J. theor. Biol. (1999) 197, 371–392
Article No. jtbi.1998.0881, available online at http://www.idealibrary.com on
The Application of Mass and Energy Conservation Laws
in Physiologically Structured Population Models of
Heterotrophic Organisms
S. A. L. M. K*†, B. W. K*  T. G. H‡
*Department of Theoretical Biology, Vrije Universiteit, de Boelelaan 1087, 1081 HV
Amsterdam, The Netherlands and ‡Department of Ecology & Evolutionary Biology,
University of Tennessee, Knoxville, TN 37996, U.S.A.
(Received on 5 March 1998, Accepted in revised form on 26 November 1998)
Rules for energy uptake, and subsequent utilization, form the basis of population dynamics
and, therefore, explain the dynamics of the ecosystem structure in terms of changes in
standing crops and size distributions of individuals. Mass fluxes are concomitant with energy
flows and delineate functional aspects of ecosystems by defining the roles of individuals and
populations. The assumption of homeostasis of body components, and an assumption about
the general structure of energy budgets, imply that mass fluxes can be written as weighted
sums of three organizing energy fluxes with the weight coefficients determined by the
conservation law of mass. These energy fluxes are assimilation, maintenance and growth, and
provide a theoretical underpinning of the widely applied empirical method of indirect
calorimetry, which relates dissipating heat linearly to three mass fluxes: carbon dioxide
production, oxygen consumption and N-waste production. A generic approach to the
stoichiometry of population energetics from the perspective of the individual organism is
proposed and illustrated for heterotrophic organisms. This approach indicates that mass
transformations can be identified by accounting for maintenance requirements and overhead
costs for the various metabolic processes at the population level. The theoretical background
for coupling the dynamics of the structure of communities to nutrient cycles, including the
water balance, as well as explicit expressions for the dissipating heat at the population level
are obtained based on the conservation law of energy. Specifications of the general theory
employ the Dynamic Energy Budget model for individuals.
7 1999 Academic Press
1. Introduction
The importance of ecological energetics, a
confluence of ecology and thermodynamics,
has been widely recognized since Lindeman’s
pioneering efforts (Lindeman, 1941, 1942) but
these concepts go back at least to Lotka (1924),
†Author to whom correspondence should be addressed.
E-mail: bas.bio.vu.nl
0022–5193/99/007371 + 22 $30.00/0
who was interested in a ‘‘law of maximum
energy’’ for biological systems. Wiegert’s synopsis (Wiegert, 1976) covers a trophic structure
energetics perspective of ecological systems
along with some of its inherent difficulties.
Because energy limitations frequently control
populations, the coupling between mass and
energy fluxes is fundamental to ecological
energetics. The application of both mass and
7 1999 Academic Press
372
. . . .    .
energy conservation laws to community dynamics has been considered too complicated to be
practical. For example, Berryman et al. (1995)
state
‘‘However, it can be argued that strict adherence to the
laws of conservation may unnecessarily constrain
predator–prey theory because predators do not always
kill their prey and sometimes kill without eating.’’
We think, however, that this does not hamper
the application of conservation laws; kills not
eaten combine with deaths via ageing or other
causes, in an explicit flux. In our opinion,
population dynamical theories and models
could benefit considerably by making explicit use
of conservation laws. The discipline of ecology
has few basic principles it can rely on, so it is
important to not dismiss the valid ones too
readily.
This paper presents a generic framework to
account for both conservation of mass and
energy in heterotrophic organisms (animals and
micro-organisms), a discussion of autotrophic
organisms (plants) is beyond the scope of this
paper. The theory can be extended to include
autotrophs (Kooijman, 1998; Kooijman &
Nisbet, 1999), however, their metabolic versatility involves more state variables. The interaction
between the individuals being restricted to simple
competition, our treatment can be judged as just
a step towards a more realistic (and complex)
modelling framework (Grover, 1997). We
demonstrate that energy uptake and utilization
govern the dynamics of the structure of the
ecosystem in terms of changes in standing crops
and size distributions of individuals and we
indicate that mass fluxes are the basis of
functional aspects of ecosystems, explaining the
roles of populations in the ecosystem. The
stoichiometry of metabolic transformations at
the population level is developed from an
energetics perspective; this indicates that mass
and energy fluxes are intimately linked to each
other through the concept of homeostasis of
body components. Consequently, the rules for
energy uptake and use do not allow supplementary assumptions on mass fluxes, such as
respiration or nitrogen waste, without creating
inconsistencies. In Reiners’ opinion (Reiners,
1986) models for energy and mass fluxes in
ecosystems are complementary, and can be
developed independently. He does recognize that
these models have many points of intersection;
we will show here that the intersection is
complete for heterotrophic organisms; mass and
energy fluxes should be considered as two aspects
of the same concept.
We believe that ecological energetics has a
wider applicability than its name suggests,
because of the coupling between energy and
mass. Many populations are limited by the
availability of nitrogen rather than energy
(White, 1993). In an energetic framework, this
situation translates to a reduced efficiency of the
conversion of food into assimilation energy,
because of the nutritional inadequacy of the
food. A more appropriate name for ecological
energetics could refer to limitations by a single
component of the resource.
Energy fluxes are generally difficult to measure
directly. Attempts to measure energy fluxes via
respiration rates (i.e. carbon dioxide or oxygen
fluxes, also called ‘‘metabolic rates’’) have
created problems, because a careless mapping of
respiration rates to energy fluxes can easily yield
incorrect budgets. Although actual growth in
biomass during the short period of the measurement of respiration rates can be negligibly small,
this does not imply that energy investment into
the growth process is also negligibly small.
Respiration is frequently, but incorrectly,
identified with routine metabolic costs, conceived as maintenance costs. The interpretation
of respiration rates in terms of energy investments has been the source of a long standing
problem in animal physiology: why is the slope
of the regression line of the log metabolic rate
against the log body weight about 0.75? Withers
(1992) calls this ‘‘one of the most perplexing
questions in biology’’. A sound approach to the
relationship between respiration and energy
fluxes provides a solution for this problem
(Kooijman, 1986, 1993).
Although the specification of energy fluxes
(known as ‘‘powers’’) is dependent upon the
individual model employed [such as the Dynamic
Energy Budget (DEB) model (Kooijman, 1993)],
the coupling between energy and mass fluxes
holds for a broad class of models representing
uptake and utilization of resources. We here
  / 
show how the coupling follows from consistency
arguments given a short list of seemingly simple
and ‘‘harmless’’ assumptions about the general
structure of the budget model. This powerful
result can be rephrased by pointing to the far
reaching implications of these assumptions; if
these implications do not apply, the general
assumptions should be reformulated, and their
harmlessness is deceptive.
As an example of the application of the theory
for mass fluxes in ecosystems we consider a
myxamoeba–bacteria–glucose food chain in the
chemostat. Experimental data by Dent et al.
(1976) have been used for parameter estimation,
and the mass fluxes have been analysed (Kooi &
Kooijman, 1994b; Kooijman & Kooi, 1996). We
here present the transfer of mineral compounds,
carbon dioxide, water, oxygen and nitrogen
waste that are implied by the organic fluxes. The
predator as well as the prey propagate by binary
fission and this justifies the use of a simplified
version of the (DEB) model (Kooi & Kooijman,
1994a). With this model for the individuals, the
steps from the physiology of the individual to
fluxes at the population level and further to
multi-species systems can be made on the
assumption that conspecific individuals only
interact by competition (Kooi & Kooijman,
1995). Although this simple assumption about
interactions seems to apply in the artificial
environment of a well-mixed chemostat in this
case, most less artificial environments require the
modelling of elaborate forms of interactions,
spatial structure, and many other effects of the
local environment. Many of these complicating
phenomena affect population dynamics via
feeding (while effects on reproduction, development, growth and survival follow from effects on
feeding), which means that the equations
describing population dynamics should be
adjusted. Most of the theory that is presented
here about mass-energy coupling, however, still
applies, given the feeding flux. The formulation
of realistic models for population dynamics in
these more complex situations is beyond the
scope of this paper.
We first summarize the mass fluxes and
dissipating heat flux for an individual. Ratios
between these mass fluxes represent stoichiometric coefficients. Theory on the stoichiometry
373
of metabolic transformations has been developed
in the microbiological literature, which selects
the food uptake flux as a reference. Because
embryos do not eat and grow at the expense of
reserves, these ratios are not practical for
animals with an embryonic life stage. A
stoichiometry on the basis of the reserve flux has
been worked out (Kooijman, 1995), but in this
paper a substantial simplification is obtained by
working with fluxes directly and avoiding the use
of stoichiometric coefficients. Following fluxes
through individuals, we consider fluxes for a
population based on these individuals which are
assumed to interact only via competition for the
same resource. We then discuss mass fluxes in
food chains as a step towards ecosystems. We
demonstrate that this approach is extremely
convenient for analysing the processes of
nutrient cycling at the ecosystem level, provided
only that a model for the energetics of an
individual is prescribed.
