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Aquatic Ecosystems
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RESEARCH REPORTING SERIES
Research reports of the Office of Research and Development,
U.S. Environmental Protection Agency, have been grouped into
five series. These five broad categories were established to
facilitate further development and application of environmental
technology. Elimination of traditional grouping was consciously
planned to foster technology transfer and a maximum interface in
related fields. The five series are:
1. Environmental Health Effects Research
2. Environmental Protection Technology
3. Ecological Research
4. Environmental Monitoring
5. Socioeconomic Environmental Studies
This report has been assigned to the ECOLOGICAL RESEARCH STUDIES
series. This series describes research on the effects of pollution
on humans, plant and animal species, and materials. Problems are
assessed for their long-and short-term influences. Investigations
include formation, transport, and pathway studies to determine the
fate of pollutants and their effects. This work provides the technical
basis for setting standards to minimize undesirable changes in living
organisms in the aquatic, terrestrial and atmospheric environments.
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ARLIS
Alaska Resources
Library & Information Services
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EPA-660/3-75-012
MAY 1975
.MODELING DYNAMICS OF BIOLOGICAL AND CHEMICAL
COMPONENTS OF AQUATIC ECOSYSTEMS
by
Ray R. Lassiter
SoQtheast Environmental Research Laboratory
National Environmental Research Center-Corvallis
u.s. Environmental Protection Agency
Athens, Georgia 30601
ROAP 03ACQ, Task 09
Program Element 1BA023
ARLIS
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Alaska Resources Library & Information Setvices
Library Building, Suite 111
3211 Providence Drive
NATIONAL ENVIRONMENTAL RESEARCH CENTER
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
CORVALLIS, OREGON 97330
Fot Sale by the National Technical Information Service
U.S. Department of Commerce, Springfield, VA 22151
Anchorage, AK 99508-4614
ABSTRACT
To provide capability to model aquatic ecosystems or
their subsystems as needed for particular research goals,
a ~deling strategy was developed. Submodels of several
processes common to aquatic ecosystems were developed or
adapted from previously existing ones. Included are sub-
models for photosynthesis as a function of light and depth,
biological growth rates as a function of temperature,
dynamic chemical equilibrium, feeding and growth, and
various types of losses to biological populations. These
submodels may be used as modules in the construction of
models of subsystems or ecosystems. A preliminary model
for the nitrogen cycle subsystem was developed using the
modeling strategy and applicable submodels.
This report was submitted in partial fulfillment of
ROAP 03ACQ, Task 09 by the Southeast Environmental Research
Laboratory in Athens, Georgia, under the sponsorship of the
u.s. Environmental Protection Agency. Work was completed
as of May 1974.
ii
CONTENTS
Sections Page
1 I
II
III
IV
v
VI
Conclusions and Recommendations
Introduction
The Modeling Process
Problem Definition
System Diagram
Process Mechanisms
Physical Constraints
Use of Models
Process Models
3
4
4
4
5
6
7
9
Response of Biological Rates to Physical Factors 9
Algal Growth as a Function of Light 9
Biological Rates as a Function of Temperature 14
Interactions of Organisms and Their Chemical
Environment 24
Dynamic Chemical Equilibrium 26
Microbial Growth 29
Energy Expense, Predation Rates and Growth
of Small Biophagous Metazoa 33
Respiration, Death, and Excretion 36
Inhibition 37
A Preliminary Model for the Nitrogen Cycle 39
Problem Definition 39
System Diagram 39
Process Mechanisms 43
Estimation of Parameters 47
Analysis of Preliminary Results 48
References so
iii
FIGURES
!IQ.
1 General shape of the graph of biological rate versus 17
temperature as described by equation 13.
2 Fit of equation 13 to data for growth rate of the 19
planktonic green alga, Chlorella pyrenoidosa, over a
temperature range.
3 Fit of equation 13 to data for growth rate of the wood 20
destroying fungus, Ganode~ aplapatvm, over a
temperature range.
4 Fit of equation 13 to data for growth rate of the 21
aquatic snail, Lvmpaea stagnalis, over a temperature
range.
5 Fit of equation 11 to data of rate of luminescence in 22
the luciferin-luciferase system over a temperature
ranget o.
6 Fit of equation 13' to data of rate of luminescence in 23
the luciferin-luciferase system over a temperature
ranget o.
7 Components and transfers important in biogeochemical 25
cycles.
8 Pattern of pH fluctuations during course of algal bloom 28
and termination of the bloom by phosphorus limitation.
9 Diagram of system components and transfers for the 40
nitrogen cycle as depicted by equations 28 through 32.
10 Forrester diagram of the nitrogen cycle showing more 41
explicitly the processes and points of influence of the
components.
11 Relative steady state concentrations of four chemical 49
components of the nitrogen cycle model.
iv
ACKNOWLEDGMENTS
Many persons at the Southeast Environmental Resear~h
Laboratory and elsewhere generously spent a great deal of
their time with me discussing topics relevant to this
research. Without their help much of this and previous
research would not have been possible. I am grateful to
them, collectively and individually.
The support of Dr. Walter M. Sanders, Chief of the
Freshwater Ecosystems Branch, and of Mr. George L. Baughman,
Deputy Chief, is gratefully acknowledged. Of particular
value has been the atmosphere conducive to research, which
they have actively encouraged.
I thank Mr. James Hill for the initial construction of
Figure 10.
The untiring efforts of Mrs. Shirley Hercules in editing
and of Mrs. Carlyn Haley in typing are sincerely
appreciated.
v
SECTION I
CONCLUSIONS AND RECOMMENDATIONS
Models describing the dynamics of complex aquatic eco-
systems may be efficiently developed using a five-step
systematic approach.
1. Define the aquatic ecosystem in terms of the
problem to be solved.
2. Construct a diagram of the ecosystem representing
the system variables and the relationship between
t~m.
3. Develop mathematical descriptions of individual
physical, chemical, and biological processes
involved.
4. Assemble the system model using the submodels while
observing the principles of conservation of mass
and energy.
5. Compare macroscopic properties of the model with
observed properties of the real world to check for
validity of the representation.
Although the submodels developed in this report are
detailed descriptions of individual ecosystem processes,
they may be insufficient to represent some specific natural
aquatic systems. Thus, to construct a representation of a
given system, submodels for those required processes not
covered herein will have to be obtained from other sources
or through additional developmental research.
The detailed submodel developed herein, describing
biological rates as functions of temperature, provides a
representation more useful for ecological models than other
available submodels. The fidelity of this function is
better over a wide range of temperature than other models
examined, provided adequate parameter estimates are
obtained.
The nitrogen cycle is a mechanistically complex
subsystem of aquatic ecosystems. Within the limitations of
the simplifying assumptions used and the lack of good
estimates for parameters, the behavior of the model for this
cycle was considered to be reasonable. An inhibition
function proved to be a key element in the functioning of
this model. Properly divided into spatially distinct
subsystems and used with reasonable parameter estimates, the
nitrogen cycle model will provide a representation of the
nitrogen cycle that is suitable for simulation of larger
-1-
aquatic ecosystems. Further development work should be done
on the nitrogen cycle model to allow it to be included in
larger ecosystem models to enhance the capability to predict
phenomena related to nitrogen compounds in the environment.
-2-
SECTION II
INTRODUCTION
Research on ecological systems. ~.g •• the fate of
pollutants in aquatic ecosystems. may be significantly aided
by the use of mathematical models. Such models are
frequently thought of as large. complex. relatively
incomprehensible computer programs. In fact these programs
are not models in the sense used here. A model is a
mathematical structure used to represent some real-world
process. Computers may be used to make the computations
and. in fact. have made it possible to utilize large simul-
taneous equation models. Computer programs of models permit
widespread use and testing of model concepts.
No single model can be general enough to be adequate for
every research need. Therefore. our goal is to develop a
capability for constructing models to fulfill specific needs
that arise in research on the fate of pollutants in aquatic
ecosystems.
In the course of working to develop this capability. a
systematic approach to the construction of models emerged.
To allow construction of models for fate of particular
pollutants several terms or equations describing various
ecosystem processes were developed or modified from others.
These terms or equations. called submodels. may be used as
modules in constructing equations that describe rates of
change at the population or compartment level. A system of
equations that describes the appropriate compartments then
may be combined to form a model of some subsystem of a total
ecosystem.
-3-
SECTION III
THE MODELING PROCESS
The process of constructing mathematical models is more
difficult for natural systems than for man-made systems.
The former requires an understanding of the way in which
nature functions and a knowledge of the functioning units,
together with the ability to express the interrelation of
these functions in mathematical terms. Although with man-
made systems the requirements are similar, the system
components are known so that functions are usually under-
stood. For natural systems the essential information may
not be available. The element of discovery through
scientific inquiry therefore accompanies virtually all
modeling of natural systems.
Every mathematical model of ecological systems includes
hypotheses that are subject to scientific investigation. In
fact a model of ecosystem phenomena may be constructed as a
complex hypothesis itself, the hypothesis being that the
model is an adequate representation of some aspects of the
system behavior as required for a stated purpose. If
quantitative outputs are required, the model requires
parameter estimates for calibration to a specific situation,
and these are obtainable only from data. Only after such
calibration can testing begin. The model may be tested
relative to another model (is it better?) or against some
performance criterion (is it adequate?). Models are neither
"good" nor "bad" apart from the purpose they are intended to
serve, and testing must be done with this in mind.
