1
A Model of hydrological and
biological interactions in a
Water Course
Ulrich Goluke
Control Data Corporation
Advanced Modeling and Simulation Group
Bloomington, Minnesota
Abstract
The model described in this report is meant to show how some of
the practical problems of combining hydrological and biological
processes can be addressed, how models can be used to examine speci-
Fic questions, and along what lines the present model ought to be
developed to eventually arrive at a ‘useful policy tool.
Introduction
A model is best thought of as a map. As different maps serve
different purposes, so do different models serve different programs.
And just as a tourist would be i11-advised to use an industrial zoning
map, so should a model not be used for purposes other than the ones it
was designed for.
Ours is a mode? to demonstrate the feasibility of addressing the
needs outlined in the Abstract through a formal system dynamics simuta-
tion model, The basic assumptions derive from that.
Assumption
First, we assume that the continuous river system can be broken
down into discrete reaches. The desired resolution can be achieved by
making the reaches smaller and smaller, thereby using more and more of
them to cover a given section of the river.
275
Second, we assume that the biological processes that take place in
the river part and the lake part of a watercourse are the same. What
differs is the speed with which material is moved and the strata where
the processes take place. We model this by using basically the same
structure for all interactions in the water but vary the parameters
from river to lake.
Third, we assume that the degree of diversity does not change over
the course of our investigation.
Fourth, we assume that it is sufficient to keep track of biological
energy, matter and nutrients, and of how hydrological conditions affect
these three.
Fifth, we assume that the value of our model will be greater if
the parameters chosen reflect to some extent a rea? river system.
Therefore, we chose to model the outflow from a eutrophic lake, a
mesotrophic river part and an oligotrophic lake. This situation exists
in reality in the Halden watercourse from the outflow from Bjoerkelangen
to Roednesjoen.
Overview
As a first approximation to reality we can view the biology and
the hydrology of a watercourse as two planes at angles to each other.
The biological plane represents in this visualization a potential
of (biological) production. The interaction with the hydrological plane
determines then how much of the potential will be realized. Passing
through a waterfall, for example, none of the potential will be
realized on account of the fast flowing waters in a lake, on the other
hand, most if not ail will be realized.
We have further assumed that the interaction of both planes takes
place along the line AB in Figure 1. The line corresponds in the formal
model to the set of state variables (i.e. the variables that accumulate
all differences in the rates of change). Each state variable has, in
addition to in- and outflow rates determined by biological causes, in-
and outflow rates determined entirely by hydrology.
In this first model we omit all interactions of the type CD. An
example of that would be the relationship between flow and turbidity
and mixing of nutrients, The strength and significance of these types
of causal relations will be explored in later versions of the model.
Conceptually, therefore, our model looks like Figure 2, n reaches,
Vinked by hydrology, are identical in structure, but different in
parameter. Each reach contains as many organisms, currently grouped
‘into trophic levels, as desired and keeps track of as many nutrients
as necessary. Each trophic level in turn is split into compartments,
initially only mass and energy. The model is, in short, a matrix with
the following elements: reaches (2), trophic levels or species (3),
living matter compartments (2), and nutrients (2). (The numbers in
brackets refer to the current level of resolutions.) Repeated solution
of the difference equations, currently once every eight hours of
simulated time, moves all matrix elements through time.
The assumption of uniformity within a reach allows us to treat both
biological and hydrological processes in a simplified manner, The
biological reality is reduced to an interaction between matter and
energy and a cycling of nutrients. Energy controls the growth, decay
and consumption of matter, whereas matter is the medium of transporting
HypRoLesy
gtoveny
Figure 1: Biological and hydrological planes.
The entire watercourse is then divided into reaches. The criterion to
separate one reach from another derives from the assumption of
uniformity within each reach. Whenever the conditions in the river
change significantly, a new reach should be started. In the present
model we chose the crudest interpretation of "significant" and merely
differentiate between the river part and the lake part of the watercourse.
276
Figure 2: Stacked Reaches connected by Hydrology
energy through the biological system. Once again we have in this
demonstration model chosen the simplest of worlds: matter is divided
into three trophic levels and dead organic matter. Of all possible
nutrients we only keep track of nitrogen and phosphorous, which are
judged to be the only limi
ing ones in Norwegian rivers.
Trop! Levels
In each trophic level we keep track of various attributes.