2. Mass Fluxes for Individuals
Table 1 gives the man symbols and notation;
Table 2 lists all assumptions about general
aspects of mass and energy fluxes.
 
Our approach is to represent and follow the
transfer of chemical elements, because elements
obey conservation laws; compounds generally do
not. For illustration, we follow the four most
abundant elements in living systems, C, H, O and
N, but this list can be extended readily because
each new element comes with a corresponding
balance equation. We delineate two sets of
chemical compounds:
2.1.
‘‘Mineral’’ (M)
C
carbon dioxide
H
water
O
oxygen
N
nitrogen waste
‘‘Organic’’ (O)
X substrate (food)
V structural body mass
E
reserves
P
product (faeces)
The structural body mass (V) and reserves (E)
constitute the individual (Assumption 1 in
Table 2), the other organic compounds and the
minerals define the chemical environment of
the individual. For simplicity’s sake, water in the
nitrogen waste (urine) is included in its chemical
. . . .    .
374
T 1
List of main symbols. The symbols in the dimension-column stand for t time, e energy, (
number (i.e. C-mole). Vectors and matrices are denoted in bold face; the notation nT means
the transpose of n. All rates are designated with dots. The symbol * is used as a placeholder,
for which another symbol can be substituted
Symbol
Dimension
Interpretation
X, V, E, P
C, H, O, N
7
Indices for
compounds
6
Food, body mass, reserves, product ($O)
Carbon dioxide, water, oxygen, nitrogen waste ($M)
Mm , M, Mh
Gm , G, R
A, D, C, T
7
Indices for
energies
6
Maturity maintenance, somatic —, endothermic heating —,
maturity growth, somatic —, reproduction,
assimilation, dissipative, catabolic, heat
n*1*2
nM, nO
J*, JM, JO
M*, M*0, M*m
MX , MK
pt*, pt
m*, mM, mO
m*1*2
h
t, a
f
e
l, lb , lp , la
ht, hta , hte
R
f, fe
F, Fe
N, Ne
—
—
(t − 1
(
(l − 3
et − 1
e( − 1
e( − 1
(e − 1
t
—
—
—
t−1
(t − 1
t−1
(l − 1t − 1
(l − 3
Number of atoms of element *1 in compound *2 per C-atom
Matrix of chemical indices of minerals, organic compounds
Flux of compound *, —of minerals, organic compounds
Mass of compound * (* = E,V), initial mass, maximum mass
Density of mass of food, —as saturation constant
Power *, the three basic powers ptA , ptD , ptG
Chemical potential of compound *, —of minerals, organic compounds
Power *2 per flux of mass *1
Matrix of coefficient that weigh powers to obtain mass fluxes
Time, age
Ingestion as fraction of its maximum, given l: scaled function response
Reserve density as fraction of its maximum
Boby length as fraction of its maximum, —at birth, puberty, division
Hazard rate, —for ageing, —for embryos
Reproduction rate
Stable age distribution of juveniles + adults, —of embryos
Relative frequency density of juveniles + adults, —of embryos
Density of number of juveniles + adults, —of embryos
‘‘composition’’, as is done for methane in faeces
(which is relevant for mammals). Faeces includes
bile and enzymes that are excreted in the gut,
since these excretions are tightly coupled to the
feeding process. In contrast to the ‘‘static’’
energy budget tradition, urine production (the
nitrogen waste) is not tightly coupled to the
feeding process, because maintenance processes
contribute via protein turnover.
Food for micro-organisms is usually called
‘‘substrate’’, and faeces ‘‘metabolic products’’.
These products generally do not originate from
substrate directly, but indirectly through the
metabolic machinery of the organism. This
problem is addressed by including such products
into the overheads of the three basic energy
fluxes (Assumption 3 in Table 2). The number of
different products can be extended in a
straightforward manner. Not only bacteria and
fungi produce compounds that are excreted into
the environment, many animals do this as well
(e.g. mucus, moults, milk). The (energy/carbon)
T 2
Assumptions about general aspects of energy and mass fluxes through an individual
1. The amounts of structural body mass and reserves are the state variables of the individual; body mass and reserves are
invariant in composition (strong homeostasis assumption).
2. Food is converted into faeces; food and faeces are invariant in composition.
3. Assimilates derived from food are added to reserves, which fuel all other metabolic processes. These processes are
classified into three categaories: synthesis of structural body mass, of (embryonic) reserves (i.e. reproduction), and
processes not associated with net synthesis. Products that leave the organism may be formed in direct association with
any or all of these three categories of processes, and with the assimilation process.
4. If the individual propagates via reproduction (rather than via division), it starts in an embryonic stage that initially has
a negligibly small structural body mass, but a substantial amount of reserves.
  / 
substrate for micro-organisms can be poor in
nitrogen, such that nitrogen must be taken up
from the environment, rather than excreted. The
compound ‘‘nitrogen waste’’ (this terminology is
appropriate for metazoans living on protein-rich
food) should be identified by ‘‘nitrogen source’’;
the sign of the flux defines uptake or excretion.
Bacteria that live on glucose as energy source will
have a negative nitrogen waste flux. Nitrogen
(and/or oxygen for aerobic organisms, and/or
cabon dioxide for bacteria feeding on methane)
is assumed to be available in sufficient quantity.
The theory can be extended, however, to include
simultaneous limitations, (cf. Zonneveld, 1996,
1997; Zonneveld et al., 1997; Kooijman, 1998).
Assumptions 2 and 3 in Table 2, when pushed
into the extreme, imply that chemical potentials
of the organic compounds per C-mole are
constant, which is consistent with the biochemical literature (Westerhoff et al., 1983). The
motivation of these assumptions can be based on
the idea that structural body mass and reserves
mainly consist of polymers (polysaccharides,
lipids and proteins), which do not take part in
the metabolism directly, while the concentration
of monomers, which are directly involved in
metabolism, is low and constant. Food (substrate) is digested intracellularly, or in the gut,
which also represents a constant chemical
environment. Products are also formed intracellularly; variations in concentrations in the
environment are assumed not to affect energy
considerations for individuals.
  fi
Assumption 3 implies that the relationships
between powers and mass fluxes involve three
groups of basic powers:
2.2.
F
GṗA J
G
ṗ 0 gṗD h
G
fṗG G
j
assimilation power (coupled to food intake)
= dissipating power (no net synthesis of biomass). (1)
anobolic power (somatic growth)
A variety of metabolic processes contributes to
dissipating power; it is sufficient at this point to
assume that the dissipating power is a function
375
of the two state variables, V and E, of the
individual and not delineate the representation.
Most of the dissipating power leaves the
thermodynamic system, consisting of the individual and relevant organic and mineral compounds, as heat, while a portion leaves the
system as nitrogen waste or (other) products.
Part of the growth and assimilation power will
also contribute to dissipating heat because of the
overhead costs; growth and assimilation do not
occur with 100% efficiency (see below).
Reproduction power ptR has a special status
because reserves of the adult female are
converted into reserves of the embryo, each of
which have the same composition by virtue of
Assumption 1 in Table 2. The efficiency of this
conversion is denoted by kR , which means that
(1 − kR )ptR is dissipating and kRptR returns to the
compound class ‘‘reserve’’, but now of the
embryo. The amount of reserves allocated to
reproduction during a very small time increment
is very small, not nearly enough to make one
embryo. This property, shared by all timecontinuous models, necessitates the existence of
a buffer of reserves with destination reproduction. Reproduction itself, i.e. the conversion of
the reserves in this buffer to embryos, is treated
as an instantaneous event. The overhead costs of
the reproduction event are taken into account in
the allocation to reproduction through the
parameter kR .
In the considerations below, the fluxes of
reserves and reserves in the reproduction buffer
are added. This makes sense biologically,
because the buffer is still in the individual; the
reason for the addition is that the assumptions in
Table 2 imply that the sum of both fluxes will be
shown to be a weighed sum of the three basic
powers, however, this does not necessarily hold
for each of the fluxes separately. (It does not hold
in the DEB model, for instance.) Mineral fluxes
depend only on the sum, so there is no need to
treat them separately. From a chemical point of
view, reproduction does not represent a transformation, because reserves are converted into
reserves with an identical composition, while the
overhead costs for reproduction contribute to
dissipating power. The inclusion of reproductive
investment into the reserve flux allows generalization to organisms that propagate via division
. . . .    .