PROBLEM DEFINITION
A difficult but signally important step in designing and
constructing mathematical models is defining the system to
be represented. This definition must consist of at least
the set of required outputs, a statement of acceptable error
of estimate, and a description of the real-world system
generating the analogous real-world outputs. For some
purposes other items may be needed, such as identification
of controllable variables. With this information modeling
may proceed into construction stages.
SYSTEM DIAGRAM
To clarify the relationships among system components it
is advantageous to construct diagrams of the systems.
Simple block and arrow diagrams may be adequate. Forrester
diagrams' allow more explicit representation of rates and
influencing factors. odum•sz energy circuit language may
-4-
also be useful. Whatever the scheme, graphical
representations allow elaboration of the system from the
initial definition to include a~l identifiable factors of
potential significance to the problem.
The system processes and components necessary to provide
the required outputs must be represented. The number of
these model components depends upon the compromise reached
between resolution and economy of resources. It is
frequently easier to include many possib~e components and to
delete unnecessary ones later than to begin with a minimal
set and add as model development proceeds. Several
analytical techniques3 are available for preliminary
analysis of the graphs. These techniques may aid in
choosing the optimum set of components to represent the
system.
At each stage of model development approximations
necessarily are made, each implying same assumption about
the real world. All differences between the real world and
the model are due entirely to the lack of correspondence
between the assumptions and the rea~ world. Res~ts from
mathematical analyses of the mode~ may therefore be equated
with reality subject only to constraints imposed by the
assumptions. Recognition and understanding of the assump-
tions are therefore necessary to eva~uate results from a
model.
PROCESS MECHANISMS
For mathematical ana~ysis linear mode~ approximations
may be convenient, but when both the pathway of materia~s
and the effects of their concentration are of interest, more
complex, nonlinear mode~s wil~ frequent~y be used.
Application of such models requires that the modeler know
enough about the processes to construct expressions for the
model system variables to adequately describe the inter-
actions among the real-world system entities. Such
expressions may merely portray observed behavior at some
level of resolution or they may imply comp~ex, real-world
interactions that are either hypothesized or known.
In ecosystem models the expressions are used to form
terms of the equations. Whereas the terms represent inter-
actions, the equations represent real-world components such
as populations, aggregations of similar populations, or
trophic levels. Especially where the functional groupings
represent entities such as populations or groupings of
similar populations, the form of the terms is determined by
the way the modeler mathematically describes the
physiological or kinetic properties of these interacting
real-world entities. For any interaction the term must
appear in at least two equations, often called donor and
-5-
recipient equations (~.g., predator and prey). The term in
a particular equation takes an appropriate sign and is
modified by coefficients for efficiency of transfer
(utilization, etc.) and for stoichiometry. Letting D =
donor and R = recipient, assuming a simple rectangular
hyperbolic description of growth of R on D, and expressing
them in like units, such an interaction can be illustrated
as follows:
dD
"d't
dR
"d't
A
= -
=
pR D
-y-( K+D ) +
+ ...
where p = growth rate constant
K = constant for half-maximal growth rate of R on D,
and
Y = yield of R per unit of D consumed, 0 < Y ~ 1.
The term ~D/(K + D) describes a simple "physiology• of R
with respect to D. The interaction is described by the
couple, i·~·· the appearance of the term in both equations.
In summary, modeling ecosystems using a mechanistic
approach involves writing equations, each of which consists
of terms that appear in one or more equations. The couples
(or n-tuples)of terms represent interactions, and they are
constructed so as to describe the physiology or kinetics in
the detail appropriate to the problem. No matter how
complex, all models are approximative. They may therefore
be judged only by their value after application to their
intended purpose.
PHYSICAL CONSTRAINTS -CONSERVATION OF MASS AND ENERGY
Ecological systems are constrained in nature by the
availability of matter and energy. Models of such systems
must be constrained by analogous physical principles. Mass
and energy budgets must be maintained if models are to be
realistic with respect to principles of conservation of mass
and energy.
In the preceding example of an interaction, another
equation is required for conservation of mass. The loss of
donor is proportional to p/Y, but gain by recipient is pro-
portional to p alone. Hence an amount proportional to
-6-
(1/Y)-1 is unaccounted for. Another equation for a by-
product, B, will account for all the mass:
dB dt =
A
l!R
1 D
(-y--1) ( K+D) + •••
The equation for the by-product may be ignored when it is of
no interest to the model.
The most important benefit of the principle of con-
servation of mass in modeling is the realistic constraint
that it places upon model behavior. In the above example
by-product accumulates as donor disappears and as recipient
increases. Because total mass remains constant the changes
in the three components are correctly phased in time. In
general the mass balance constraint ensures correctly phased
behavior of model components, an important characteristic of
ecosystem models.
USE OF MODELS
Simulation, the calculation of system behavior as a
function of time, and mathematical analysis may be used to
examine or analyze models for their properties. Models are
analyzed for two basic reasons. In their development,
analysis is done to compare macroscopic properties of the
model to known properties of the real-world system to check
for misrepresentation. When the model has been judged to be
an accurate representation of the real world for the
purposes intended, it is analyzed to seek properties of the
real system by observing properties of the model. For a
complex nonlinear model mathematical analyses may be
difficult or impossible, and therefore examination of
simulation results is the more frequently used approach.
Simulation is frequently used to obtain responses of the
model to specific conditions, !·~·· answers to "what if"
questions.
The constituent submodels can also be used separately if
information about a single subcompartment only is needed,
~.g., chemical equilibrium or accumulation of toxic sub-
stances in an organism. such computations may also be done
as a preliminary check on parameter values before doing a
large simulation.
A model may serve as a complex hypothesis about the real
world. It can be used to plan experiments, the goal of
which is to provide data to modify the model, if necessary,
to better describe the real world.
-7-
Models discussed are structured in detail to facilitate
research, but future models may be made simpler by using
generalizations obtained from research suggested by the
detailed models. Simplifying assumptions are frequently
made because of insufficient knowledge, but simplifications
resulting from research insights can lead to more useful and
reliable models.
-8-
SECTION IV
PROCESS MODELS
Many different strategies are used to construct
ecosystem models. Process modeling is one such strategy.
In it. mathematical expressions are constructed to describe
mechanistically at some chosen level of resolution the
various processes taking place in the real world. ~.g.,
transfer, transformation, birth, and death. These
expressions, termed submodels, may then be used as modules
in the construction of larger subsystem, ecosystem models.
The processes are described mechanistically, to permit
valid predictions beyond the range of the calibration data
where mechanisms are understood, and to permit incorporation
of testable hypotheses where they are not.
RESPONSE OF BIOLOGICAL RATES TO PHYSICAL FACTORS
The physical factors of light and temperature are
important environmental variables in ecosystems. Because
the process of photosynthesis is the basic biochemical
reaction supporting life it is important to pay particular
attention to its description in constructing a mechanistic
model for an aquatic ecosystem. Accurate computation of the
rate of primary production of organic material, especially
for lakes or slowly flowing water bodies, is necessary for
accuracy in other computations related to living organisms.
Temperate zone temperatures vary widely through the
seasons. Simulations of ecosystems over long time intervals
therefore require accurate descriptions of responses of
various rates to temperature changes.
~1 Growth as a Function of Light
Light is one of the principle factors affecting plant
growth. Because many of the materials frequently dissolved
or suspended in aquatic systems absorb or scatter light,
light entering at the surface is attenuated as it penetrates
the water. Light intensity therefore is a function of depth
and of water content. Hence. in modeling the response of an
aquatic ecosystem to light, absorption and turbidity must be
included. Because plant pigments are sensitized by light of
discrete wavelengths, it may be necessary to include
expressions for the differential light absorption
characteristics of the various materials present.
The influence of light was represented in two ways. The
simplest was to use a scaled and truncated sine wave as a
-9-
crude means of describing algal growth as it varies with
light intensity throughout the daylight cycle•.
Hutchinsons presented a more explicit representation of
light pene~ration using an integrated form of Beer's Law.
It includes direct light absorption by water and absorbing
materials (color) • and attenuation by scattering
particulates. Light intensity I at depth z may be expressed
as
= (1)
where I 0 = intensity at the surface,
= extinction rates per unit depth, (products of an
extinction coefficient and concentration of the
light adsorbing substance, indicated by
subscript i:
w = water,
c = color,
P = particulate turbidity.
The extinction rates for color and turbidity vary widely.
The rate for particulate turbidity may be written as np = hP
where P is a turbidity measure and h is an extinction
coefficient. P may be subdivided into turbidity by non-
living and living particles. If the non-living particles
can be assumed constant with time for a particular situation
and the light-absorbing living particles are all algae, then
the equation has only one variable, algal density.
Separating the algal density from the density of non-living
particles, the expression for the exponential extinction
rate, nT, becomes
=
where Ci = the concentration of chlorophyll a in the ith
group of algae (mg m-3),
hnPn = the extinction rate for non-living particle
turbidity, and
he = the extinction coefficient for chlorophyll a.
-10-
{2)
A further modification of the equation for Iz incor-
porating the non-vertical light path results in
Iz =
-1.19TITZ
I 0 e (3)
in which 1.19z is the distance traveled by the average beam
of light in reaching depth z. Poole and Atkinss found the
factor 1.19 to apply throughout the day because of the
varying contribution to the total incident light by direct
sunlight and by skylight.
