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Microbial Prey-Predator System inchemostat
B.W. Kooi
Department of Theoretical Biology
VU University, Amsterdam The Netherlands
Email: [email protected]
URL: http://www.bio.vu.nl/thb
Wed 17 Chap 9: individuals–population–ecosystem
• From individual to population
• Unstructured vs structured populations
• Structured population model of dividers
• Transient versus ultimate behaviour, pseudo steady states
• Analysis of glucose–bacterium Escherichia coli–cellularslime–mold Dictyostelium discoideum
• One life-stage model, juveniles, discrete reproductionwithout buffer, adults do not grow
Fri 19 Chap 9: individuals–population–ecosystem
• Structured population model of rotifers population
• Analysis of algae Chlorella pyrenoidosa and rotifer Bra-chionus calyciflorus
• Two life-stage model, juveniles and adults, adults donot grow, discrete vs continuous reproduction
• 2-stage chemostat experiments with two separate chemo-stat where the outflow from the first is the inflow intothe second
• 1-stage chemostat experiments on a mixed culture ofthe algae and rotifer
G. F. Webb, Theory of nonlinear age-dependent population dynamics, Marcel Dekker,
New York, Basel, 1985
Metz, J.A.J. and Diekmann, O., The dynamics of physiologically structured popu-
lations, Springer-Verlag, Berlin, 1986
A.M. de Roos, A gentle introduction to physiologically structured population mod-
els, Structured-Population models in marine, terrestrial, and freshwater systems, S.
Tuljapurkar and H. Caswell, 119-204, Chapman & Hall, New York, 1997
R.M. Nisbet, Delay-Differential Equations for Structured Populations, Structured-
Population models in marine, terrestrial, and freshwater systems, 89-118, S. Tul-
japurkar and H. Caswell, Chapman & Hall, New York, 1997
J.M. Cushing, An Introduction to Structured Population Dynamics, Society for In-
dustrial and Applied Mathematics, Philadelphia, 1998
O. Diekmann and J.A.J. Metz, How to lift a model for individual behaviour to the
population level?, Phil. Trans. R. Soc. B, 365, 3523-3530, 2010
A.M. De Roos, O. Diekmann, P. Getto and M.A. Kirkilionis, Numerical equilibrium
analysis for structured consumer resource models, Bull. Math. Biol., 72, 259-297,
2010
A.M. de Roos and L. Persson, Population and Community Ecology of Ontogenetic
Development, Monographs in Population Biology 51, Princeton University Press,
Princeton, 2013
Tri-trophic food chain in chemostat
X0: Nutrient, glucose
X1: Prey, bacterium Escherichia coli
X2: Predator: cellular slime–mold Dictyostelium discoideum
Dilution rate: D = 0.064 h−1
Glucose concentration inflow: Xr = 1 mgml−1
Data from;
V.E. Dent, M.J. Bazin and P.T. Saunders
Behaviour of Dictyostelium discoideum Amoebae and Escherichia coli Grown to-
gether in Chemostat Culture
Arch. Microbiol., 109:187-194, 1976
Consumer
Nutrient
Producers
chemostat environment
Function, derivative and ODE
XP
X
Y
YΔ
Δ
Function
Y = f(X)
Derivative at point P where X = XP
dY
dX= lim
X→XP
ΔY
ΔX= lim
X→XP
f(XP +ΔX)− f(XP )
ΔX
In biology the variable is often a function of time t, saypopulation size N(t). Then the derivative at point tP be-comes
dN
dt= lim
t→tP
ΔN
Δt= lim
t→tP
f(tP +Δt)− f(tP )
Δt
In an ordinary differential equation ODE we have
dN
dt= F(N)
Not restricted to a point but defined on an interval e.g.t ≥ 0
Problem is well posed if the function F(N) is smooth andtogether withInitial condition:
N(0) = N0
Function of multiple variables, partial derivative and PDE
Function
Z = f(X, Y )
Function of multiple variables, partial derivative and PDE
Partial derivative w.r.t. X at point P = (XP , YP )
∂Z
∂X= lim
X→XP
ΔZ
ΔX= lim
X→XP
f(XP +ΔX, YP )− f(XP , YP )
ΔX
and similar for Y
In population biology the variable is often a population den-
sity m(t, a) of time t and age a. Then the partial derivative
∂m
∂t= lim
t→tP
Δm
Δt= lim
t→tP
m(tP +Δt, a)−m(tP , a)
Δt
In an partial differential equation we have
∂m
∂t+
∂m
∂a= F(m(t, a)) = −δm(t, a)
Not restricted to a point but defined in a two dimensional
space Ω = (t ≥ 0, ab ≤ a ≤ ad)
Time interval (t, t+Δt) for age-class (a, a+Δa)