376
T 3
Assumptions of the DEB model, in addition to the ones listed in Table 2
5. The transition from embryo to juvenile initiates feeding. The transition from juvenile to adult initiates reproduction and
ceases maturation. Transitions occur when the cumulated energy invested in maturation exceeds a threshold value.
6. Somatic and maturity maintenance are proportional to body volume, but maturity maintenance does not increase after
a given cumulated investment in maturation. Heating costs for endotherms are proportional to surface area.
7. The feeding rate is proportional to surface area and depends hyperbolically on food density.
8. The reserves must be partitionable, such that the dynamics are not affected, and the energy density at steady state does
not depend on structural body mass (weak homeostasis assumption).
9. A fixed fraction of energy, utilized from the reserves, is spent on somatic maintenance plus growth, the rest on maturity
maintenance plus maturation or reproduction (the k-rule).
delineates these relationships for the DEB model
for a three-Stage–three-Dimensional (3S–3D)
isomorph, i.e. an individual that does not change
in shape during growth, and reproduces via eggs,
which implies the three stages embryo, juvenile
and adult. Since the DEB model requires that
food uptake is proportional to surface area
(Assumption 7 in Table 3), volume/mass
conversions should be considered; we present
them in Appendix A.
Table 5 presents the DEB model for a
one-Stage–one-Dimensional (1S–1D) isomorph,
i.e. an individual that grows only in length, so
that surface area is proportional to volume, and
divides into two identical daughter individuals.
This simplifies the DEB model considerably.
Bacteria and fungi are interesting examples of
1S–1D isomorphs. Table 5 also specifies the three
basic powers for the well-known Marr–Pirt
(Marr et al., 1969; Painter & Marr, 1968; Pirt,
but do not allocate to reproduction. Division is
treated here as an instantaneous event, which
occurs when the individual reaches a threshold
size. Details about the division only play a role
at the population level because individuals are
followed up to the division event. Division is
assumed to produce two identical daughter
individuals, but this restriction can be relaxed
in several ways, without affecting the main
argument.
2.2.1. Specific budget models
Table 3 gives the assumptions of the DEB
model, in addition to the ones listed in Table 2.
These assumptions have been underpinned
mechanistically and tested against empirical data
(Kooijman, 1993, Hanegraaf 1997). The three
groups of powers should be specified as functions
of the state of the individual. Figure 1 displays
the energy fluxes. As an example, Table 4
X
Food
P
Faeces
A
Heat
Storage
Mh
M
Somatic
work
Gametes
C
Mm
Maturity
work
R
G
Gm
Volume
Maturity
F. 1. Energy fluxes through a heterotroph. The rounded boxes indicate sources or sinks. All powers contribute to
dissipating heat, but this is not indicated in order to simplify the diagram. The powers ptX = JtXmX and ptP = JtPmP for ingestion
and defecation ‘‘connect’’ the individual with the environment. The DEB model assumes that JtX AJtP AptA . The dissipating
power is ptD = ptMm + ptM + ptMh + ptGm + (1 − kR )ptR .
  / 
377
T 4
The powers as specified by the DEB model for a 3S–3D isomorph of scaled length l and scaled reserve
density e at scaled functional response f 0 X/XK + X, where X denotes the food density and XK the
saturation constant. Relationships are given in the diagram 1. The table presents scaled powers, where
mE denotes the chemical potential of the reserves. Parameters: g investment ratio, kM maintenance rate
coefficient, k partitioning parameter for catabolic power, lh scaled ‘‘heating length’’. Ectotherms do not
heat, i.e. lh = 0. Implied dynamics for e q l q lb: d/dt e = f − e/lktM g and d/dt l = e − l − lh /e/g + 1
ktM /3
Power
mEMEmktMg
Embryo
0 Q l E lb
Juvenile
lb Q l E lp
Adult
lp Q l Q 1
Assimilation, ptA
0
fl 2
fl 2
Catabolic, ptC
g+l
el 2
g+e
el 2
el 2
Somatic maintenance, ptM
kl 3
kl 3
kl 3
Maturity maintenance, ptMm
(1 − k)l 3
(1 − k)l 3
(1 − k)lp3
Endothermic heating, ptMm
0
kl 2lh
kl 2lh
Somatic growth, ptG
e−l
kl 2
e/g + 1
kl 2
Maturity growth, ptGm
e−l
(1 − k)l 2
e/g + 1
(1 − k)l 2
0
Reproduction, ptR
0
0
(1 − k)(l 2
g + l + lh
g+e
g + l + lh
g+e
e − l − lh
e/g + 1
e − l − lh
e/g + 1
kl 2
e − l + lhe/g
e/g + 1
1965) and the Monod model (Monod, 1942) for
microbial growth.
e − l + lhe/g
+ l 3 − lp3 )
e/g + 1
chemical indices of the organic compounds for
carbon equal to 1 and refer to their amounts as
C-moles. The homeostasis Assumptions 1 and 2
in Table 2 are equivalent to the condition that
the chemical indices do not change.
Let J* denote the rate of change of the
compound * as a result of utilization (J* Q 0) or
 
Let n*1*2 denote the chemical index of
compound *2 for element *1. We will choose the
2.3.
T 5
The powers as specified by the DEB model for a 1S–1D
isomorph of scaled length l and scaled reserve density
e at scaled functional response f. We take ptMh = ptR = 0,
so that ptD = ptM + ptMm + ptGm, and 2 − 1/3ld Q l E ld. An
individual of structural volume V 0 MV/[MV] takes up
substrate at rate [JXm]fV. The implied dynamics for e
and l: d/dt e = f − e/ld ktMg and d/dt l = ktMl/3 e/ld − 1/
e/g + 1. We also present the Marr–Pirt and the Monod
models in the same notation
Power
mEMEmktMg
DEB
Marr
Monod
Assimilation, ptA
l f/ld
l f/ld
l 3f/ld
Dissipating, ptD
l3
l3
0
e/l − 1
l3 d
e/g + 1
l 3f/ld − l 3
l 3f/ld
Somatic growth, ptG
3
3
. . . .    .
378
production (J* q 0) by the individual. The
conservation of mass states
F1
G0
0=G
2
G0
f
0 0 nCN J FJC J
2 0 nHN G GJH G
1 2 nON G GJO G
GG G
0 0 nNN j fJN j
1
1
1 J F JX J
F1
Gn
nHV nHE nHP G G JV G
HX
+G
G GJ + J G (2)
n
n
n
n
GnOX nOV nOE nOP G G E J RG
f NX NV NE NP j f P j
This can be summarized in matrix form as
O = nM− 1JM + nOJO. Thus the fluxes for the
‘‘mineral’’ compounds can be written explicitly
as
JM = −nM− 1nOJO
(3)
with
n
F
0
0
− CN
G 1
nNN
G
nHN
−1
2
0 −
G 0
2nNN
n−1
M =G
n
G −1 −4−1 2−1 4nNN
G
1
0
0
0
G
n
NN
f
J
G
G
G
G (4)
G
G
G
j
and n 0 4nCN + nHN − 2nON .
We will now explain why the organic fluxes JO
follow from the basic powers pt via
F
J
J F−m−1
0
0 J
G JX G G 0 AX
−1 G
0
m
V
GV
G
= G −1
G
JO 0 G
−m−1
mE
JE + JR
−m−1
E
E
G
G
G
G
−1
f JP j f m−1
mDP
m−1
AP
GP j
F
GṗA J
G
× gṗD h 0 hṗ (5)
G
fṗG G
j
where mE is the chemical potential of the reserves,
and m*1*2 denotes the power *2 per flux of mass *1,
i.e. the coupling between mass and energy fluxes.
The m*1*2s serve as model parameters.
The substrate flux JX = −ptA /mAX follows from
Assumptions 1 to 3 in Table 2, which imply a
constant conversion coefficient from food to
assimilation energy. We quantify assimilation
energy by its fixation into reserves, so reserves
are formed at a rate ptA /mE , where mE stands for
the chemical potential of the reserves. The ratio
mAX /mE equals the C-moles of food ingested per
C-mole of reserves formed. The amount of work
that can be done by ingested food is mXJX ; a part,
ptA , is fixed into reserves, a part, ptAmP /mAP , is fixed
into product, and the rest dissipates as heat and
mineral fluxes associated with this conversion.