To assess the integrated effect of light on the growth
of a population of photosynthesizing cells dispersed
throughout a water column of the eu~otic zone, a function
relating growth rate to light intensity is needed. Steele•
presented such a model for photosynthesis as a function of
light intensity as follows:
p =
1-ai
ap Ie m
where a is a scaling constant, and
Pm is maximum photosynthesis rate.
(4)
No rationale or development for the model was presented, but
the equation fits several sets of photosynthesis-light
curves.
The equation can be derived from the following:
1. The rate of change of p with a unit change in I
depends both on the value of p and on the deviation
of I from the optimum I (!m):
2. The rate of change of p with a unit change in I is
inversely proportional to I, i·~·• the
photoinhibition effect.
The following equation expresses these assumptions:
dp
di
= kp (1-I/Im)
I
Integration yields the following:
p = Pa(I/I }k k(1-I/I) m e m
-11-
(5}
Steele 1 s6 equation (4) is obtained by letting k = 1 and
1/Im =a. Equation (5), because it contains the parameter
k, which can be fit to data, provides a better fit than does
steele's equation. However, the form of Steele's equation,
because of its tractability, is more useful in developing
expressions for the average growth in a water column. When
equation (5} was fit by least squares to widely differing
data sets, k varied from about 0.87 to about 1.67. For wide
variation in light intensities, computations using equation
(5) will give better results than will equation (4).
Steele's equation may be integrated over the depth of
the euphotic zone (or other chosen depth} in order to obtain
the average photosynthesis rate. substituting equation (3}
into equation (4), integrating over the depth, ze, of the
euphotic zone, and averaging, the necessary expression is
obtained
p=
- e
Bannister7 discussed the variable nature of Pm and the
relatively constant nature of another parameter mm, the
maximum quantum yield. Using steele's equation Bannister?
derived Pm in terms of mm and c (mg chlorophyll/m3) :
12 ~ I h C/e m m c
where 12 atomic weight of carbon, and
he = the rate of absorption of light by chlorophyll
a.
substituting Bannister•s expression into equation (6) gives
p in terms of I, C, and the parameters ?'IT, m , I , and I : m o m
p=
-12-
(6)
(7)
This expression assumes that all algae are alike. The
average rate of photosynthesis for the jth algal group is
(p • Cj/C). Therefore the expression for the rate of photo-
synthesis by the jth group is
Pj =
12 -mimjhcCj
(-I /I -1 •1 9n z \e 0 mje T e -Io/Imj)
- e
1.19nTze
This expression can be used for simulations or other compu-
tations utilizing the average photosynthesis rate for a
column of euphotic zone water under a meter square of
surface. However, an expression for the rate of carbon
fixation per unit biomass, w, is usually more convenient for
simulation. To obtain an expression for w using equation
(8) , a relationship between chlorophyll and biomass (B) is
needed:
cj = bjBj
where bj = chlorophyll per unit biomass of population j.
Substituting for c and dividing by B gives
w. = J
12-I (-I/ -1.19n • mjhcbj e o Imje Tze
1.19nTze
- e
-Io/Imj)
Finally the specific growth rate subject only to light limi-
tation, 0 ~ must be obtained in terms of w and the
stoichiometric coefficient relating total biomass to carbon,
na:
"~ lJ = naw (9)
The relationship of p~to the specific growth rate, l.l• in the
context of nutrient and temperature limitations will be
discussed separately and a tentative expression for 1.1 will
be developed in a following section.
-13-
(8)
In the above discussion only phytoplankton dispersed
uniformly throughout the water column are considered. Other
kinds of plants are also important in aquatic ecosystems.
In fact, in streams attached algae and rooted macrophytes
may account for the entire productive capacity. A
development similar to the one above would result in an
expression similar to equation (9) for the growth of
macrophytes. Whereas uniformly dispersed phytoplankton are
easily represented in equations for light extinction,
discrete leaves of macrophytes may cause some problems in
representation. However, a term for total leaf area could
be used as a light extinction variable as described by
Bloomfield~ ~!·• in a model describing macrophyte growth
in an ecological context.
Biological Rate a~Egnction of Temperature
Temperature is a principle abiotic factor in providing
niche requirements for organisms. The existence of upper
and lower limits outside which biological processes fail
points to the likelihood of a temperature response with a
optimum for these processes.
Much of the work with biological rates as functions of
temperature makes use of the concept of Q10 • Q 10 is the
ratio of rate constants of biological reactions at two
temperatures, 10 degrees apart. Assuming that the rate is
exponentially related to temperature over the interval of
interest, the following expression• describes the
relationship between rate constants and temperatures:
=
where T 1 and T2 may differ by any amount.
Given a Q 10 value and a rate constant (k 1 ) at some
temperature (T,), one can determine the rate constant, k 2 ,
at some other temperature, T2 •
=
However, the assumption that the rate increases expon-
entially with temperature is not valid over a large portion
of the range of temperatures that an organism will tolerate.
Instead the biological rate increases with increasing
temperature to a maximum at an optimum temperature (T ) , m
-14-
then falls off to zero at some upper limiting temperature
(TL).
Three different sub-models or equations have been
developed to describe the dependence of biological rates on
temperature, all of which are characterized by a temperature
optimum with an associated maximum rate. one was developed
for use in ecosystem models for the Eastern Decidious Forest
Biome of the International Biological Program (IB~•. Its
form is rather complex:
k = (
TL-T \X eX[1-(TL-T)/(TL-Tm)
TL-Tm 7
where X= W2(1 + ~«1 + 40/W) ]2
400
W = (ln Q10 ) (TL-T)
and all other symbols are as previously defined.
(10)
Since it contains only two parameters and a Q10 , this
model is not able to represent rates over a wide variety of
temperatures. However, it is useful around the optimum
temperature.
A second equation was presented by Johnson ~t al.&o.
Based upon transition state theory, this model assumes that
the rate of an enzyme reaction is a function of the
activation energy and the equilibrium between the native and
denatured forms of the enzyme. It has the form
k = c T
1 +
-~Ht/RT
e
~S/R -~H/RT
e
where c is a scaling constant,
~H* is the heat of activation for transition state
intermediates,
(11)
~H is the heat of activation for the reaction for which
k is the rate coefficient,
~s is the entropy of activation of the same reaction,
~d
R is the gas law constant.
-15-
This model may be fit to biological rates over wide
temperature ranges; however, its parameters are thermo-
dynamic quantities, which are difficult to interpret at the
ecological level.
Another model, presented by Lassiter and Kearns•, was
developed in response to a need for a simple model,
applicable over a wide temperature range. The model takes
into consideration the following factors:
• Rate coefficients are always positive, approaching
zero asymptotically with respect to values of an
external stimulus; thus a change in the rate in
response to an external stimulus must be proportional
to the magnitude of the rate itself.
• A biological rate constant reaches maximum at an
optimum temperature; therefore, the rate of change of
the constant is hypothesized to be proportional to
the deviation of temperature from optimum.
• The rate, which diminishes when the optimum tempera-
ture is exceeded (its rate of change becomes
negative), decreases more rapidly as the upper limit
is approached. An inverse relationship between the
rate of change and deviation from the maximum
temperature (TL) is hypothesized.
The following equation incorporates the above con-
siderations:
dk
dT = (12)
Integration of equation (12) with k = km when T = Tm results
in the following expression•:
Although the equation contains four parameters, it fits
only data that conform generally to the assumptions used to
derive the equation. The graph of the equation is always
skewed to the right (Figure 1), the amount of skew depending
upon the difference between Tm and TL. No other type of
data can be represented by the model.
-16-
LLJ
t-
<( a::::
TEMPERATURE
Figure 1. General shape of the graph of biological
rate versus temperat'.lre as described by
equation (13).
-17-
The model has been fit to several sets of published
data, ~.g., growth rates of Chlorella sp.''• egg production
by several species of wood fungitz, growth of four species
of aquatic snailst3, and luminescence from certain
luciferin/luciferase reactionsto. Figures 2 through 4 give
examples of fits to published data. The equation is a good
representation for all data thus far examined on biological
rates versus temperature.
The shape of the graph for the equation is similar to
that for equation (11). Figures 5 and 6 show least squares
fits of equations (11) and (13) to the same data set (Figure
8.25, Johnson ~S ~.to). The fits are not markedly
different. Equation (11) may be more meaningful in an
analytical sense in that parameters are thermodynamic
quantities, which permit biological rates to be compared on
fundamental grounds. However, for ecological modeling it
may be more meaningful to use the parameters of equation
(13) for the function relating growth or other biological
rate to temperature.
Because Q10 data have been so frequently reported, they
represent a source of information that may be used in
deriving the parameters for equation (13). For example,
given a Q10 computed from rates obtained over the
temperature range Tt to T2 and reasonable estimates for Tm
and TL, values for Km and a may be computed.
Using the two rates k 1 at T 1 and k 2 at T 2 or any two
rates in the range over which Q10 is valid, and the
estimates of Tm and TL, the constant, a, may be computed by
the following equation:
a = (14)
The optimum rate, km• may then be computed by substituting a
into equation (13) with k = k 1 at T1 or k = k 2 at T2 • Using
k 1 and T1 , the following expression may be used to determine
~=
-a(T 1 -Tm) -a(TL-Tm)
= k 1 e [ (TL-T 1 )/(TL-Tm)1 (15)
-18-
UJ
1-
~
0::::
133
~ 66.5 :s:
0
0::::
(!)
0
10 27.5
TEMPERATURE, deg C
45
Figure 2. Fit of equation (11) to nata for growth rate
of the planktonic green alga, Chlorella
pyrenoidosa, over a temperature range.