a+Δaa
δm(t, a)
m(t, a) m(t, a+Δa)
a
Within time interval (t, t+Δt) for age class (a, a+Δa)
(m(t+Δt, a)−m(t, a)
)Δa = m(t, a)Δt−m(t, a+Δa)Δt
− δm(t, a)ΔaΔt
Divide by ΔaΔt gives
m(t+Δt, a)−m(t, a)
Δt+
m(t, a+Δa)−m(t, a)
Δa= −δm(t, a)
Then with limΔt → 0 and limΔa → 0 gives
∂m
∂t+
∂m
∂a= −δm(t, a)
Problem is well posed if the function m(t, a) is smooth and
together with
Initial condition at t = 0:
m(0, a) = m0(a)
Boundary condition at a = ab and a = ad:
m(t, ab) = m0(ab) , m(t, ad) = m0(ad)
For dividers:
reproduction
mortality
a+Δaa
δm(a)
m(a)
m(ab) m(ad)
d
m(a+Δa)
a
b
Model formulation
Populations described using unstructured models then:
State of the population is described by one or a few vari-
ables (number, biomass)
Later this turns out to be realistic for dividers (microbial
cells, worms)
Nevertheless we formulate now a structured model
Dividers:
One life-stage population where individuals divide at species
specific size into two equal newborns. Hence, no allocation
to reproduction
Individual growth model (i-state)
System of two ODE’s for the volumetric-length li being the
cubic root of the volume and the energy reserves density
ei both as functions of the time t
d
dtei = ge(li, ei) = vi
fi−1,i − ei
lid
dtli = gl(li, ei) =
viei − kmigili3(ei + gi)
fi−1,i(t) =xi−1(y)
ki−1,i + xi−1(t),
where i = 0 for nutrient and i = 1,2 for prey and predator,
respectively
Individual reproduction model
Individual propagates by binary fission into two equal parts
ldi: length at division
Two equal new individuals occurs at lengthlbi = 2−1/3 ldi
with the same energy density ei as the mother individualebi = edi
b d
juvenile
adult
reproduction
Population growth model (p-state)
ni(t, ei, li): denote the density of individuals having energy
density ei and length li at time t
∫ ebea∫ lblani(t, ei, li) dli dei: number of individuals per volume of
reactor with an energy density between ei = ea and ei = eband a length between li = la and li = lb at time t.
Individuals are taken out of the population at a constant
probability rate pi,i+1 per individual per unit of time. This
term takes mortality, predation as well as dilution into ac-
count.
Population PDE reads:
∂
∂tni(t, ei, li) =− ∂
∂li(ni(t, ei, li)gl(li, ei))
− ∂
∂ei(ni(t, ei, li)ge(li, ei))
− pi,i+1 ni(t, ei, li) ,
where gl and ge are given the growth rate of structure and
reserves
We assume that the energy density for all individuals is
equal and denoted by ei(t) independent of li. Then, the
PDE reduces to
∂
∂tni(t, li) = − ∂
∂li(ni(t, li)gl(li, ei))− pi,i+1 ni(t, li) ,
ni is at the individual level a function of time t and size lionly, but at the population level also a function of ei
Boundary condition for the hyperbolic partial differentialequation reads
ni(t, lbi) gl(lbi, ebi) = 2ni(t, ldi) gl(ldi, edi) .
ni(t, lbi) = 2ni(t, ldi)ei − kmigildi/vi
ei − kmigilbi/vi.
Hence, the fission is tied to the growth process
b d
juvenile
adult
reproduction
In what follows some statistics will be of particular interest:
the total number of individuals
Ni(t) ≡∫ ldi
li=lbi
ni(t, ei, li) dli ,
the mean surface area
El2i ≡ Ni(t)−1
∫ ldi
li=lbi
l2i ni(t, ei, li) dli ,
and the mean biovolume
El3i ≡ Ni(t)−1
∫ ldi
li=lbi
l3i ni(t, ei, li) dli .
Ecosystem model (e-state)
Formulation of the interaction between the abiotic and
biotic populations and the environment
Nutrient supply
Wash-out of the populations from the reactor
l
e
vH
P
e
l
vH
p0,1
glucose
f0,1
J
J
f1,2
p1,2
bacterium Escherichia coli
slime-mold Dictyostelium discoideum
l
e
P
e
l
p0,1
glucose
f0,1
J
J
f1,2
p1,2
bacterium Escherichia coli
slime-mold Dictyostelium discoideum
Nutrient–prey interaction
Using these notions the coupling between the nutrient and
the prey is given by
(p0,1 −D)X0 = I0,1f0,1N1(t)El21
p0,1: is the rate the nutrient leaves the reactor because of
consumption by prey and wash-out
D: wash-out
Functional response:
f0,1(t) =X0(t)
k0,1 + x0(t)
Prey–predator interaction
Prey uptake rate of the predator (i = 2) equals the biovolume-drain rate of the prey (i = 1) and we take into accountthe losses due to medium throughput D.