The ratio mAX /mAP equals the C-mole of food
ingested per C-mole of product that is derived
directly from food; products can also be formed
indirectly from assimilates, via the processes of
growth and maintenance, explaining the product
flux JP .
If the individual is a metazoan and the product
−1
−1
is interpreted as faeces, we take mDP
= mGP
= 0.
Faeces production is then coupled to food intake
only. Alcohol production by yeasts that live on
glucose is an example of product formation
−1
−1
where mDP
$ 0 and mGP
$ 0. Carbon dioxide and
water partially serve as faeces here; this is taken
into account by (3) and (5), via the coefficients
for the assimilation flux. At this point, molecular
details about the process of digestion being intraor extra-cellular are not needed. This knowledge
only affects the precise interpretation of the
coefficients in h.
The body mass flux JV = ptG /mGV follows from
Assumptions 1 and 2 in Table 2, which imply
that a constant amount of energy, mGV , is invested
per C-mole of structural body mass. Note that mV
is actually fixed in a C-mole of structural body
mass, so that mGV − mV dissipates (as heat or via
products that are coupled to growth) per C-mole.
The flux of (parent) reserves is given by
JE = mE− 1 (ptA − ptC ), because reserve energy is
generated by assimilation and used by
catabolism, i.e. the sum of all other metabolic
powers (Assumption 3 in Table 2). The catabolic
power can be written as ptC = ptD + ptG + kRptR .
The flux of embryonic reserves (reproduction),
JR = mE− 1kRptR , appears as a return flux to the
reserves because embryonic reserves have the
same composition as that of the parent as a
consequence of the homeostasis Assumption 1.
The sum of the (parent) reserve and embryonic
reserve fluxes amounts to JE + JR = mE− 1
(ptA − ptG ). This completes the derivation of (5).
  / 
.
–1 J X
.
.
10 ( J E + J R )
Structure biomass
Reverses
.
1 JP
Faeces
.
flux J *
Food
.
40 J V
379
l
.
2 JC
.
–2 J O
Water
Oxygen
.
10 J N
Ammonia
.
J*
Carbon dioxide
.
2 JH
Scaled length l
F. 2. The organic fluxes JtO (top) and the mineral fluxes JtM (bottom) for the DEB model as functions of the scaled
length l at abundant food (e = 1 for l q lb ; 0 Q l Q 1). The various fluxes are multiplied by the indicated scaling factors
for graphical purposes. The stippled curve separates adult from embryonic reserves (reproduction). The parameters: scaled
length at birth lb = 0.16,—at puberty lp = 0.5 (both indicated on the abscissa), scaled heating length lh = 0 (ectotherm),
energy investment ratio g = 1, partition coefficient k = 0.8, reproduction efficiency kR = 0.8. The coefficient matrices are:
F
0
0 J
F
G−1.5
G
G10
0
0
0.5
G, nM = G2
mE h = G
1
−1 −1
G
G
G
0
0 j
f 0.5
f0
Figure 2 illustrates the fluxes of organic and
mineral compounds, JO and JM, of the DEB
model as a function of the structural body mass
(i.e. scaled length, see next section), at abundant
food. The embryonic reserve flux is negative
because embryos do not eat. The growth just
prior to birth is reduced because the reserves
become depleted. The switch from juvenile to
adult implies a discontinuity in the mineral
fluxes as functions of size (not age), but this
discontinuity is negligibly small. The reason of
the discontinuity is that energy invested in
development dissipates because it is not fixed in
mass, while energy invested in reproduction is
(partly) fixed in (embryonic) reserves.
 . 
The reserves mass and the structural body
mass relate to the fluxes as ME (a) = ME0 + f0a
JE (t) dt and MV (a) = MV0 + f0a JV (t) dt. Assumption 4 in Table 2 states that the initial value of
2.4.
0
2
1
0
0
0
2
0
0J
G
3
G,
0
G
1j
F
1
1
1
1J
G1.8
G
1.8 1.8 1.8
nO = G
G.
0.5 0.5 0.5 0.5
G
f0.2 0.2 0.2 0.2G
j
the structural body mass is negligibly small, i.e.
MV0 = 0. The mass of reserves of an embryo in
C-moles at age 0, ME0, might be introduced as a
model parameter, but the DEB model obtains
the value from the constraint that the reserve
density of the embryo at birth equals that of the
mother, i.e. e(ab ) = f.
The changes in structural body mass, MV , and
reserve mass, ME , relate to the powers as d/dt
MV = JV = ptG /mGV and d/dt ME = JE = (ptA − ptC )/
mE . If the model for these powers implies the
existence of a maximum for the structural body
mass, MVm , and for the reserve mass, MEm , it
proves convenient to replace the state of the
individual, MV and ME , by the scaled length
l 0 (MV /MVm )1/3 and the scaled energy reserve
density e 0 MEMVm /(MVMEm ). The change of the
scaled state then becomes
d
pt /m
l = G2/3 GV1/3
dt
3MV MVm
. . . .    .
380
and
0
1
d
MVm
ptA − ptC ME ptG
e=
−
.
dt
MVMEm
mE
MV mGV
The reproduction rate, in terms of number of
offspring per time, is given by Rt = JR /ME0. The
three basic powers, supplemented with the reproductive power, therefore, fully specify the
individual as a dynamic system.
   
We introduced the dissipating power ptD , which
is an element of pt, to quantify a group of powers,
such as maintenance, that is not allocated to
biomass production. In addition to this energy
loss, heat dissipates in association with the
processes of assimilation and growth because
these processes are less than 100% efficient. The
total dissipating metabolic heat ptT follows from
the energy balance equation
2.5.
0 = ptT + mTOJO + mTMJM
= ptT + (mTO − mTMnM− 1nO)mpt (6)
where
mTM 0 (mC mH mO mN ) and mTO 0 (mX mV mE mP )
designate the chemical potentials of the various
mineral and organic compounds, respectively.
The thrust of the formulation is that the energy
allocated to reserves and structural body mass
appears as parameter values, while the energy
fixed in these masses is given by the chemical
potentials, the differences appearing as dissipating heat, i.e. overhead costs.
The dissipating metabolic heat contributes to
the thermal fluxes to and from the individual.
The individual loses heat via convection and
radiation at a rate ptTT = 4vtT 5 (Tb − Te )V2/
3 + 4vtR 5 (Tb4 − Te4 )V2/3. Here Te denotes the
absolute temperature in the environment, including a relatively large sphere that encloses the
individual. For radiation considerations, the
sphere and individual are assumed to have gray,
opaque diffuse surfaces. Tb is the absolute
temperature of the body; V 2/3 is the body surface
area; 4vtT 5 is the thermal conductance and
4vtR 5 = os is the emissivity times the Stefan–
Boltzmann constant s = 5.67 × 10 − 8 Jm − 2 s − 1
K − 4; see, for instance (Kreit & Black, 1980). The
body temperature does not change if ptT = ptTT .
This relationship can be used to obtain the body
temperature, given knowledge about the other
components. Most animals, especially the
aquatic ones, have a high thermal conductance,
which gives body temperatures only slightly
above the environmental ones. Endotherms,
however, heat their body to a fixed target value,
usually some Tb = 312 K, and have a thermal
conductance as small as 4vtT 5 = 5.43 J cm − 2
hr − 1 K − 1 in birds and 7.4–9.86 J cm − 2 hr − 1 K − 1
in mammals, as calculated from Herreid II &
Kessel (1967), (see Kooijman, 1993).
Most endotherms are terrestrial and lose heat
via evaporation of water at a rate ptTH . Here we
can use the relationship ptT = ptTH + ptTT to obtain
the thermoneutral zone: the minimum environmental temperature at which no endothermic
heating is required (ptMh = 0). Alternatively, we
can deduce the heating requirements at a given
environmental temperature. To this end, we first
consider the water balance in more detail, to
quantify the heat ptTH that goes into the
evaporation of water. The individual loses water
via respiration at a rate proportional to the use
of oxygen, i.e. JHO = dHOJO , (see Kooijman, 1995;
Verboven & Piersma, 1995), and via transpiration, i.e. cutaneous losses. The latter route
amounts to 2–84% of the total water loss in birds
despite the lack of sweat glands (Dawson, 1982).
Water loss JHH via transpiration is proportional
to surface area of the body, to the difference
between vapour pressure of water in the skin and
the ambient air, to the square root of the wind
speed, and depends on behavioural components.