-19-
L&.J ......
<(
0:::
~ 5.14
0
0:::
(.!)
0 ~------------~----------~
5 22
TEMPERATURE, deg C
39
Figure 3. Fit of equation (13) to data for growth rate
of the wood destroying fungus, Ganoderma
aplanatum, over a temperature range.
100
a..!
1-
<(
a:=
:I:
~
0
~ 50.1
a..! >
1-
~
a..! a:=
0.227 L......----------1'---------~
10 20 30
TEMPERATURE, deg C
Figure 4. Fit of equation (13) to data for growth rate
of the aquatic snail, Lymnaea stagnalis,
over a temperature range.
-21-
103
u.J
1 u z
u.J u
I
V')
u.J z
~ 52.6 ::::>
...J
LL.. / 0
u.J .....
< 0::::
2.51 ~
281 296 311
TEMPERATURE, deg K
Figure 5. Fit of equation 0.1) to data of rate of
luminescence in the luciferin-lvgiferase
system over a temperature range •
-22-
103 I I ~6
~ J>6\
~ l ~ c!
~ 52.6 ::::>
....J
LL.
0
~ I fo ~ ~ /6
.,I!
2.51 1\-ft~
281 296 311
TE('APERATURE, deg K
Figure 6. Fit of equation (13) to data of rate of
l1xminescence in the luciferin-luciferase
system over a temperature rangelO.
-23-
To solve equation (13) for all four parameters. rates at
four temperatures are needed.
If a larger set of data for rate versus temperature is
available. the parameters may be better estimated by least
squares. This technique'• was used to fit the equation to
the growth. egg production. and luminescence data cited
previously.
For modeling aquatic ecosystem processes. equation (13)
is easily used and has been found to describe adequately the
response of rate processes to temperature fluctuations.
INTERACTIONS OF ORGANISMS AND THEIR CHEMICAL ENVIRONMENT
The biota are controlled by the availability of chemi-
cals and the chemicals in turn are transformed by the
organismszt. A description of the ecosystem therefore
requires a set of simultaneous differential equations for
both biota and chemicals. all highly coupled.
The biota influence their physical environment in simple
ways such as reducing light intensity by shading.
Biological and chemical processes. however. seem to be
linked in much more complex and subtle ways. Many chemical
reactions occur so rapidly that they may be considered to be
always at equilibrium with respect to the much slower
biological processes. Yet these biological processes over a
period of time may exert a great influence on the chemical
system. The molar ratio (16:1) of nitrogen to phosphorus in
the sea is presumably an example of such influence resulting
from biological processes operating over geological timets.
Descriptions of several processes involving interactions
between the biota and their chemical environment were
included as components for the ecosystem models:
• inorganic chemical equilibrium
• production and decomposition of organic compounds.
• exchange of gaseous materials at the air-water
interface. and
• food-web transfers of the aquatic biological
community.
Figure 7 shows schematically components and transfers that
are important in biogeochemical cycles. Although pollutants
are generally thought of in terms of their effects on the
environment. a model for the fate of pollutants must in
addition consider them as materials subject to transfer and
transformation by the above processes.
-24-
~~ c= A~~;ere" -\ _
....... -----=-----
Air/Waterlnterface ~ ~ ~~-----"""-
Inorganic Chemical -I Producers I .. I Consumers Equllibrl um
~ ~ ~leCoLosJ
~~
Fi~1re 7. Components and transfers important in
biogeochemical cycles.
-25-
The dissolved materials in aquatic systems comprise a
complex medium wherein photochemical. biochemical, and other
chemical processes occur. It is beyond the scope of most
dynamic models to represent this complexity in more than a
simple way.
To develop a capability to model chemical equilibria
dynamically, a simple representation of the complex real
world was chosen that would include materials of rather
certain importance to ecosystem processes. Various
techniques have been presented for ccm~uting concentrations
of materials in complex solution equilibria16,,7,ta, but for
this submodel a simpler computational scheme was needed that
would allow the chemical equilibria to be maintained
dynamically throughout a simulated time period. The
technique chosen was an iterative search for [H+] that
results in achievement of charge balance.
Ionization fractions or distribution coefficients as
functions of the equilibrium constants and hydrogen ion
concentration were computed for each ligand. For exam~le
the carbonate series is characterized by three distribution
coefficients, a 0 • a 1 , and a 2 :
-1
[H 2 C0 3 *J ( K1 K 1 K2 ) a = 1 + [H+J + [H+]2 0 CT
[HCo 3 -J ( [H+] [K2 J) _,
a, = = --+1+--
CT K 1 [H+J
co = ( [H+] [H+] r 3 a = = --+ + 1 2 CT K1K2 K2
where C = (H CO*]+ [HCO -]+(CO =]
T 2 3 3 3
(H 2co 3 *] = (00 2 ] + (H 2 00 3 ].
The ai are found simply by substituting a value for [H+].
Given a distribution coefficient, ai, and the total con-
centration, CT, the concentration of any form. Ci, can be
computed:
=
-26-
Distribution coefficients are used in this manner to compute
the concentrations of all the ligands considered.
concentrations of the free metal ions are computed using
concentrations of the ligands and solubility products.
subject to the constraint that the total metal present
remain constant. If computation indicates precipitation.
the solids are assumed to remain in microdispersed form.
readily available for solution when conditions permit. The
key to all computations is [H+]. [H+] is varied by the
iterative search routine until the objective function. a net
charge equation. is found to be arbitrarily close to zero.
at which point the computations are complete.
For the chemical equilibrium system to be maintained
dynamically. the totals of materials used and regenerated
may be changed each computational interval. Differential
equations for co 2 • available phosphate. and NH 3 (discussed
later) are used to compute these additions and subtractions.
The largest set of metals and ligands utilized included
sodium. calcium. magnesium. carbonates. phosphates.
sulphates. sulfide. acetate. ammonia. and nitrate.
Equilibrium constants were obtained from various sources
including Kernt• and Stumm and Morgants.
Aquatic ecosystems were simulated using the model of
Lassiter and Kearns• with and without chemical equilibrium.
The pattern of biomass production. especially for primary
producers. was different for the two cases. Without
chemical equilibrium only one form of a chemical nutrient
could be assumed, and therefore as might be expected. growth
of producers and corresponding depletion of the nutrient
proceeded faster. When the model included chemical
equilibrium. the chemical nutrients were apportioned among
several chemical species with a corresponding lowering of
the concentration of the available form. As the chemicals
were used they were reapportioned by equilibration and
therefore made available in a buffered fashion to the
organisms.
variation in pH was induced by the algal growth compon-
ents in the model similar to the manner in which it occurs
in nature. One curve presented• showed an abrupt halt in pH
fluctuations with the development of phosphorus limitation.
Another simulation with higher decomposition rates (higher
rate of mineral recycling) also resulted in low phosphorus
concentrations terminating a bloom. However. with the
higher recycling rates the pH fluctuations did not abruptly
cease. but instead decreased rapidly in magnitude. These
results are shown in Figure 8.
Both studies illustrate the importance of a dynamically
varying chemical equilibrium subsystem as part of models
directed toward study of nutrient-algal relationships. The
-27-
8.2
8.0
7.8
a 7.6
7.4
7.2
7.0
1 2 3 4 5
TIME, days
Phosphorus-li rnited
phytoplankton growth
resulting in smaller ~H
excursions
6 7 8
Figure 8. Pattern of pH fluctuations during course of
algal bloom and termination of the bloom by
phosphorus limitation. Smooth curve was
drawn through points (dots} produced by a
computer program for a dynamic chemical
equilibrium model linked to a differential
equation model of phytoplankton dynamics.
-28-
9
fluctuations in pH are as much a part of these relationships
as are the fluctuations in either nutrient concentration or
algal density, and perhaps carry greater implications for
other, less directly related, reactions such as those of
specific pollutants.
The chemical equilibrium submodel has two major
disadvantages. First, equilibrium constants have not been
corrected for temperature. Falls and Vargazo list some of
their equilibrium constants as functions of temperature, and
data such as these will be useful. second, the method is
not general enough to allow relative ease of specification
of the system. The system described above will probably be
abandoned and replaced by a streamlined subset of another
model such as that presented by Morel and Morganl7, which
overcomes at least the latter problem. The use of the
present system has, however, provided experience with a
dynamic equilibrium model, and its use has shown that it is
feasible to represent dynamic chemical equilibria con-
currently with biological processes in ecosystem models.
Microbial Growtb
One important microbial activity in ecosystems is
decomposition, the recycling of nutrients from organic back
to inorganic forms. A representation of this feature of the
ecosystem is necessary in models that are intended to
accurately represent the cycling of materialsZI.
Decomposition is a function of growth rate, and therefore an
expression for microbial growth is needed.
Microbes in general (algae and bacteria for present
purposes) require for growth an energy source, inorganic
nutrients, and a suitable chemical and physical environment.
For algae, which are autotrophic, the energy source is
light, a physical factor (discussed previously). An
equation was derived for algal growth as limited by light
(energy) alone (equation 9). Bacteria, because they feed on
non-living organic material, are considered to be not only
heterotrophic, but more specifically, saprophagouszz. They
derive their energy from the transformation and utilization
of organic materials. Before an equation can be derived
that describes microbial growth in general as a function of
inorganic nutrient concentrations, an expression for
bacterial growth as limited by energy sources, analogous to
equation (9) for algae, must be developed.
