species and 5
resources, the oscillations
can be chaotic
(“competitive chaos”):
Huisman & Weissing, 1999
How important are “supersaturated coexistence”
and “competitive chaos” in the natural world?
High amplitude cycles → close approach to
extinction (even when uniformly persistent).
Heteroclinic cycles are not uniformly persistent
(close to extinction for long periods of time).
“Supersaturated coexistence” and “competitive chaos”
are highly parameter dependent (ordering relations and
values).
Schippers et al. (2001): Randomly parameterized species usually not
supersaturated:
Also, adding a randomly parameterized species to a supersaturated
assemblage often reduces no. coexisting species to ≤ no. resources.
Huisman et al. (2001): Supersaturated coexistence more likely when
certain tradeoffs restrict randomized parameterization of species:
Scenario 1 – completely random.
Scenario 5 – ordering relations for R* and q from original work.
Scenario 3 – ordering relation where low R* ↔ low q.
Huisman et al. argued that scenario 3 was theoretically plausible and
had some empirical support.
Roelke & Eldridge (2008): Supersaturated coexistence is less likely in
spatially variable systems (gradostat-like), and is altered by periodic
forcing:
Multiple nutrients and microbial interactions other than
competition, e.g. predation (Grover 2003)
Z
C
zooflagellate
B
bacteria
N
P
dissolved nutrients
recycling fluxes omitted
bacteria that have been ingested
Photo by T.H. Chrzanowski
Governing population dynamic equations:
dB
  B B  DB  mZ  aBZ
dt
dZ
  Z Z  DZ  mZ Z
dt
B
Z
B, Z
m B, m Z
D
a
= bacteria density
= zooflagellate density
= growth functions
= mortality rates (constant)
= dilution rate (constant)
= attack rate (constant)
Growth functions combine Droop’s Equation
with Liebig’s Law of the Minimum:
min


Q j ,i
max
i  i 1  max 
j  Q

 j ,i
i = B, Z
j = C, N, P

 
 
Bacterial quota dynamics (from Thingstad 1987):
dQ j , B
dt
V j ,B
 V j ,B  BQ j ,B  R j ,B
uptake
"use in growth"
for j  C , N , P
release
 [ j ]  Q max
j ,i  Q j ,i
V 
 K  [ j ] 
 Q max  Q min
j ,B
j ,i
 j ,i
maximal 
max
j ,B
uptake
R j ,B
Michaelis-Menten
saturation term

 for j  C , N , P

decrease with quota
0 for j  N , P

   g  B QC , B   m  QC , B  QCmin
,B 
 growth term
maintenance term

Note: [j] denotes dissolved concentration of element j (=C,N,P).
Note: Release of carbon (RC,B) is respiration and disappears from the
system. There is mass conservation for N and P, but not C.
Zooflagellate quota dynamics :
dQ j ,Z
dt
 aBQ j , B   Z Q j ,Z  R j ,Z
ingestion
R j ,Z
"use in growth"
for j  C , N , P
release
 Q max
j ,Z  Q j ,Z
 aBQ j , B  ae j BQ j , B  max
 Q  Q min
j ,Z
 j ,Z
ingestion



assimilation, decreases with quota

 Q max
j ,Z  Q j ,Z
 aBQ j , B 1  e j  max
 Q  Q min
j ,Z

 j ,Z

 
 