To maintain homeostasis, the animal has to
drink at a rate JH − JHO − JHH . The heat loss by
evaporation amounts to ptTH = mTH (JHO + JHH ),
with mTH = 6 kJ mol − 1. Within the thermo-neutral zone, endotherms often control their body
temperature by evaporation, through panting or
sweating, which affects the water balance, and
consequently, requires enhanced drinking. The
present reasoning can be used to work out
the quantitative details.
2.5.1. Thermodynamic constraints
Given the assumption of constant partial
molar chemical potentials, and, therefore, con-
  / 
stant partial molar entropies, for the organic
compounds, the second law of thermodynamics
implies that the processes of assimilation,
dissipation and growth are exothermic. Hence,
we can decompose the dissipating heat into three
positive contributions ptTT 0 (ptTA ptTD ptTG ), with
ptTT 1 = ptT and 1T = (1 1 1), which follow from the
balance equations for these three processes
−1
OT = ptTT + (mTO − mTMnMM
nO)h diag(pt)
(7)
where diag(pt) is the diagonal matrix with the
elements of pt on the diagonal. The sum of the
three equations (7) returns the overall balance
equations (6), because diag(pt)1 = pt. We see that
the heat that dissipates in connection with a basic
power, is proportional to that power. We also see
that the three factors that multiply the basic
powers in these three balance equations, should
all be negative. This implies a constraint for each
column of h. In a combustion frame of reference,
where mM = 0, these constraints translate to
mTOh Q OT. This constraint leaves us with the
problem to obtain the chemical potentials mO,
which we need for the chemical environment
within the organism. The method of indirect
calorimetry can be used for this purpose
(Kooijman, 1995). This method relates the
dissipating heat to the mineral fluxes as
ptT = mTT JM with regression coefficients
mTT 0 (mCT mAT mOT mNT )
(8)
2 (60 0 −350 −590) kJ mol − 1
(9)
for aquatic animals that use ammonia as
N-waste (Brafield & Llewellyn, 1982). Small
corrections have been proposed for birds
(Blaxter, 1989) and mammals (Brouwer, 1958).
The energy balance (6) and the mineral fluxes (3)
lead to the desired result
mTO = mTT nM− 1nO
(10)
in a combustion frame of reference. This
relationship is sufficient, and perhaps necessary,
to obtain the chemical potentials of complex
organic compounds in a complex environment.
If knowledge of these potentials is available
(from previous experiments, for instance), (10)
381
can be used as a constraint on the chemical
coefficients nM and nO.
 
The three basic powers form a basis for a
vector space of mass fluxes. The reserves play the
important role of rendering the three powers
independent. Without reserves, these three
powers only span a two-dimensional vector
space. The Marr–Pirt model for microbial
growth is an example of such a model. Table 5
shows that ptA = ptD + ptG holds for this model.
Within this framework, product formation
should be taken as a weighted sum of two
powers, such as maintenance (and, therefore,
biomass) and growth, as has been proposed by
Leudeking & Piret (1959). The Monod model
(see Table 5) has neither reserves nor maintenance. It is an example of a one-dimensional
vector space for mass fluxes, with ptA = ptG and
ptD = 0. Within this framework, product formation should be taken proportional to growth
(or assimilation) only. The Marr–Pirt model
represents a special case of the DEB model, and
the Monod model represents a special case of the
Marr–Pirt model. We can generalize on the
assumptions of Table 2 in several ways, one
being to add the reproduction flux as a fourth
basic flux to the vector space of mass fluxes. This
increase in flexibility is bought at a cost of
additional parameters.
Mineral and organic fluxes can be partitioned
into the contributions by assimilation, dissipation power and growth in a simple way. When we
write J* = J*A + J*D + J*G for *$4M, O5, and
collect these fluxes in two matrices, we have
2.6.
JM* = −nM− 1nOJO* and JO* = h diag(pt)
(11)
where diag(pt) represents a diagonal matrix with
the elements of pt on the diagonal, so that
diag(pt)1 = pt, and JtM*1 = JM, JO*1 = JO.
If food input is too small for too long a period,
the reserve flux JE will eventually become too
small to ‘‘pay’’ the maintenance costs. The model
should specify what happens in those situations.
One possibility is that the individual dies by
starvation, another is that it shrinks, i.e. JV Q 0.
In the latter case, we might assume that the
. . . .    .
382
structural biomass mineralizes instantaneously,
which contributes to a mineral flux.
We conclude that all mass fluxes, mineral and
organic, as well as the dissipating heat, are
weighted sums of the powers ptA , ptD and ptG . It is
the task of the model for resource uptake and use
to specify these three basic powers as functions
of the state variables of the individual. Because
linear functions of linear functions are linear as
well, dissipating heat is a weighted sum of three
mineral fluxes, which is the basis of the widely
applied method of indirect calorimetry. Its
empirical success can be conceived as support for
the general assumptions about the basic structure of budgets.
3. Mass Fluxes in Populations
The step from individual to population
requires a specification of the interactions
between the individuals. The simplest specification is feeding on the same resource, which
implies competition. The sections below discuss
the mass and energy fluxes in transient states;
steady states are discussed in Appendix B.
  
We now consider a population of parthenogenetically reproducing individuals that develop
through embryonic, juvenile and adult stages.
Sexually reproducing animals can be included in
a simple way, as long as the sex ratio is fixed. The
population structure, derived from the collection
of individuals that compose the population, is
based upon individual characteristics. We will
assume the existence of a maximum amount for
structural body mass and reserves for individuals, and use the scaled length l, the scaled
reserve density e and the age a to specify the state
of the individual, but the techniques to model the
dynamics are readily available for an arbitrary
number of state variables (Hallam et al., 1990,
1992).
Suppose that a population of individuals lives
in a ‘‘black box’’ and that the individuals only
interact through competition for the same food
resource. Food is supplied to the black box at a
constant rate htXMX , where htX has dimension
3.1.
time − 1 and MX is the amount of food (in C-moles
per black box volume). Eggs are removed
from the black box at a rate hte ; juveniles and
adults are harvested with a rate ht randomly, i.e.
the harvesting process is independent of the
state of the individuals (age a, reserves e, size l).
Furthermore, the ageing process harvests juveniles and adults at a state-dependent rate hta , which
is beyond experimental control; (see Kooijman,
1993 for a specification of the ageing process that
is consistent with the DEB model). Individuals
harvested by the ageing process leave the black
box instantaneously.
The present purpose is to study how food
supplied to the black box converts to body mass
and reserves that leave the box in the form of
harvested individuals when the amounts of
oxygen, carbon dioxide, nitrogen waste and
faeces in the black box are kept constant. This
implies that these mass fluxes to and from the
box equal the use or production by all
individuals in the box. We do not remove food,
which implies that the amount of food in the box
depends on both the food supply and the
harvesting rates of individuals.
In summary, the conversion process has three
control parameters: htX , hte and ht and our aim is
now to evaluate all mass fluxes in terms of these
three control parameters, given the properties of
the individuals. This result is of direct interest for
particular biotechnological applications, and for
the analysis of ecosystem behaviour provided we
write the control parameters as appropriate
functions of other populations and/or environmental processes and specify the processes of
degradation of faeces and dead individuals to
recycle the nutrients that are locked into these
compounds.
We will use the index + to refer to the
population, to distinguish fluxes at the population level from those at the individual level.
Embryos are treated separately from juveniles
and adults, not only because we allow different
harvesting rates for both groups, but also
because they do not eat, and therefore do not
interact with the environment via food.
Given the initial conditions Fe (0,a,e,l) and
F (0,a,e,l), the change of the densities of
embryos and of juveniles plus adults over the
  / 
state space is given by the McKendrick-von
Foerster hyperbolic partial differential equation:
0
1
0
1
1
1
d
F (t,a,e,l) = −
(Fe (t,a,e,l) l
1t e
1l
dt
−
1
d
Fe (t,a,e,l) e
1e
dt
−
1
F (t,a,e,l)
1a e
(12)
0
1
0
1
−
1
d
F(t,a,e,l) e
1e
dt
−
1
F(t,a,e,l) − ht
1a
(13)
where faa12 fll12 fee12 F(t,a,e,l) de dl da is the number
of individuals (juveniles plus adults) having an
age somewhere between a1 and a2, a scaled energy
density somewhere between e1 and e2 and a scaled
length somewhere between l1 and l2. The total
number of juveniles plus adults equals N(t) = f0a
fl1b f01 F(t,a,e,l) de dl da. The total number of
embryos likewise equals Ne (t) = f0a f0lb f0a
Fe (t,a,e,l) de dl da, where lb is the scaled length
at birth (i.e. the transition from embryo to
juvenile).