One approach to the development of an equation for
bacteria is to consider free energy changes in net equations
for the transformation of particulate organic matter into
bacterial biomass. Assume that the organic material in the
system may assume four different forms --particulate
-29-
organic matter (OHn)r dissolved or broken down organic
matter (OMd), oxidized organics (CO~ and other inorganic
ions), and bacterial biomass. The free energy changes
associated with the transformation of materials from OMp to
biomass (6Gpd' /!Gdc' and !!Gel) are as follows:
-6Gpd Breakdown or
OMP OMd Dissolution
-f.Gdc Oxidation
biomass Assimilation
The bacteria may take part in either or both of the first
two processes and, by definition, take part in the
assimilation process. The associated free energies of the
reactions may then be utilized by the bacteria for metabolic
processes (growth, activity, etc.). The above equations,
however, provide no information concerning the efficiency
with which the bacteria can make use of the available
energy. For use in computations of growth rates estimates
of these efficiencies are needed as well as information on
the standard free energies of formation, 6G 0 , of the OM ,
OMd, bacterial biomass, and inorganic nutrients. Rate P
expressions for each of the processes are also required.
However, since the details of this approach have not
been worked out, an empirical approach, which makes use of a
rectangular hyperbolic function, has been used to describe
bacterial growth as it is limited by energy availability
alone. The concentration of total organic material (OM ) is
used as a measure of energy availability. T
= (16)
in which ~~ = growth rate limited by energy alone,
K8 = the conc~tration of OMT that gives rise to a
growth rate, ~/2,
i = maximum growth rate assuming no nutrient or
energy limitations.
Given equation (9) for autotrophs and equation (16) for
saprophages, equations may be formulated to describe growth
rates for microbes in general as limited by inorganic
nutrient concentration. To do so, equations must be
-30-
developed that take into consideration the various forms of
the nutrient elements, carbon, nitrogen, and phosphorus.
Organisms can utilize nutrients only in specific forms. A
knowledge of these preferences is essential since environ-
mental conditions and microbial populations exert a large
effect on the distribution of nutrients among the various
possible forms.
Some such information is currently available. From
their observations that green algae grow more rapidly in
acidic media, Emerson and Green23 and osterlindz• concluded
that these algae prefer dissolved co 2 and/or carbonic acid
to other forms of inorganic carbon. Keenanzs reported that
blue-green algae grew better at high pH's. Although he
interpreted this as a direct pH dependence only, it may
possibly be interpreted as a dependence on carbonate ion
concentration. Little is said in the literature about the
relative availability of ammonia and ammonium ion. However,
judging from the pH values at which microbial growth optima
occur, NH 4 + is the more easily assimilated form. Based on
similar information concerning the nutrient, phosphorus, the
mono-and diprotonated forms of orthophosphate may be
assumed to be preferred by microorganisms.
organisms are known to utilize both nitrate and ammonia
for growth. Some show a preference for nitrate and some for
ammonia, although the reasons for these preferences are not
always clear. Ammonia is believed to be, in general, more
easily assimilated than nitrate because less energy is
required for its conversion to amines.
When the specific chemical forms of the nutrients
required for growth are known, their concentrations may be
used in algebraic expressions relating growth to nutrient
concentration. Many workers•,z•,z7,ze,z• have used an
expression for the chemical nutrients that is comparable to
the single factor of the Monod growth equation but is a
product of several factors, one for each nutrient
1l = n .. n
n
i•1 (K s
Si -)
i
in which n = total number of nutrients
si = concentration of nutrient, i
Ki = the concentration, si, that gives rise to
that is one half of its maximum value (p).
-31-
(17}
Another expressionzt 27 uses the minimum of n quantities,
Si/(Ki + Si), i = 1, 2, ••• , n, in place of the product of
then quantities in equation (17).
Droop3·o discussed another model incorporating
Caperon 1 s3t suggestion that growth occurs by utilization of
nutrients from internal pools. Droop argued for a linear
relation between uptake rate and nutrient concentration in
the cell, or cell quota (Q), in units of grams of nutrient
per gram of biomass. Plots of uptake rate versus cell quota
were linear, intersecting the Q-axis at a point,~ (> 0),
termed the "subsistence quota.• At large values of cell
quota (Q), he found that the asymptotic value of p equals
the slope of the plot of uptake versus cell quota. Based on
this observation he derived the relation
= ~ Ill
1 -
kq
Q
in which p ~ is the maximum growth rate resulting from
internal nutrient concentration.
Three properties of the equation are important:
lJ/lJ~ 1 as Q becomes large,
lJ 0 as Q kq,
lJ = Is lJ~ when Q = 2kq•
The first two properties are reasonable biologically, but
the third is not. It seems unreasonable to expect that, in
general, for a nutrient concentration of one half the half-
saturation constant(~), growth will cease. Nevertheless,
Droop has addressed the problem of describing algal growth
with more care than previous workers, and his model or some
variation may be found to be an improvement over previous
ones for many uses. His formulation for algal growth in
response to multiple nutrients is
(18)
As Droop pointed out, this model predicts luxury uptake
when nutrient concentration is high, but does not set an
upper limit to the extent of this luxury uptake.
Equation (18) differs from equation (17) in several
ways. It describes growth as occurring from internally
stored nutrients. More significantly, it includes the
-32-
concept of a subsistence quota and, because it differs
algebraically, some of the constants must be interpreted
differently. However, in spite of these differences,
results from the two equations do not differ greatly;
equation ( 11) , which requires fewer computations, is usually
preferable.
For some green algae and for bacteria, both inorganic
nutrients and small soluble organic molecules are used as
substrates for growth. Thus some of the factors of equation
(17) may include dissolved organic material. Algae, for
example, are known to assimilate several sugars (including
phosphorylated forms), fatty acids, and some amino acids3Z,
and to grow well using urea (apparently directly) , amino
acids, and amides (for some forms by extracellularly
liberating ammonia)33.
Energy Expens~, Predation Rates, and growth Rates Of Small
Biophagous Metazo!
Small metazoa (~.q., rotifers and crustacea) found in
the plankton and periphyton are heterotrophic, and many feed
upon living material. Because of these habits· they are
potentially important in the aquatic ecosystem, affecting
·nutrient cycles via their roles in the food web. Wiegert
and OWenzz termed such organisms "biophagesn to distinguish
them from saprophages, organisms that feed upon non-living
organic material.
Submodels describing processes related to feeding and
growth of biophages assume no age or size structures in the
populations. Energy expense has usually been assumed to be
constant, but the effects of the environment and of food
availability on energy intake and expense may be important
variables in the growth of motile forms.
Three types of models have been developed that may be
used to describe the feeding and growth of small biophages.
Lassiter and Hayne3• presented a finite difference model,
which considered that energy expense sets satiation level
(maximum intake rate per unit time). The model predicted
that when food was plentiful, the biophages were satiated
and more energy was taken in than was expended. The
difference was applied to growth of the population. The
model is pedagogically useful, but it is difficult to use in
differential equation models.
other types of models relate growth directly to food in-
take without considering energy expense. Many of these
models employ a single compartment to represent each trophic
levelZ7,ze,z•. These models are based on the assumption
that a rectangular hyperbola adequately describes the
feeding and growth rates of biophages. Lassiterzt extended
-33-
these formulations to include multiple populations of
biophages. The equation for growth rate, Gi, of the ith
biophage population is given by
Gl.· = llA B i i
A
m
K· + r P· ·B· l. • 1 l.J J J•
in which lli is maximal growth rate,
(19)
Pij is a weighting constant defining the predation
rate by the ith biophage on the jth prey type
relative to the rate on any other prey type,
Bi is biomass of biophage,
Bj is biomass of prey,
Ki is the biomass of prey as modified by the
weight, p, that result in half maximal biophage
growth, and
m is the number of prey populations.
Feeding of small biophages results in losses to
microbial populations. To compute the rate of death caused
by predation on a microbial population, first the fraction
of growth of a biophage population resulting from predation
on a specific prey is needed. For biophage i feeding upon
microbial organism k, the fraction is
= m
r P·jB· j=1 l. J
(20)
Therefore the death rate to population k due to predation by
r predators is
r
D =L
i=1
in which nki is a stoichiometric coefficient relating a unit
of prey biomass to a unit of predator. The stoichiometric
coefficient is useful when two or more nutrient cycles are
-34-
simultaneously of interest. It is a number relating the
difference in empirical formulae of ~redator and prey, and
it has the effect of setting the maximum yield. Using this
approach the yield coefficient, Yi, which usually represents
the yield of predator per unit prey ingested, becomes an
assimilation efficiency, a unitless number in the interval
(0,1). For a discussion of predator-prey and other
stoichiometry in ecosystem models (including yield), see
Lassiterzt.
The third type of modele makes use of a submodel in
which the constant, K, of the rectangular hyperbola is
replaced by a term that is linearly related to biophage
density. This expression has the effect of representing
direct interference among predators competing for prey,
whereas equation (19) incorporates only competition for the
same prey population.
Neither equation (19) nor Bloomfield's submodel
considers energy expense as a variable. However, either of
the submodels could be rearranged to include energy budget
equations if such could be developed. TO do so, the yield
coefficient in the submodels would have to be made a
function of energy expense and intake.