There is only a coupling via biovolumes conversion and notvia the energy storages. For the coupling between the preyand the predator we have
(p1,2 −D)N1(t)El31 = I1,2f1,2N2(t)El22
where p1,2 is the rate the prey leave the reactor becauseof consumption by predator and wash-out, and
f1,2 =N1(t)El31
k1,2 +N1(t)El31
Because we do not consider predation of the top-predator
we have
p2,3 = D
that is, the predators leave the reactor only because of
wash-out.
Finally the equation of continuity for the nutrients reads
d
dtX0 = Dxr − p0,1X0 .
From full structure to no structure
linear chain trick: MacDonald (1978), see also Metz &
Diekmann (1989) and Cushing (1989).
We assume that the energy density for all individuals will
approach the value ei(t) = fi(t) , also when fi(t) is still a
function of time t. Then, using integration by parts the
PDE formulation reduces to
d
dtNi = −pi,i+1Ni
d
dtNiEl2i = 2
∫lilini(t, ei, li)gl(li, ei) dli − pi,i+1NiEl2i
d
dtNiEl3i = 3
∫lil2i ni(t, ei, li)gl(li, ei) dli − pi,i+1NiEl3i .
where
gl(li, fi) =vifi−1,i − kmigili
3(fi−1,i + gi)
Use of this equation yields
d
dtNiEl3i =
(vi fi−1,i
fi−1,i + gi
El2iEl3i
− gikmi
fi−1,i + gi− pi,i+1
)NiEl3i .
Now we define the overall population growth rate μi−1,i so
that
dNiEl3idt
= (μi−1,i − pi,i+1)NiEl3i .
with
μi−1,i =vi fi−1,i
fi−1,i + gi
El2iEl3i
− gikmi
fi−1,i + gi.
This result can be used to eliminate the El2i−1 term in the
inter-level condition for instance
(p1,2 −D)N1(t)El31 = I1,2f1,2N2(t)El22
and we obtain
pi−1,i −D = Ii−1,i(fi−1,i + gi)μi−1,i + gikmi
vi
NiEl3iNi−1El3i−1
.
Now we are able to define the variable Xi used for the food
density in the expression for the functional response fi−1,i
precisely being the biovolumes Xi = NiEl3i for i = 1,2.
Then we end up with a set of three coupled differential
equations for the substrate density X0 and the total bio-
volume of the prey X1 and predator X2.
In the example discussed here we have
X0: glucose
X1: bacterium Escherichia coli
X2: cellular slime–mold Dictyostelium discoideum
However, the growth rate μi−1,i
μi−1,i =vi fi−1,i
fi−1,i + gi
El2iEl3i
− gikmi
fi−1,i + gi
depends on the length distribution of the individuals
To facilitate the transition from the individual level to the
population level we set the energy density at the value it
would have at constant food densities f , such that e = f
The PDE, where ni is a function of time t and size li,
reduces to
∂
∂tni(t, li) = − ∂
∂li
(ni(t, li)
d
dtli
)− pi,i+1ni(t, li) ,
while the boundary condition reads
ni(t, lbi) = 2ni(t, ldi)vifi−1,i − kmigldivifi−1,i − kmiglbi
.
The method of separation of variables yields the solution
ni = ni0 e(μi−1,i−pi,i+1)t
(vifi−1,i − kmigili
3(fi−1,i + gi)
) ln 2
lnvifi−1,i−kmigilbivifi−1,i−kmigildi
−1
.
where μi−1,i is the overall population growth rate
μi−1,i =kmigi
3 (fi−1,i + gi)
ln 2
lnvifi−1,i−kmigilbivifi−1,i−kmigildi
.
The constant ni0 is given by the initial length distribution
for t = 0.
Observe that the age distribution depends on the food
densities f
Quasi-stationary approach
In order to keep a dynamical description it is possible to
relax the requirement e = f and to use it as requirement
e(t) = f(t) or even to introduce a separate equations for
the reserves again
Following restriction holds
This method does not yield the solution of the arbitrary
initial value problem for the complete deb model. To solve
initial value problems with variable food supply numeri-
cally one can use for example the ‘Escalator boxcar train’
method developed in de Roos (1988)
Individual surface area is proportional to individual
volume: V1-morphs
Then the dynamics of the reserves is not eliminated but
the internal size structure is
The equations for the individual energy storage and growth
now become
d
dtei = νi−1,i (fi−1,i − ei) with fi−1,i =
Xi−1
ki−1,i +Xi−1
d
dtli =
νi−1,i ei − gi kmi
3 (ei + gi)li ,
where νi−1,i is the specific energy conductance which re-
lates to the energy conductance vi in the deb model
This gives
d
dtXi =
∫li
∫eil3i ni(t, ei, li)
νi−1,i ei − gi kmi
(ei + gi)dei dli − pi,i+1Xi .