The boundary condition at a = 0 reads, when
Rt(e, l) is the reproduction rate:
g gg
a
0
Hence, the individuals can differ at a = 0,
because e0 can depend on e, and individuals can
make state transitions at different ages and
different scaled reserves.
The dynamics for food amounts to
d
M = htXMX + JX+
dt X+
+ hta (a,e,l))F(t,a,e,l)
d
Fe (t,0,e,l) a = d(e − e0)d(l − l0)
dt
d(e − e0) is the Dirac delta function in e
(dimension: e − 1). The boundary condition at
l = lb reads:
d
d
F(t,a,e,lb ) a = Fe (t,a,e,lb ) a for all a,e
dt
dt
(15)
− hte Fe (t,a,e,l)
1
1
d
F(t,a,e,l) = −
(F(t,a,e,l) l
1t
1l
dt
383
1
0
× R (ẽ,l̃)F(t,a,ẽ,l̃)dl̃ dẽ da for all e,l
1
(16)
where MX+ denotes the food density in C-moles
per black box volume, and JX+ 0 f0a fl1b f01
F(t,a,e,l) JX (e,l) de dl da, where JX (e,l) denotes
the (negative) ingestion rate of an individual of
scaled energy reserves e and scaled length l, as
discussed in the previous section. The faecal flux
JP+ is simply proportional to the ingestion flux,
i.e. JP+/JX+ = JP /JX .
The molar fluxes of body mass and reserves
(* = V, E), are given by
J* + = hte
ggg
a
0
0
a
0
Fe (t,a,e,l)M*(e,l) de dl da
0
g gg
a
+
lb
1
lb
1
(ht+ hta (a))F(t,a,e,l)
0
× M*(e,l) de dl da.
(17)
The mineral fluxes JM + and the dissipating heat
ptT + follow from (3) and (6)
JMM + = −nM− 1nOJO +
(18)
lp
(14)
where l0 denotes the scaled length at a = 0, which
is taken to be infinitesimally small and lp the
scaled length at puberty (i.e. the transition from
juvenile to adult). The quantity e0 is the scaled
reserve at a = 0, which can be a function of e of
the mothers. The function d(l − l0) is the Dirac
delta function in l (dimension: l − 1) and similarly
ptT + = −mTMJM + − mTOktO + = (mTMnM− 1nO − mTOJ)O + .
(19)
Due to the linear relationships between mass and
energy fluxes, the mass fluxes are simple metrics
on the densities Fe and F, which are solutions of
the partial differential equations (12) and (13);
the determination of the solution generally
requires numerical integration.
. . . .    .
384
  
For dividing organisms, we assume that the
ageing rate is independent of age and include this
hazard rate into the harvesting rate ht; the state
variable age is not used, so the scaled length l and
the scaled reserve density e specify the state of
the individual. The conversion process of substrate into biomass has two control parameters:
htX and ht.
Given the initial condition F(0,e,l), the
dynamics of density F(t,e,l) is then given by
3.2.
0
1
1
1
d
F(t,e,l) = −
F(t,e,l) l
1t
1l
dt
−
0
1
1
d
F(t,e,l) e − htF(t,e,l)
1e
dt
(20)
with boundary condition
d
F(t,e,lb ) l
dt
b
d
l
l = lb = 2F(t,e,ld )
dt
b
population level for the total structural body
mass
d
d
ln MV + = ln l 3 − ht
dt
dt
where the scaled volume kinetics dl 3/dt = 3l 2
dl/dt is given by the model for individuals. The
other differential equation is at the individual
level for the scaled reserve density kinetics de/dt,
which should also be specified by the model for
individuals, see e.g. Table 5. The scaled reserve
density kinetics specifies the (nutritional) state of
a random individual.
The expressions for the dissipating heat (19),
and mineral fluxes (18) still apply here, while
JO + = hpt+ , with pt+ = fllbd f01 pt(e,l 3) F (t,e,l) de dl,
and pt(e,l3) denotes pt, evaluated at scaled energy
reserve e and cubed scaled length l3. (For 1Disomorphs it is more convenient to use l3 as an
argument, rather than l.) This result is direct
because fllbd f01 F (t,e,l) de dl = MV + /MVm , so that
pt+ = pt(e,
l = ld
for all e
(21)
where the scaled length at ‘‘birth’’ relates to the
scaled length at division as lb = ld 2 − 1/3, for
organisms that divide into two parts. These
dynamics imply that both daughters are
identical.
Suppose now that the dynamics of the scaled
reserves is independent of the scaled length, and
that the dynamics of the scaled length is
proportional to the scaled length. The scaled
reserve density then has the property that all
individuals eventually will have the same scaled
reserve density, which may still vary with time.
(The DEB model for 1D-isomorphs is an
example of such a model.) For simplicity’s sake,
we will assume that this also applies at t = 0,
which removes the need for an individual
structure. The consequence is that a population
that consists of one giant individual behaves the
same as a population of many small ones.
The partial differential equation (20) collapses
to two ordinary differential equations (Kooi &
Kooijman, 1994a), one of these is at the
(22)
gg
ld
lb
1
l 3F(t,e,l) de dl)
0
= pt(e,MV + /MVm ) = pt(e,1)MV + /MVm .
The latter equality only holds for models such as
the DEB model for 1S–1D isomorphs, where all
powers are proportional to structural body mass.
The dynamics for food amounts to
pt (e,1) MV +
d
,
MX + = htXMX + JX + = htXMX − A
mXA MVm
dt
(23)
where ptA (e,1) does not depend on the scaled
reserves e, in the DEB model.
The environment for the population reduces to
the chemostat conditions for the special choice of
the harvesting rate ht relative to the supply rate:
htXMX = ht(MX − MX + ).
4. Mass Fluxes in Food Chains of Dividers
Suppose that a food chain of dividers lives in
a chemostat, where substrate is supplied at rate
htMX , and population i feeds only at population
i − 1. We assume that all individuals have the
same composition of structural body mass and
reserves, that all are harvested at the specific rate
  / 
ht; and that the only cause of removal of an
individual from its population is by being eaten
in the food chain or harvested. We have studied
the rich dynamics of this food chain extensively
(Kooi et al., 1997b; Boer et al., 1998), and now
study the conversion process from substrate into
biomass with control parameters MX and ht.
Most literature on food chain dynamics allows
growth of the zeroth trophic level, which renders
it very difficult, if not impossible, to complete
mass and energy balances (Kooi et al., 1997a,
1998); we do not allow the substrate to grow.
The dynamics of substrate, structural body
mass and scaled reserves in a chemostat is given by
d
pt (e,1) MV + 1
M = htMX − A1
− htMX +
dt X +
mAX1 MVm 1
= ht(MX − MX + ) + JX + 1
(24a)
pt (e ,1) MV + i ptA(s + 1)(e,1) MV + (i + 1)
d
−
M = Gi i
mGVi MVmi
mAX(i + 1) MVm(i + 1)
dt V + i
− htMV + i = JV + i
+ JX[mos(i + 1) − htMV + i
(24b)
×
0
1
(24c)
where we appended index i to various symbols
(e, MV + , MVm , MEm , ptA , ptC , ptG , mE , mGV , mAX ) to
indicate the species. We take MV + (N + 1) = 0 in
(24b).
The organic, mineral and dissipating heat
fluxes for each species are given by (B.8), (18)
and (19), respectively. The various mineral fluxes
in the food chain are additive, i.e. JM+ + = a3i = 1
ktM + i = −nM− 1 nO ktO+ + . The total organic fluxes
in the food chain are given by JO+ + = a3i = 1 JO + i ,
except for the total substrate flux, i.e. the first
element of JO+ + , which should be replaced by
JX1, because only species 1 eats substrate.

We use the experimental data by Dent et al.