Instead of modifying either equation, however, yet
another equation was developed and used because of its
mnemonic value. Redefining Pij as the rate constant for
predation of biophage i on prey j, the growth rate of
biophage population i is given by
Gi
Y. B. I: p .. B.
l. l. j_ l.J J =
n~ k (K. + I: B. )
l. l. j J
in which all other terms are as previously defined. In this
submodel growth rate is a function of ~redation rate,
assimilation efficiency (yield), and the stoichiometry
relating the two. Maximal growth rate does not appear in
the formulation. Feeding is represented in this function as
the essential interaction, and growth as a result of the
interaction. Since growth is a function of yield and yield
is a function of an energy budget, growth is also a function
of an energy budget.
Assuming equation (22) for growth of predators, rate of
death caused by predation on population k is given by
r
Dk = L FikBi
i=1
I: p .. B.
j l.J J
K. + I: B.
l. j J
-35-
(22}
(23}
in which all symbols are as previously defined.
In using these submodels in an ecosystem model, other
couplings in the system must be considered. For example, at
higher energy expense rates correspondingly greater input
rates of co2 to the water from biophage metabolism must be
accounted for. Patten3s reasoned that a high degree of
control exerted by higher trophic groups over lower is a
significant and general feature of ecosystems. It was
hypothesized from computer experimentationzt that
zooplankton activity in aquatic systems may markedly affect
the behavior of every component. Thus effort spent in
accurately computing rates of growth for consumer organisms
may be rewarded by a large increase in the accuracy of
resulting predictions of such quantities as producer
standing crops and various aspects of material cycling.
Respiration, Death, and Excretion
Respiration is a general term for the metabolic
functions required to maintain the physiological integrity
of organisms. Its uses in ecosystem modeling are manifold:
mass balances for o2 and co 1 , terms in net growth equations,
and energy balance terms. A functional module for
respiration, then, will be a very much used term. It may be
a very simple function such as might be used for microbial
respiration, or it may be rather complex such as the several
terms relating to respiration in the energy budget3•. If
respiration is assumed to be a function of temperature
alone, equation (13) may be used. For respiration the
optimum temperature, Tnt, and the upper limiting temperature,
TL, usually are fairly close together, resulting in a rapid
decline in rate as the temperature increases for Tm to TL.
Two classes of anabolic processes appear to be operative
in organisms, viz., those processes involved in creating new
structure, which have their temperature optima well below
the limiting temperature, and those processes that function
in repair, which operate more effectively with increasing
physiological stress. The latter type are respiratory
processes and reach a maximum only when heat denatures the
proteins involved, the process occurring at TL.
Death to organisms may result from several causes.
Predation has already been discussed and equations have been
presented to describe it (equations 21 and 23). Temperature
may also exert lethal affects. Death rate due to
temperature may be described by the following expression:
=
T > T m
-36-
(24)
in which D is the specific death rate due to excessive
temperature, a is a scaling constant, and k is from equation
(13).
Death to natural populations also results from several
unspecifiable causes, tbe probability of which is assumed to
remain constant for each of the organisms. Therefore, the
specific death rate from causes other than excessive
temperature or predation is simply a constant in the
differential equation.
Excretion represents yet another mechanism by which bio-
logical populations may lose biomass. Algae are known to
excrete small organic compounds36 the physiological
mechanism of which, however, is unclear. The specific rate
of excretion is therefore assumed to be constant.
For consumers, excretion may result from several causes.
Metabolic products resulting from catabolic processes may be
excreted, which argues for an excretion rate proportional to
metabolic rate. The excretion of metabolic products, then,
may be represented as proportional to the temperature
dependent respiration rate.
Another process that necessarily results in excretion by
consumers is the imperfect matching of the composition of
food biomass and consumer biomass. This is a problem in
predator-prey stoichiometry, and was discussed in detail by
Lassiterz•. A stoichiometric coefficient is formed for the
units of food consumed per comparable unit of consumer
formed. This coefficient varies depending upon the com-
position of the food. It is used to compute another
coefficient, n, for excretion of an element by the consumer.
The excretion rate, Eir for population i is then simply
Ei = nG1 •
Inhibition
One other submodel has proved beneficial, viz., an
expression for the inhibition of biological processes by the
presence of some specific chemical species. This may be
actual inhibition, for example, inhibition of
denitrification by the presence of oxygen, or it could
reflect a preference, such as inhibition of uptake of No 3-
by presence of "preferred" NH 3 by microorganisms.
The change in the rate coefficient, k, of the inhibited
process with respect to the inhibiting substance is assumed
to be proportional both to the value of the coefficient and
to the concentration, x, of the inhibiting substance.
-37-
These assumptions may be rationalized on the following
bases: first the rate coefficient must remain in the
interval (0, k ) and therefore a change in response to an
external stimufus must be a function of the rate itself; and
second, the change in k with x for a particular process is
dependent upon the particular value of x, subject to change
by physiological adaptation. These assumptions may be
stated as
3k ax = -akx
which, upon integration gives
-ax 2
k = kme
in which a is a scaling coefficient and the other symbols
are as previously defined.
(25)
(26)
When two or more substances independently inhibit a pro-
cess, equation (2) can be extended to
k =
in which there are n inhibiting substances.
By suitable choice of constants, ai, the inhibition
function may be made to blend processes smoothly or to
switch a process on or off within a very small range of
concentration of the inhibitor.
-38-
(27)
SECTION V
A PRELIMINARY MODEL FOR THE NITROGEN CYCLE
In constructing a model for a system or for one of its
subsystems, some logical sequence of steps should be
followed so that errors, repetition, and the time required
for the task may be minimized. A model for the nitrogen
cycle, as a subsystem of an aquatic ecosystem, was developed
following the procedure discussed in Section III, The
Modeling Process. Because this model is viewed as a
preliminary model to be revised when incorporated into a
more comprehensive ecosystem model, the modeling process
will be used again when the revision occurs and other
submodels are included. some of the submodels developed in
the previous section were used as modules in building the
larger, nitrogen cycle model.
PROBLEM DEFINITION
Many ecosystem models have been constructed using first-
order or other approximations to only portions of the
nitrogen cycle•,zt,z6,z?,za,z•. None have attempted a
mechanistic description of the complete cycle. Textbook and
other general descriptions37,3a,39,40 indicate that
compensatory aspects of the nitrogen cycle may be important
in producing a realistic model. That is, depletion and
replenishment should be represented to accurately model
nitrogen dynamics. A nitrogen model was developed to take
into consideration the compensatory aspects of the cycle
more completely and to examine the benefits derived from the
more complete representation. The model was to be
constructed to permit examination of the influence of
pollutants on nitrogen transformations, ~.g., the influence
of mercury on nitrification rates.
The minimal set of system variables chosen included
ammonia, nitrite, nitrate, and organic nitrogen.
SYSTEM DIAGRAM
System diagrams representing the four variables helped
to define the couplings, influences, and other processes
that must be included in the model. Figures 9 and 10
represent a final version of the diagrams used in the model
development.
Ammonia is a product of the decomposition of organic
nitrogen. The rate of the decomposition process is a
function of oxygen concentration, as are most other
processes of the nitrogen cycle. Therefore oxygen is
-39-
Air Water~~~~
NH,, NO,-
suppressed N-fixing Organisms
IXtl
Organic Material lx2l
Decomposition j
IP,, P4l
NH 3 suppressed
Death ..
I Assimil~ion
L--------~-Aerobic IP 1l
NO,-lx 5l
--;;, e::
u
r!: :s ...
u Cl.> ... :s c: ... <
Cl.> <
Figure 9. Diagram of system components and transfers
for the nitrogen cycle as depicted by
equations (28) through (32).
-40-
I ,.. ...
CD--
N-flxlng Org~nlsms
lx,l -~ ; l ~ I I
I I
~
----1------------------------I
: ~E G) 'i ---Pr .,..._ x, -~ = ~-1 ~ ~
' c A ~
1 •• -~·-r..Ld ~ I r·-~:::-®--~~~;·---8-----,-,
I I '--<' I I --[!;.) ~ '----~~-------' NH, lx
1
l I_ ~ _ r~ E: --Q ... ----------I-J
~--------: :: ~----~----l~
Fi<J.l re 10.
'-----------..
' I
I
I
~ I Jl-------·P----------J I ~. ---.:: -----~ ~ .-----<-------'--------------•
I ~ il I ~I ·-------------------
~---l! i J : CD lJfL----=---~ ._ l :
~ ~-~ I NO,-Ix,l I ~ I NOr-lx,l l--A-(-)
----cv
I
I ' _,_ _________ J
----------------------------------------~
Forrester diagram of the nitrogen cycle (equations 28 through 32)
showing more explicitly the processes and points of influence of
the components.
required either as a system input or as a system variable.
For this model, it proved more useful to use it as an input
(forcing) function.
organic nitrogen is produced by four processes:
microbial assimilation of ammonia, nitrite, and nitrate, and
fixation of elemental nitrogen. Because nitrite occurs at
very low concentrations, its assimilation could be ignored
for the present purposes. Both nitrate and ammonia suppress
N-fixation, although ammonia is the more effective
suppressant3•. To keep the diagram (and later the
mathematical model) as simple as possible, only one type of
organism was represented by a dynamic variable. (For other
microorganisms, fixed population sizes were assumed.) This
organism was assumed to fix elemental nitrogen if neither
nitrate nor ammonia was plentiful and thus to replenish the
nitrogen in the system when it became depleted. Deaths from
this population and from the fixed size populations
contributed to the organic nitrogen pool.
Ammonia disappears by two processes: assimilation (pro-
ducing organic nitrogen as noted previous!~, and
nitrification. Nitrification was assumed to proceed in two
steps: (1) NH3 ~ No 2-, and (2) No 2-~ No 3-. Both use
oxygen as the oxidizing agent.