Note: For j =N,P assimilation efficiency ej = 1.
Note: Release of carbon (RC,Z) is respiration and disappears from the
system. There is mass conservation for N and P, but not C.
Dissolved nutrient dynamics :
dC
 D  Cin  C   BVC , B  mZ ZQC ,Z  mB BQC , B
dt
dN
 D  N in  N   BVN , B  ZRN ,Z  mZ ZQN ,Z  mB BQN , B
dt
dP
 D  Pin  P   BVP , B  ZRP ,Z  mZ ZQP ,Z  mB BQP , B
dt
Note: For carbon (j = C), only mortality is recycled.
Note: For nitrogen and phosphorus (j = N, P), mortality is recycled
and so is excess nutrient that is ingested but not assimilated.
Topics to investigate with multi-nutrient microbial
predator-prey model:
1. Paradox of enrichment
2. Relation of carbon decomposition to supply of
other nutrients
1a. Paradox of enrichment is much like one-nutrient models when
ratios of nutrient supplies are “balanced” for prey and predator:
1b. Paradox is modified or eliminated when enrichment of one
nutrient occurs at “unbalanced” ratios:
1c. Limit cycles in the paradox of enrichment arise only with
sufficient balance of nutrients:
2. Decomposition of dissolved organic carbon is reduced by limit
cycles and enhanced by balanced nutrients:
Adding a competitor (Grover 2004):
Z
A
C
zooflagellate
B
N
bacteria, two species
P
dissolved nutrients
recycling fluxes omitted
Paramaterized so that
species A is a better
competitor for C and
N, species B is a better
competitor for P.
Relation of asymptotic outcomes to nutrient supplies
when species B is preferred prey:
Multiple attractors (1): coexistence in a limit cycle versus
coexistence at equilibrium
Multiple attractors (2):
exclusion in a limit cycle
versus coexistence at an
equilibrium
Relation of asymptotic outcomes to relative preference of predator for
prey: < 1 means species B is preferred; > 1 means species A is preferred.
Modeling the dynamics of mixotrophs (Ph.D.
student, Ken Crane):
Mixotroph – a microorganism that can grow
phototrophically or heterotrophically.
Phototrophic growth – uses photosynthesis to produce
complex organic matter from simple constituents while
absorbing light.
Heterotrophic growth – consumes complex organic
matter (“food”), either in dissolved form or in the form of
prey or particulate matter.
Mixotrophs that prey on bacteria are very common in
oceans and lakes, and there is a great variety of species.
Flagellates that prey on bacteria
to obtain nutrients and that also
use photosynthesis to make
complex organic matter often
dominate total microbial biomass
in large parts of the ocean in
many lakes, especially those with
low nutrient supply.
chloroplasts for photosynthesis
An organism we study in the lab
is an example (Ochromonas
danica). Without light it grows
purely as a predator of bacteria.
bacteria that have been ingested
Photo by T.H. Chrzanowski
Mixotrophs typically coexist with microorganisms pursuing other
nutritional strategies
A = “algae” that use only photosynthesis (“autotrophs” or “phototrophs”)
Z = “zooflagellates” that can only eat bacteria
B = bacteria that are prey both mixotrophs and zooflagellates
There is a complex web of resource-mediated interactions
I = light, used by algae and mixotrophs
C = dissolved organic carbon, produced by algae and mixotrophs, consumed
by bacteria
P = inorganic phosphorus, consumed by algae, mixotrophs, bacteria
There is also nutrient recycling from various processes.
A
M
Z
B
I
C
P
Interesting questions:
Under what circumstances do species of these different nutritional
strategies coexist?
Under what circumstances to mixotrophs persist and become abundant?
What kind of mixotrophs do best? There is a spectrum, from mixotrophs
that are almost purely autotrophic to those that are almost purely
heterotrophic.
No published mathematical models explicitly address all the resources
involved.
A
M
Z
B
I
C
P
Formulation of bacterial dynamics:
dB
 (  B  D  mB  aBZ Z  aBM M ) B
dt
dQPB
 VPB   B QPB
dt
B  min  PB , CB 
 PB
min


Q
max
PB
  B 1 

Q
PB 

VPB
max




Q
P
max
PB  QPB
 VPB 
  max
min 
k

P
Q

Q
PB 
 PB
  PB
CB  
max
B
 C 


k

C
 BC

Note: Bacterial quota for carbon
is constant, and carbon-dependent
growth depends directly on the
dissolved organic carbon
concentration (C).
Formulation of algal dynamics:
dA
 (  A  D  mA ) A
dt
dQPA
 VPA   A QPA
dt
 A  min   PA , CA 
 PA
min


Q
max
PA
  A 1 

Q
PA 

VPA
max




Q
P
max
PA  QPA
 VPA 
  max
min 
k

P
Q

Q
PA 
 PA
  PA
CA 
 Amax

 kCA  I in 
ln 

k

I
 CA out 
I out  I in e 
  ( kbg  k B BQCB  k A AQCA
Note: Algal quota for carbon is constant,
and carbon-dependent growth is equivalent
to light-dependent growth. A well-mixed
water column of depth z is assumed.
 k M M QCM  k Z ZQCZ ) z
Formulation of zooflagellate dynamics:
dZ
 (  Z  D  mZ ) Z
dt
max
dQPZ
QPZ
 QPZ
 aBZ BQPB max
 Z QPZ
min
dt
QPZ  QPZ
Z  min   PZ , CZ 
 PZ  
max
Z
min
 QPZ

1



Q
PZ


CZ  aBZ eZ B
QCB
QCZ
Note: Zooflagellate quota for
carbon is constant.
Formulation of mixotroph dynamics combines
characteristics of algae and zooflagellates:
dQPM
dt
dM
 (  M  D  mM ) M
dt
max
 QPM
 QPM 
  VPM  (1   ) aBM BQPB  max
 M QPM
min 
 QPM  QPM 
M  min  PM , CM 
 PM
min