We would like to have
d
dtXi =
νi−1,i ei − gi kmi
(ei + gi)− pi,i+1Xi .
When f(t) is a function of time the value for e(t) for each
individual converges exponentially to same time-path, so
that e(t) can be interpreted as the reserve density of a
randomly chosen individual at time t
Mathematically this means that the support of the density
n(t, e, l) reduces one dimension n(t, l) and e(t) is now a
p-state variable
Then
μi−1,i =νi−1,i ei − gi kmi
ei + gi.
Model formulation with reserves again
Populations described using un-structured models
State of the population is described by one or a few vari-
ables (number, biomass)
Realistic for dividers (microbial cells, worms)
One life-stage population where individuals divide at species
specific size into two equal newborns
b d
juvenile
adult
reproduction
Tri-trophic food chain in chemostat
X0: glucose
X1: bacterium Escherichia coli
X2: cellular slime–mold Dictyostelium discoideum
Dilution rate: D = 0.064 h−1
Glucose concentration inflow: Xr = 1 mgml−1
Data from;
V.E. Dent, M.J. Bazin and P.T. Saunders
Behaviour of Dictyostelium discoideum Amoebae and Escherichia coli Grown to-
gether in Chemostat Culture
Arch. Microbiol., 109:187-194, 1976
Set of ordinary differential equations
d
dtX0 = D(Xr −X0)−
X0X1I0,1
k0,1 +X0
d
dte1 = ν0,1
(X0
k0,1 +X0− e1
)
d
dtX1 =
(ν0,1e1 − kM1g1
e1 + g1−D
)X1 − X1X2I1,2
k1,2 +X1
d
dte2 = ν1,2
(X1
k1,2 +X1− e2
)
d
dtX2 =
(ν1,2e2 − kM2g2
e2 + g2−D
)X2
DEB Parameter values
Var. Par. value Var. Par. value Unit
X0(0) 0.433 mgml−1
X1(0) 0.361 X2(0) 0.084 mm3 ml−1
e1(0) 1 e2(0) 1 -
k0,1 ≤ 10−5 k1,2 0.18 μgml,
mm3
mlg1 0.86 g2 4.43 -km1 0.0083 km1 0.16 h−1
ν0,1 0.67 ν1,2 2.05 h−1
I0,1 0.26 I1,2 0.26 mgmm3 h, h−1
Classification of the population models
Reserves → no yesMaintenance ↓
no Monod Droop
yes Marr-Pirt DEB
Monod
0
0.2
0.4
0.6
0.8
1
0 20 40 60 80 100120140
x0
[mg ml−1]
t [h]
glucose
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Mean cell volume
For bacteria the volume at division (Vd) depends on the
food level, see Donachie (1968) and Kooijman (2010).
dna duplication is triggered upon exceeding a fixed cell
size Vp and dna duplication lasts a fixed time period tDindependent from the food density. We assume that this
holds also for the myxamoebae, where tD refers to the
duplication time of the biggest chromosome.
Observe that the growth rate μi is independent of the vol-
ume at division Vdi, i = 1,2. This allows us to use the
model described above.
In order to simplify the equations we assume that the
growth rate μi is constant during the dna duplication and
equal to the value at the onset of the duplication (Vi = Vpi).
Then relationship between Vdi and Vpi:
Vdi = Vpi eμitDi i = 1,2
To use this relationship for the population level we have
to make assumptions about the volume distribution of in-
dividuals
Experimental data for the distribution of the volumes were
not reported in Dent et al. (1976).
As a first approximation we assume that the volume distri-
bution is proportional to V −2 which is the steady-state cell
size distribution for exponential growth with fixed division
size and division into two equal daughters.
Then we have for the Mean Cell Volume (MCV )
MCVi = ln2 Vdi = ln2 Vpi eμitDi .
0.20.40.60.81
1.21.41.61.8
0 20 40 60 80 100120140
MCV1
[μ m3]
t [h]
E. coli
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MCV2
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D. discoideum
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Literature
• V.E. Dent, M.J. Bazin and P.T. SaundersBehaviour of Dictyostelium discoideum Amoebae and Escherichiacoli Grown together in Chemostat CultureArch. Microbiol., 109:187-194, 1976
• S.A.L.M. Kooijman and B.W. KooiCatastrophic behaviour of myxamoebae. Nonlin. World, 3:77-83,1996
• B.W. Kooi and S.A.L.M. KooijmanExistence and Stability of Microbial Prey-Predator SystemsJournal of Theoretical Biology, 170:75-85, 1994
• B.W. Kooi and S.A.L.M. KooijmanThe Transient Behaviour of Food Chains in ChemostatsJournal of Theoretical Biology, 170:87-94, 1994