Decomposition of organic nitrogen can occur both aero-
bically and anaerobically. Anaerobic decomposition may
proceed by several mechanisms. For initial simplicity,
decomposition was assumed to be carried out by two fixed-
size populations of microorganisms. Both were assumed to be
facultative anaerobes, switching their metabolism from
aerobic to anaerobic according to prevailing conditions .•
One was assumed capable of using nitrite and the other
capable of using nitrate as oxidizing agents. Other
possible competing anaerobic processes were ignored.
Nitrite is produced when facultative anaerobes utilize
N0 3 -as the oxidizing agent in the decomposition of organic
matter (denitrification), and when they oxidize NH 3 (nitrification). Denitrification is said to be inversely
related to redox potential•'· However, according to
Hardy•z, the denitrifying enzymes are sensitive to oxygen
concentration; above 0.2 mg o 2/l, the enzymes are repressed.
At low oxygen concentrations, which usually occur at low
redox potentials, the denitrifying enzymes are de-repressed
and denitrification can proceed.
Nitrate arises by nitrification of N0 2-and disappears
by assimilation into microbial biomass and by denitri-
fication.
Finally, nitrogen is gained by fixation of N2, as
discussed previously, and is lost as~, the end-product of
-42-
denitrification. Thus the cycle is closed in the causal
sense. It is open, however, inasmuch as N2 is for all
practical purposes an infinite source-sink reservoir.
PROCESS M:EX:HANISMS
For this model simple constructs were used to express
the essential aspects of the system. For example, the
customary hyperbolic function (equation 17) was not used in
the description of growth rates, but the inhibition function
(equation 27) was used extensively. The equations were
coupled such that exchange of gaseous nitrogen was the only
mechanism for loss or gain for the system.
The nitrogen cycle in nature requires both oxygen-
replete and oxygen-depleted environments for all processes
to occur. All processes described in this single model were
assumed to proceed concurrently, except when inhibition or
resources limited rates to near zero. For example, oxygen
was used to switch the nitrification and denitrification
processes on and off, simulating aerobic and anaerobic
environments.
Five dynamic variables were used in the model:
x1 = organisms capable of assimilating NH 3 and N0 2-, and
of fixing N2 ; expressed as concentration,
x2 = concentration of organic material (containing N in
the same fixed proportion as x 1 ),
x 3 = NH 3 as (N],
x 4 = N0 2-as (N], and
x 5 = No 3-as (N).
Oxygen was made time varying and was labeled X£• Four
fixed-size populations were included (P 1 , P2 , P3 , and P4 ).
Nitrification steps (NH 3 ---+ N0 21 and (N0 2-~ No 3-) were
mediated by P1 and P2 , respectively. Denitrification steps
(No 3-~ No 2-) and (N0 2-~ N2 )were mediated by P3 and
P4 , respectively. Elemental nitrogen was denoted by x 0 , and
was assumed to be present at saturation concentrations.
Inhibition functions of the form of equations (26) and
(27) are denoted generically (each may differ in
coefficients) by I(Xi) or I(xi, Xj) in the following
description of the interactions and construction of
equations. Also stoichiometric coefficients are generically
represented as s. The rate coefficients for the various
processes are symbolized by k's with subscripts. The
subscripts have no value for identifying the process to
-43-
which they belong other than to allow one to distinguish
whether terms of different equations refer to the same
process.
The equation for x 1 consists of terms for assimilation
and death:
dx 1
dt = lk 00 x 0 I(x3 ,x 5 ) + k 01 x 5 I(x3 ) + k 02 x 3 - k 1 Jx 1 (28)
Note that N-fixation (first term) is inhibited by both NH 3 (x3) and No 3 -(x5 ), while N0 3-assimilation (second term) is
inhibited by NH 3 only.
Organic nitrogen was represented as a fixed portion of
organic material (10 moles of N atoms per mole of organic
material). The chemical compositions of P1 , P2• P3 , P4 • and
x1 were assumed to be identical. The equation for the
appearance and disappearance of x2 was constructed:
= k 1x 1 + k 2P 1 + k 3P 2 + k 4P 3 + k 5P4
-(k6P3 + k7P4)x2x6
-k13 I(x6) P3x2x5
-k15 I(x6) P4x2x4
in which the positive terms are inputs from deaths of the
indicated populations. the first negative term represents
aerobic organic decomposition, and the last two negative
terms, anaerobic decomposition.
Both nitrate and ammonia are readily assimilated by most
microorganisms, but ammonia is the more frequently preferred
form. Aerobic decomposition. denitrification, and nitri-
fication are all microbially mediated processes from which
the organisms obtain energy to assimilate biomass. Thus
with each of these processes there is accompanying
assimilation of ammonia or nitrate. Assimilation terms for
both ammonia and nitrate are represented as proportional to
the product of the energy yielding process and the con-
centration of NH3 (or No 3-). The equations for ammonia (x 3)
and nitrate (xs) are given below (together because of
similarity), followed by the equation for nitrite.
-44-
(29)
For ammonia:
dx 3 ~ = S [k13 I(~6} P3x2x5
+ k 15 I(x6 } P 4x 2x 4
+ (k6P3 + k7P4}x2x6]
-ko2x1x3
-[k8k12P1x3x6
+ k9k14P2x4x6
+ k10k13 I(x6)~3X2XS
+ k 11 k 15 I(x6 )P 4 x 2 x 4 Jx 3
-k12P1x3x6
The terms in the NB3 equation are
(30.1)
(30.2)
(30.3}
(30.4)
(30.5)
(30.6)
(30.7)
(30.8)
(30.9)
30.1 anaerobic decomposition using N0 3-as the oxidizing
agent,
30.2 anaerobic decomposition using N0 2-as the oxidizing
agent,
30.3 aerobic decomposition,
30.4 assimilation by x 1 ,
30.5 assimilation by P 1 accompanying the net reaction,
NB3 ~ N02-,
30.6 assimilation by P 2 accompanying the net reaction,
No2 -~ N03 -,
30.1 assimilation by P3 accompanying decomposition of
organic material using N0 3-,
30.8 assimilation by P4 accompanying decomposition of
organic material using No2-, and
30.9 direct nitrification, NH 3 ~ N0 2-.
-45-
For nitrate:
dx 5
dt = k14P2x4x6
-[S k 13 I(x6 )P 3 x 2
+ x 1 k 01 r(x 3 )
+ k81 I(x3>k,2P1x3x6
+ k91 I{x3)k,4P2x4x6
+ k101 I(x3)k13P3x2x5
The terms in the N0 3 -equation are
31.1 nitrification, No 2-~ N0 3-,
31.2 denitrification, N0 3-~ N0 2-,
31.3 assimilation by x 1 ,
{31.1)
(31.2)
(31.3)
(31.4)
{31.5)
(31.6)
(31.7)
31.4 assimilation by P 1 accompanying the net reaction,
NH 3 ~ N0 2 -,
31.5 assimilation by P 2 accompanying the net reaction,
N0 2----+ N0 3-,
31.6 assimilation by P 3 accompanying decomposition of
organic material using N0 3-,
31.7 assimilation by P 4 accompanying decomposition of
organic material using N0 2 •
The NH 3 and the N0 3-equations are similar in some
respects and different in others. Three modes of decom-
position are direct sources for NH 3 (in the real world there
are more than three), and only one process, nitrification,
results in N0 3-. All the loss terms are similar, except for
the inhibition of NOl-loss by the presence of ammonia (in
the real world the d fference is not so distinct).
-46-
The equation for nitrite is given by
dx4
dt = k12P1x3x6
+ S k 13 I(x6 )P 3x 2x 5
-S k14P2x4x6
- s k 15 I{x 6 )P 4x 2x 4
(32)
The nitrite equation includes no assimilation term. The
first term is input by nitrification, NH 3 ~ N0 2-, and the
second is input by denitrification, No 3-~ N0 2-. The
third term represents loss via nitrification, N0 2-~ No 3-,
and the fourth, for denitrification, N0 2-~ N2 •
These equations, even without hyperbolic or other des-
criptive factors, are uncomfortably complex. The
complexity, however, is necessary if the model is to include
the couplings and influences depicted in Figures 9 and 10.
The numerous influences upon most of the rates give rise to
long terms in the equation; the highly coupled nature of the
cycle gives rise to many terms in each equation.
ESTIMATION OF PARAMETERS
Ideally values for parameters for such a model would be
determined from data from applicable experiments. All of
the parameters however, were merely estimated. Using a
model such as this in which each term contains several
factors, each of which is constrained to non-negative
values, it becomes very clear that parameter estimates are
strong functions of the form of the model. Initial
parameter estimates ranged from 0.0012 for the rate
constants for assimilation of ammonia and nitrate by
denitrifying organisms to 1.1 x 1012 for the rate constant
for decomposition of organic material by those same
organisms. To one used to thinking in terms of first-order
rate constants, these values (especially the latter) may
seem absurd. Had the factors appeared as rectangular
hyperbolae, so that each factor was unitless and constrained
to the interval (0, 1), the parameter estimates would have
approximated first-order rate constants.
