Microbial Prey-Predator System in chemostat · The dynamics of physiologically structured...

Microbial Prey-Predator System inchemostat

B.W. Kooi

Department of Theoretical Biology

VU University, Amsterdam The Netherlands

Email: bob.kooi@vu.nl

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,

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

Function

Y = f(X)

Derivative at point P where X = XP

dX= lim

X→XP

ΔX= lim

X→XP

f(XP +ΔX)− f(XP )

In biology the variable is often a function of time t, saypopulation size N(t). Then the derivative at point tP be-comes

dt= lim

t→tP

Δt= lim

t→tP

f(tP +Δt)− f(tP )

In an ordinary differential equation ODE we have

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 )

∂X= lim

X→XP

ΔX= lim

X→XP

f(XP +ΔX, YP )− f(XP , YP )

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

∂t= lim

t→tP

Δt= lim

t→tP

m(tP +Δt, a)−m(tP , a)

In an partial differential equation we have

∂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)

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)

m(t, a+Δa)−m(t, a)

Δa= −δm(t, a)

Then with limΔt → 0 and limΔa → 0 gives

∂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(ab) m(ad)

m(a+Δa)

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

dtei = ge(li, ei) = vi

fi−1,i − ei

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

juvenile

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

juvenile

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

glucose

bacterium Escherichia coli

slime-mold Dictyostelium discoideum

glucose

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

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

dtNi = −pi,i+1Ni

dtNiEl2i = 2

∫lilini(t, ei, li)gl(li, ei) dli − pi,i+1NiEl2i

dtNiEl3i = 3

∫lil2i ni(t, ei, li)gl(li, ei) dli − pi,i+1NiEl3i .

gl(li, fi) =vifi−1,i − kmigili

3(fi−1,i + gi)

Use of this equation yields

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

dNiEl3idt

= (μi−1,i − pi,i+1)NiEl3i .

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

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) = − ∂

(ni(t, li)

)− 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

where μi−1,i is the overall population growth rate

μi−1,i =kmigi

3 (fi−1,i + gi)

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

dtei = νi−1,i (fi−1,i − ei) with fi−1,i =

Xi−1

ki−1,i +Xi−1

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

dtXi =

∫eil3i ni(t, ei, li)

(ei + gi)dei dli − pi,i+1Xi .

We would like to have

dtXi =

(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

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

juvenile

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

dtX0 = D(Xr −X0)−

X0X1I0,1

k0,1 +X0

dte1 = ν0,1

k0,1 +X0− e1

dtX1 =

(ν0,1e1 − kM1g1

e1 + g1−D

)X1 − X1X2I1,2

k1,2 +X1

dte2 = ν1,2

k1,2 +X1− e2

dtX2 =

(ν1,2e2 − kM2g2

e2 + g2−D

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,

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

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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

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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

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