(1976) as an example of the application of our
4.1.
d
M = (MX − MX + )ht− [JXm 0,1]f0,1MV + 1,
dt X +
(25a)
d
kt e − ktM1g1
M
= MV + 1 E1 1
− htMV + 1
dt V + 1
e1 + g1
(25b)
kt e − ktM2g2
d
= MV + 2 E2 2
− htMV + 2,
M
e2 + g2
dt V + 2
(25c)
J
J
eiptGi (ei ,1)
= E + i − ei V + i
MVmimGVi
MEmi
MVmi
MVmi
for i = 1,2, . . ., N
MV + i
theory for mass fluxes in food chains. They grew
myxamoeba (Dictyostelium discoideum), which
fed on bacteria (Escherichia coli), which fed on
glucose (CH2O) in a chemostat at 25°C. Figure 3
gives their observations on cell volumes during
the transient states of the chemostat following
inoculation, together with our model fits based
on eqns (24a–24c) and the specifications of
Table 5. We used the equations for 1S–1Disomorphs, rather than the fully structured
dynamics of 1S–3D-isomorphs, since this simplifying approximation holds well for organisms
that divide on two parts, (cf. Kooijman & Kooi,
1996; Kooijman et al., 1996). The resulting
dynamics for a chemostat with throughput rate
ht and concentration of glucose in the feed MX ,
are
− [JXm 1,2]f1,2MV + 2,
pt (e ,1) − ptDi (ei ,1) − ptGi (ei ,1)
d
e = Ai i
MEmimEi
dt i
−
385
d
e = ktEi (fi − 1,i − ei ) for i = 1,2,
dt i
(25d)
where ei (t), i = 1,2 denote the scaled reserve
densities, [JXmi − 1, i ] the maximum specific ingestion rate and fi − 1,i = MV + (i − 1)/(MK(i − 1) +
MV + (i − 1)) for i = 1,2, is the scaled Holling type
II functional response. The parameters are listed
in Table 6. Figure 3 gives the model fits to the
data, and the phase diagrams. These dynamics
imply very realistic predictions for the changes in
mean cell size (Kooijman & Kooi, 1996).
To arrive at mass fluxes, we assumed that
(nC* nH* nO* nN*)T = (1 2.6 0.9 0.2)T
(26)
for * standing for the structural biomass and the
reserves of the bacteria and the myxamoeba, and
the faeces of the myxamoeba. The bacteria did
not produce any products or faeces; ammonia
(H3N) was the N-waste/source. To couple energy
−1
−1
and mass fluxes, we took mPX
/mAX
= 0 for the
. . . .    .
386
(a)
V c [E m ] 2 ml hr
mm 3
0.2
.
P2
.
,
,
mm 3
V c [E m ] 1 ml hr
0.4
P1
(b)
0.9
0.6
0.3
0.0
0.0
0
50
100
150
0
(c)
1.0
f 1,2 , e 2
f 0,1 , e 1
1.0
0.5
0.0
50
100
150
0
50
100
150
100
150
0.5
M X+1
,
0.5
[M X ] 1 [M V ] 2
M V+2
,
ml
[M V ] 1
[M X ]
150
ml
mm 3
ml
mm 3
,
mg
,
M V+1
100
1.0
,
M X+
50
(f)
1.0
0.0
0.0
0
50
100
150
0
(g)
(h)
0.2
ml hr
[M E ] 2 V c
.
0.1
.
.
0.1
.
[M E ] 1 V c
.
J C+2 , –J O+2,–J N+2
,
mm 3
ml hr
0.2
,
mm 3
150
0.5
(e)
.
100
0.0
0
J C+1 , –J O+1,–J N+1
50
(d)
0.0
0.0
0
50
100
t (hr)
150
0
50
t (hr)
F. 3. Transient behaviour of a glucose–bacterium–myxamoeba food chain in a chemostat of volume Vc . (a, b) Energy
Fluxes: assimilation (– – –), dissipation (....) and growth (——); (c, d) functional response (——) and scaled energy density
(– – –); (e, f) biovolume densities of prey (....) and predator (——), and the data by Dent et al. (1976): glucose (w) and
E. coli (W) in (e), E. coli (w) and D. discoideum (W) in (f); (g, h) CO2 (——) production and O2 (– – –) consumption and
NH3 (....) consumption/production rates.
  / 
387
T 6
The parameter estimates for the DEB food chain model, as applied to the date
by Dent et al. (1976). The reactor volume was 200 ml, the temperature 25°C,
the concentration in the feed 1 mg ml − 1, and the throughput rate h = 0.64 hr − 1.
For glucose we have 1 Cmol = 30 mg and for the bacteria and myxamoeba
both 1 Cmol = 31.8 mm 3, where we assumed that the specific mass equals 1 mg
mm − 3. For simplicity’s sake, we introduced ktE 0 ktMg/ld (see Appendix A)
Parameter
Value
Units
Interpretation
MX + (0)/[MX ]
MV + 1(0)/[MV ]1
MV + 2(0)/[MV ]2
e1(0)
e2(0)
MK1/[MX ]
MK2/[MV ]1
g1
g2
ktM1
ktM2
ktE1
ktE2
0.58
0.46
0.070
1 (def)
1 (def)
0.0004
0.18
0.86
4.43
0.0083
0.16
0.67
2.05
Initial glucose concentration
Initial E. coli density
Initial D. discoideium density
Initial E. coli reserve density
Initial D. discoideum reserve density
Saturation constant E. coli
Saturation constant D. discoideum
Investment ratio E. coli
Investment ratio D. discoideum
Maintenance rate coefficient E. coli
Maintenance rate coefficient D. discoideum
Specific energy conductance E. coli
Specific energy conductance D. discoideum
[JXm 0,1]
0.65
[JXm 1,2]
0.26
mg ml − 1
mm3 ml − 1
mm3 ml − 1
—
—
mg ml − 1
mm3 ml − 1
—
—
hr − 1
hr − 1
hr − 1
hr − 1
mg
mm3 hr
hr − 1
−1
−1
bacteria and mPX
/mAX
= 0.2 for the myxamoeba.
Finally, we need the conversion coefficients
[MX ]1/[ME ]1 = 1.5, [MX ]2/[ME ]2 = 2.5 and [MV ]i /
[ME ]i = 1 for i = 1, 2 (see Appendix A). The
results are given in Fig. 3(g,h). The respiration
quotient, i.e. the carbon dioxide flux over the
oxygen flux, is almost constant at a value 1.05.
5. Discussion
We have shown the short list of general
assumptions (Table 2) about energy budgets has
far-reaching consequences for the coupling of
energy and mass fluxes in biota. The most strict
form of the concept homeostasis assumes that
biomass as a whole does not change in chemical
composition. We employ a relaxed version by
application of homeostasis to structural biomass
and reserves separately. Because the amount of
reserves can vary with respect to structural body
mass, the individual can change in relative
frequency of chemical elements in a particular
way. More complex models might distinguish
more types of structural body mass and/or
reserves to relax the concept of homeostasis even
more. This is necessary to accommodate the
metabolic versatility of plants, for instance. If the
Maximum ingestion rate E. coli
Maximium ingestion rate D. discoideum
number of types is equal to, or greater than, the
number of elements, no restriction exists on the
relative frequencies of elements in biomass. An
increase of the number of types of compounds
rapidly becomes counterproductive, however,
because the parameter values could be very
difficult to obtain in a reliable way. Some models
follow particular compounds. These models are
less likely to be useful at the level of the whole
individual, because of the staggering amount of
compounds that are present in an organism.
We showed how theory for energy budgets
provides a theoretical underpinning of the
method of indirect calorimetry. It seems to be
essential to delineate one reserve and one
structural component. Both more simple and
more complex models (with more components)
are inconsistent with the method of indirect
calorimetry. Although this type of statement is
hard to prove formally, it would not surprise us
when the set of general assumptions in Table 2
turns out to be the only one that is consistent
with this method. We also have shown that the
oxygen consumption that is associated with the
feeding (i.e. assimilation) process can be
understood and quantified using mass balance
considerations, which is considered to be an
388
. . . .    .
open problem in the physiological literature
(Withers, 1992). The present derivation simplifies
the one presented in Kooijman (1995). With
simple supplemental assumptions, energy budgets can be used to quantify evaporation of water
in terrestrial animals, which not only affects
drinking behaviour, but also represents an
important aspect of the thermal balance,
especially for endotherms. This opens the route
to advanced models for the control of body
temperature.
For simplicity’s sake, we excluded complex
situations such as when temporary absence of
oxygen affects energetics, and the method of
indirect calorimetry must be changed. Simultaneous limitations by energy and mass can be
included, but require more elaborate models that
have additional state variables to accommodate
such simultaneous limitations (Kooijman, 1998).
The step from individuals to populations
involves a number of assumptions about
interactions between individuals, and between
individuals and their local environment, that is
here kept as simple as possible. We stress that
(Kooijman, 1993) the ‘‘introduction of a
structure does not necessarily lead to realistic
population models due to the effects of many
environmental factors’’. One of the assumptions
implicitly made here is spatial homogeneity, as is
the case in a well stirred chemostat. Furthermore, the large number assumption makes it
possible to eliminate stochastic fluctuations at
the individual level and to model the population
dynamics deterministically using a set of PDEs
coupled with ODEs. Sometimes a structured
population dynamic model describing them can
be reduced to an equivalent delay differential
equation (DDE) or even an ODE model. Many
realistic extensions can be incorporated in the
presented formulation, which frequently do
complicate the analysis of the dynamics considerably, but hardly affect the mass–energy
coupling that has been discussed here. This is
because the coupling is effectuated inside
organisms, and mass and energy fluxes at the
population level represent simple additions of
those for individuals. This is why elaborate
interactions between individuals, for instance, do
not affect the conclusion that mass fluxes are
weighted sums of three energy fluxes. This does
not hold, however, for all possible forms of
complicating phenomena.