-47-
ANALYSIS OF PRELIMINARY RESULTS
The results indicate that this model or a similar one
should be a part of a larger ecosystem model if part of the
purpose of the larger model is to describe the transient
behavior of dissolved chemicals. Steady state results for
the four chemical components for both aerobic and anaerobic
conditions are shown in Figure 11. output from the model
indicated that the net rates of change for some processes
were of the right order of magnitude to be included in a
larger model. A1 so, the behavior of the components was
shown to be interdependent so that each component is
affected by every other component either directly or
indirectly.
The parameter estimates may have been badly in error.
such a possibility is suggested by the relatively high
steady state N0 2-values under aerobic conditions and the
relative stability of NH 3 under a change from aerobic to
anaerobic conditions. The high aerobic No 2-values are
partially explainable by the failure of the model to contain
assimilation terms. The high turnover rate of NH 3 resulting
from inputs and outputs under both aerobic and anaerobic
conditions may help explain its stability.
The whole cycle may be considered to be comprised of the
anaerobic (denitrification) subsystem, and the aerobic
(nitrification) subsystem with N-fixation occurring in both.
For denitrification to occur, nitrification must already
have occurred. Tusneem and Patrick•3 showed that for
continuously flooded soils denitrification rate was depen-
dent upon the size of the aerobic layer in which nitrifi-
cation occurred. This model operates in the same manner,
i·~·, a switch to anaerobic conditions obviously results in
little denitrification if little N0 3 -has been formed by
nitrification.
Further work is needed to improve the model. Nitrite
assimilation should be added, and the whole model should be
divided into aerobic and anaerobic coupled subsystems.
Improved parameter estimates should be made. Subsequently,
it will be incorporated into an ecosystem model coupled to
chemical equilibria and tested with dynamically varying
organism and dissolved o 2 concentrations. coupled with
models for pollutants shown to affect the metabolism of one
or more of the types of organisms active in the cycle,
perhaps it may yield estimates of system impact of the
pollutant and shorten the investigative process.
-48-
:z
0
1-
<(
0::::
1-:z
LLJ u :z
0 u
. 1-·-·-·-·........ , t-----,--~· . ,
No3-
No2-
NH3
Organic Material
\
I
t-···-···-···, ,' . , 1---------·-·t . .
,
, , ,
\ ,'' ,
~,
I •
\ I
I .
\
\ .
~ .. !...-...••·-····-1
TIME
Figure 11. Relative steady state concentrations of four
chemical components of the nitrogen cycle
model. Results show a change from aerobic to
anaerobic conditions.
-49-
SECTION VI
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complex Chemical Equilibria. Ind. Eng. Chem. §Q:27-57,
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17. Morel, Francois, and James Morgan. A Numerical Method
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19. Kern, David M. The Hydration of carbon Dioxide. J.
Chem. Educ. 1!:14-23, January 1960.
-51-
20. Falls, c. Paul, and Louis P. Varga. Chemical Equili-
brium Models of Lake Keystone, Oklahoma. Environ. SCi.
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21. Lassiter, Ray R. The Effect of Higher Trophic Level
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Available Resources and Population Density in
Terrestrial vs. Aquatic Ecosystems. J. Theor .• Biol.
12:69-81, 1971.
23. Emerson, R., and L. Green. Effect of Hydrogen Ion con-
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11:157-168, 1938.
2q. Osterlind, Sven. Growth of a Planktonic Green Alga at
various Carbonic Acid and Hydrogen Ion Concentrations.
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Phosphorus. J. Envir. Eng. Div., Proc. Amer. soc.
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26. Chen, c. w., and G. T. Orlob. Ecologic Simulation of
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Walnut Creek, Calif. u.s. Department of the Interior.
August 1971. q5p.
21. Rich, Linvil G., John F. Andrews, and Thomas M. Keinath.
Mathematical Models as Aids in Interpreting Eutro-
phication Phenomena. Clemson University, Clemson, sc.
1971. 25p.
28. Schofield, William Rodney. A Stochastic MOdel of a
Dynamic Eco-system in a One-dimensional, Eutrophic
Estuary. Ph. D. Dissertation, Virginia Polytechnic
Institute and State University. Blacksburg, Virginia.
December 1971. 150p.
29. O'Connor, Donald J., Robert v .. Thomann, and Dominic M.
DiToro. Dynamic water Quality Forecasting and Manage-
ment. Manhattan College. New York, NY. EPA-660/3-73-009. u.s. Environmental Protection Agency. August 1973. 20lp.
30. Droop, M. R. some Thoughts on Nutrient Limitation in
Algae. J. Phycol. 2:26q-212, 1973.
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31. caperon, John. Population Growth in Populations of
Organisms Limited by Food Supply. Ecology. !!(5):715-
722, 1967.
32. Danforth, William F. Substrate Assimilation and Hetero-
trophy. ID: Physiology and Biochemistry of Algae,
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33. Syrett, P. J. Nitroqen Assimilation. 1n: Physioloqy
and Biochemistry of Algae, Lewin, R. A. (ed.). New
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34. Lassiter, R. R., and D. w. Hayne. A Finite Difference
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In: Systems Analysis and Simulation in Ecoloqy, Patten,
B. c. (ed.). New York, Academic Press, 1971. p. 368-
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35. Patten, B. c. Need for an Ecosystem Perspective in
Eutrophication Modeling. In: Modeling the Eutrophica-
tion Process, Proceedings of a workshop held at Utah
state University, Logan, Utah, September S-7, 1973,
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1. Utah Water Research Laboratory, OOlleqe of
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36. Fogq, G. E. Nitroqen Fixation. 1n: Physiology and
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Academic Press, 1962. p. 161-170.
37. Alexander, M. Introduction to Soil Microbiology. New
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38. Alexander, M. Microbial Ecoloqy. New York, John Wiley
and Sons, Inc., 1971. 511p.
39. Mechalas, B. J., P. M. Allen III, and w. M. Matyskiela.
A Study of Nitrification and Denitrification.
Enviroqenics, El Monte, Calif. Federal Water Quality
Administration, Department of the Interior, Cincinnati,
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40. Loehr, R. c., T. B. s. Prakasam, E. G. Srinath, and Y.
D. Joo. Development and Demonstration of Nutrient
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January 1973. 340p.
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41. Patrick, William H., Jr. Nitrate Reduction Rates in a
Submerged Soil as Affected by Redox Potential. Seventh
International Congress of Soil Science, Madison, Wise.,
1960. p. 494-499.
42. Hardy, R. w. F., and R. D. Holsten. Global Nitrogen
Cycling: Pools, Evaluation, Transformations, Transfers,
Quantitation and Research Needs. Central Research
Department, E. I. DuPont Nemours and Co. Wilmington, DL.
Mimeographed article distributed at a symposium, "The
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Quality Management Implications." u.s. Environmental
Protection Agency. October 2-4, 1972. 27p.
43. Tusneem, M. E., and w. H. Patrick, Jr. Nitrogen Trans-
formations in Waterlogged Soil. Dept. of Agronomy,
Agric. Expt. Stat., Louisiana State University, Bull. No.
657, June 1971. 75p.
-54-
TECHNICAL REPORT DATA
(Please read IIUli:Uctions on the revene before completing)
1. REPORT NO. 12. 3. RECIPII~NT'S ACCESSION' NO.
EPA-660/3-75-012
4. TITLE AND SUBTITLE
MODELING THE DYNAMICS OF BIOLOGICAL AND
CHEMICAL COMPONENTS OF AQUATIC ECOSYSTEMS
5. REPORT DATE
Januarv 1Q7C\
6. PERFORMING ORGANIZATION CODE
7. AUTHOR(S)
Ray R. Lassiter
9. PERFORMING ORG '\NIZATION NAME AND ADDRESS
Southeast Environmental Research Laboratory
u.s. Environmental Protection Agency
College Station Road
Athens, GA 30601
12. SPONSORING AGENCY NAME AND ADDRESS
8. PERFORMING ORGANIZATION REPORT NO.
10. PROGRAM ELEMENT NO.
1BA023
11. CONTRACT!GRANT NO.
Southeast Environmental Research Laboratory u.s. Environmental Protection Agency
College Station Road
13. TYPE OF REPORT AND PERIOD COVERED
Task Milestone Report
14. SPONSORING AGENCY CODE
Athens, GA 30601
15. SUPPLEMENTARY NOTES
16. ABSTRACT
To provide capability to model aquatic ecosystems or their sub-
systems as needed for particular research goals, a modeling strategy
was developed. Submodels of several processes common to aquatic eco-
systems were developed or adapted from previously existing ones.
Included are submodels for photosynthesis as a function of light and
depth, biological growth rates as a function of temperature, dynamic
chemical equilibrium, feeding and growth, and various types of losses
to biological populations. These submodels may be used as modules in
the construction of models of subsystems or ecosystems. A preliminary
model for the nitrogen cycle subsystem was developed using the modeling
strategy and applicable submodels.
17. KEY WORDS AND DOCUMENT ANALYSIS
Ia. DESCRIPTORS
Computerized Simulation
Photosynthesis
Phytoplankton, Zooplankton
Water Chemistry
Nitrogen cycle
Limnology
18. DISTRIBUTION STATEMENT
Unlimited
EPA Form 2220·1 (9·73)
b. IDENTIFIERS/OPEN ENDED TERMS lc. COSATI Field/Group
Predator-prey models
Inhibition mo(lel
Microbial Growth Rate
Algll Growth Rate
Aquatic Ecosystem Modbl
Temperature Related
Growth
06/06
06/13
08/0B
19. SECURITY CLASS (This Report) 21. NO. OF PAGES
55
20. SECURITY CLASS (This page) 22. PRICE
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