Q
max
PM
  M 1 

Q
PM 

VPM  V
max
PM
CM
 kCM  I in

ln 

 kCM  I out
max
 P  QPM
 QPM 

 max
min 
 k PM  P  QPM  QPM 
 Mmax

QCB

(1


)
a
e
B

BM M
QCM

I out  I in e 
  ( kbg  k B BQCB  k A AQCA
Note: Mixotrophy parameter  ranges
0 to 1 and defines the mixture of
predation and photosynthesis.
 k M M QCM  k Z ZQCZ ) z
Formulation of dissolved organic carbon dynamics:
dC
 D (Cin  C )   B BQCB  eA max(0, CA   PA ) AQCA  eM max(0, CM   PM ) M QCM
dt
Term 1
Term 2
Term 3
Term 4
 mB BQCB  mM M QCM  mA AQCA  mZ Z QCZ
Terms 58
Term 1 – Chemostat supply
Term 2 – Excretion of organic carbon when algae are nutrient limited
Term 3 – Excretion of organic carbon when mixotrophs are nutrient limited
Terms 5-8 – Recycling of organic carbon from dead organisms
Carbon fluxes from organisms are regarded as respiration and are dissipated from
the system. There is no mass conservation constraint for carbon.
Formulation of dissolved phosphorus dynamics:
dP
 D ( Pin  P )  BVPB  AVPA   M VPM
dt
Terms 2  4
Term 1
 mB B QPB  mM M QPM  mA AQPA  mZ Z QPZ
Terms 58
 (1   ) aBM
min
min
 QPM  QPM

 QPZ  QPZ

B M QPB  max

a
B
Z
Q
BZ
PB 
min 
max
min 
Q

Q
Q

Q
PM
PM
PZ
PZ




Term 9
Term 10
Term 1 – Chemostat supply
Terms 2-4 – Uptake
Terms 5-8 – Recycling of phosphorus from dead organisms
Term 9 – Recycling of phosphorus ingested by mixotrophs but not assimilated
Term 9 – Recycling of phosphorus ingested by algae but not assimilated
All fluxes of phosphorus from organisms appear as dissolved phosphorus. There
is a mass conservation constraint for phosphorus.
Numerical analysis strategy
Examine bifurcations as mixotrophy parameter  varies from 0 to 1.
Fix all other biological parameters, which are assigned so that
generalist mixotrophs are poorer competitors than specialist algae
and zooflagellates, for the specialists’ resources. Mixotrophs are
poorer competitors for P and light than algae, and poorer
competitors for bacterial prey than zooflagellates.
Examine various combinations of resource supply parameters (Cin,
Pin, Iin, z, kbg).
System is very stiff -- use Matlab ode23s solver (modified
Rosenbrock triple, one step implicit method with new Jacobian
every time step).
Four groups of coexistence sequences for 0 <  < 1:
1.
2.
3.
4.
Persisting species
Resource supply conditions
BAZ – BAM – BAMZ – BAZ
High P supply
BAZ – BAM – BM – BMZ – BAZ
Low P supply (Pin) and Low C supply (Cin)
BAZ – BM – BAM – BM – BMZ - BAZ
Low P supply (Pin) and High C supply (Cin)
Various others
Dark (low Iin with high kbg and high z)
Mixotrophs persist under most resource supply conditions unless they are too
specialized ( near 0 or 1).
Mixotrophs often outcompete either algae or zooflagellates when P supply is
low.
All four nutritional types typically coexist when P supply is high and  for
mixotrophs is intermediate.
The most abundant mixotrophs are usually those with relatively high 
(mostly photosynthetic, getting supplemental nutrition from prey). Microbial
ecologists think this is the most common and abundant mixotrophic strategy.
“Eutrophic” Lake – High P Supply
A. Bacteria
B. Algae
1012
1013
1012
1011
1010
109
108
107
106
105
104
103
102
3
Density (cells / m )
3
Density (cells / m )
1014
1011
1010
109
108
107
0.0
0.2
0.4
0.6
0.8
1.0
0.0
C. Mixotrophs
0.4
0.6
0.8
1.0
0.6
0.8
1.0
D. Zooflagellates
1011
3
3
Density (cells / m )
1011
Density (cells / m )
0.2
1010
109
108
1010
109
108
107
106
105
0.0
0.2
0.4
0.6
 Parameter
0.8
1.0
0.0
0.2
0.4
 Parameter
P-limited growth
C-limited growth
Alternation during limit cycle
“Oligotrophic” Lake – Low P Supply
B. Algae
1012
3
Density (cells / m )
3
Density (cells / m )
A. Bacteria
1014
1013
1012
1011
1010
109
108
107
106
105
104
103
102
1011
1010
109
108
107
0.0
0.2
0.4
0.6
0.8
1.0
0.0
C. Mixotrophs
0.4
0.6
0.8
1.0
0.6
0.8
1.0
D. Zooflagellates
1011
3
3
Density (cells / m )
1011
Density (cells / m )
0.2
1010
109
108
1010
109
108
107
106
105
0.0
0.2
0.4
0.6
0.8
 Parameter
1.0
0.0
0.2
0.4
 Parameter
P-limited growth
C-limited growth
Alternation during limit cycle
Conclusions:
1. Multiple resource models needed for realism in many
applications.
2. Complex dynamics can arise (multiple attractors, chaos,
heteroclinic cycles, supersaturated coexistence, long transients).
3. Practical implications of complex dynamics unclear. With
realistic parameters dynamics are often simpler (equilibria and
limit cycles).
4. Classical ecological principles often emerge (resource
partitioning, predator-mediated coexistence).
5. Challenges include incorporating still more complexity (numbers
of species & resources, food web structure), spatially variable
habitats, numerical analysis problems.