We conclude that open systems, such as
populations of living organisms, do not hamper
the application of energy and mass balance
equations. Indeed, the use of such balance
equations can be of great help to model
populations realistically. Although we acknowledge the fact that the incorporation of dynamic
energy budgets does not automatically lead to
realistic population models, we do believe that
useful population models should be consistent
with the principles of these budgets.
The authors like to thank Cor Zonneveld, Paul
Hanegraaf and Hugo van den Berg for stimulating
discussions.
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APPENDIX A
Volume–Mole Conversions
There does not seem to exist one most useful
notation for energetics. Volumes are handy in
relation to surface areas, which are needed for
the process of food/substrate uptake in the DEB
model, while moles are handy for mass fluxes.
Coefficients [M ] convert volume to C-moles
*
(dimension: mole volume − 1; the brackets [] refer
to volume − 1, while the braces 45 refer to surface
area − 1). The energy–mass coupler m*1,*2 couples
energy flux *1 to mass flux *2 (dimension energy
per mole). The chemical potential m also has
*
dimension energy per mass, but cannot be
interpreted as ratio of fluxes. The mass–mass
coupler y*1, *2, also known as yield or stoichiometric coefficient, is a ratio of mass fluxes and
taken to be constant, just like other couplers. We
−1
have
y*1,*2 = y*2,*1
,
y*1,*2y*2,*3,* = y*1,*3
and
−1
h*1*2 = m*2*1 is a mass–energy coupler. Volumes
are indicated with V, masses in C-moles with M,
structure-specific masses with m = M /MV .
*
*
Mass fluxes in C-moles per time are indicated
with J (the dot refers to time − 1), structure*
specific mass fluxes with jt = J /MV . Energy
*
*
fluxes (i.e. powers) are indicated with pt. Index X
refers to food, P to product (faeces). The
following conversions between volume-based
and mole-based quantities hold, where the
. . . .    .
390
dimensions are indicated with l (length), m (mass),
e (energy), t (time).
Reserve mass
Structure volume
Maximum reserve
M
V= V
[MV ]
l3
Maximum struc. v.
Vm =
0 1
vt
ktMg
3
=
0
kyVX 4JXAm 5
MVm
=
[MV ]
ktM [MV ]
1
l3
e
Maximum reserve
Em = [Em ]Vm = mEMEm
e
Maximum reserve density
Energy requirement per structural volume
e
l3
[EG ] = mGV [MV ]
Scaled length
0 1 0 1
MV
MVm
1/3
V
Vm
=
1/3
—
mAX =
mAP =
mE =
[MV ]
= mEkgyVE
[Me ]
—
e
l 2t
4ptAm 5 = mAX 4JXAm 5
Specific maintenance
e
l 3t
[ptM ] = ktMmGV [MV]
Energy conductance
4ptAm 5
4J
5
= yEX XAm
[Em ]
[Me ]
l
t
mGV =
[EG ]
y [MV ]
= EV
k[Em ]
k [Me ]
—
Maintenance rate
ktM =
[ptM ]
= jEMyVE
[EG ]
1
l
[Em ]
[Me ]
m
[EG ]
= E
[MV ] yVE
mE
j
= PA
mAP jEA
—
Structural mass
MV = V[MV ]
m
e
m
e
m
m
m
yPX =
mAX jPA
=
mAP jEG
m
m
Structure
coupler
Reserve
yVE =
mE
j
= VE
mVG jEG
m
m
Food
coupler
Reserve
mE
j
= XA
mAX jEA
m
m
Specific assimilation flux
m
mt
Specific maintenance flux
m
mt
Assimilation flux
JEA = jEAMV = JXAyEX
m
t
Food flux
Maximum structural mass
0
e
m
Product
coupler
Food
jEM = ktMyEV
ME
e [Em ]
=
MV mE [MV ]
MVm = Vm [MV ] =
4ptAm 5 mE
=
4JPm 5 yPE
jEA = jXAyEX
Relative reserves
mE =
e
m
Product
coupler
Reserve
yXE =
Investment ratio
g=
4ptAm 5
m
= E
4JXm 5 yXE
Growth
coupler
Structure
Maximum spec. assimilation
vt=
m
Assimilation
coupler
Food
yPE =
Reserve density
e = mE
Em
[M ]
= MVm e
[MV ]
mE
m
Reserve chemical potention
e
l3
[Em ] = mE [Me ]
l=
MEm =
ey
E
= MV EV
kg
mE
Assimilation
coupler
Food
Reserve energy
E = mEME
ME =
1
kyVX 4JXAm 5
[MV ]
ktM [MV ]
3
m
JXA = jXAMV
m
t
  / 
APPENDIX B
Steady-state Fluxes
This appendix describes the steady-state
fluxes. The gist of the argument is that, if
environmental conditions change slowly with
respect to the structure of the population in
terms of the frequency distribution of the
individuals over the state values, the population
can be treated as a super-individual with
relatively simple rules for mass and energy fluxes.
Such a simplification is useful if the population
model is conceived as a module in an ecosystem
model.
  
At steady state the easiest approach is to relate
the states of the individuals to age. We no longer
use the density F(t, a, e, l), but, instead, the
relative density f(t, a) = F(t, a)/N(t). This
relative density no longer depends on time at
steady state, so we omit the reference to time. We
will write J (a) for the flux of compound * with
*
respect to an individual of age a, where ab is the
age at birth and ap the age at puberty. These ages
might be parameters, but the DEB model obtains
them from MV (ab ) = MVb and MV (ap ) = MVp .
The characteristic equation applies at steady
state:
ME0 = exp4−hteab 5
g
a
exp4−hta −
g
a
hta (a1)da15JR (a)da
391
We introduce the expectation operators Oe and O,
i.e. OeZ 0 f0ab Z(a)fe (a) da and OZ 0 faab Z(a)f(a)
da, for any function Z(a) of age.
The harvesting rates of organic compounds
equal their mass fluxes, i.e.
FJX + J
GJ G
JO + 0 G V + G = hNOpt
J
G E+G
fJP + j
F
−htXMX
J
GN O ht M + NO(h
G
t+ hta )MV
e e e
V
=G
G (B.4)
N O ht M + NO(ht+ hta )ME
G e e e E
G
f
j
htXMXmAX /mAP
The number of juveniles plus adults in the
population, N, and of embryos, Ne , are given by
N=
NOJR
JX +
and Ne = (1 − exp4−hteab 5)
OJX
hteME0
  
The population growth rate must be zero at
steady state. We use this to solve the value of the
scaled functional response, i.e. f = (ktM + ht)/(ktM /
ld − ht/g) in the case of the DEB model. This
model has the nice property that e = f at steady
state; it then follows that MX + = MKf/(1 − f),
where MK is the saturation constant of the
Holling type II functional response in C-mol per
reactor volume.
The stable age distribution amounts to
(B.1)
f(a) = 2ht exp4−hta5 for a$[0,ht− 1 ln 2]
(B.5)
We use the characteristic equation to solve for
the food density MX + , so the scaled functional
response is f. Given this food density, the
trajectories of the state variables are fixed.
The age distributions of embryos and juveniles
plus adults are given by
The number of individuals in the population,
the total structural body mass, and the organic
fluxes are given by
×
ap
fe (a) =
0
hte exp4−htea5
for a$[0,ab ] (B.2)
1 − exp4−hteab 5
(ht+ hta (a))exp4−hta − f0a hta (a1)da15
f(a) =
faab exp4−ht− f0a hta (a1)da15da
for a$[ab ,a]
(B.3)
N=
JX +
MV +
=
OJX ld3MVm ln 2
MV + 0 NOMV =
htXMX [MV ]
f[JXm ]
JO + = hpt+ = hpt(f,1)
MV +
MVm
(B.6)
(B.7)
(B.8)
The mean mass per individual is thus OMV =
MV + /N.
392
. . . .    .
The relative contributions of the three basic
powers depend on the substrate density, and
therefore on throughput rate. Hanegraaf (1997)
gives a detailed analysis of mass and energy
transformations in chemostats at steady states,
including mixed substrates, fermentations and
product formation.