Currently, only mass and energy are included, but in the future, stored
nutrients, energy compartments and the like may be added.
We felt it necessary to include matter and energy even in this
demonstration model, because we believe the usual linear relationship
between the two to be wrong ina small, but crucial area, namely in
the phase relationship between them.
Energy is fixed, through photosynthesis for primary producers, and
through consumption and energy transfer for the other levels, stored
and then released. The bulk of it is used for maintenance: respiration
and maintaining the structural and biochemical integrity of the organism.
Some is used for growth. Other uses, notably community energy expendi -
ture and energy expenditure for defensive strategies are not yet
accounted for. In any case, however, it is the availability of energy
that determines an organism's maintenance, growth, and consumption
strategies. Matter is intractably connected to the energy balance, for
it provides the mechanisms to fix more energy while at the same time
maintaining it constitutes the most significant energy drain. Time
lags between growth of mass, fixing of energy, utilization of energy
277
7
and further changes in mass seemed to us important enough to trace the
relationships in detail.
—_
or
Niven paso Jw
ater
‘ ey
\ { ;
oon
Figure 3: General Trophic Level Structure
Figure 3 gives a general impression of the relationships included
in each trophic level. Mass increases exponentially in the absence of
death at a certain rate (i.e. doubling time). reflecting environmental
conditions such as temperature, nutrient availability etc. Growth, in
the long run, i.e. during a season, is balanced by deaths, either
through grazing or simply through natural death. Phase shifts between
growth and death lead to the usual abundance in the summer and the
near absence of mass in the winter.
The level, or amount, of mass causes a certain amount of energy to
be fixed, although the increase of energy {s also related to the amount
stored, relative to the needs. In other words, if an organism has
starved for some time, it will use more of its remaining energy to
secure more food (Calow: 30-36), thus raising its energy fixed per unit
mass. Energy will be decreased primarily through maintenance expendi-
ture. In the model we assume that if energy is limited other uses will
be curtailed to allow the maintenance of existing organisms. This is
achieved by manipulating the amount allocated to each of the rasks.
The residual of energy is available for growth. Death of matter
reduces both the mass and the amount of energy stored.
Nutrients
The role of nutrients in the demonstration model is limited. We
keep track of nitrogen and phosphorous in their soluble form. We
make allowances for the seasonal turnover in lakes that move nutrients
from the bottom where they tend to be released up to the epilimnion
where they tend to be used by primary producers. We ink concentration
to growth by Michaelis-Menton type relationships but loose track of
nutrients once they are taken up in biomass.
At a later stage we believe it to-be very fruitful to track
nutrients even in the biomass which will then allow for a more realis-
tic representation of the relationships between concentration and
uptake, nutrients-in-biomass concentration and growth, and the luxury
consumption relationships.
278
Hydrology
Since hydrological models are rather far advanced we have tried to
avoid the mistake of reinventing the wheel. Instead, we have concen-
trated on the question of linking hydrology and biology and have
provided only a very rough hydrology section. Principally, we generate
average flows in each reach and keep track of the volume. By knowing
the mass of organisms, grouped into trophic levels, we know the
amounts entering and leaving a reach. At present, all water enters
reach 1, flows through the system, and leaves reach 2. We have not yet
allowed for evaporation, diversion, runoff and the like.
Nodel_ Extensions
Work on extending the model must be precedéd by a thorough analy-
sis of the purpose-of the extension. Assuming for the time being that
a more accurate policy tool is the purpose, the following guidelines
may be of help.
Biology
It is easy to include more and more detail, but whether this will
automatically lead to a better model is not clear, As maps can become
confusingly complex, so can simulation models. The suggestion below
should, therefore, be read with these reservations in mind.
Nutrients
One obvious extension is the inclusion of more, or different,
nutrients. A prime candidate is oxygen. Care should be taken not to
include all nutrients, but only those that make, or are expected to
make, a significant contribution to a change in the behavior of the
system.
279
Energy
Currently, energy is biological energy on the population level,
even though the concept of maintenance and growth are mapped over
directly from the organism Jevel. A useful question to explore is the
significance of other uses of energy at the population level: growth
through division vs. growth through reproduction; defensive strategies;
maintenance of structural and biochemical integrity; changes in energy
uses over the life of the population; and the like. Equally interesting
is the problem of mapping biological energy over to the ecosystem level
to allow direct comparison between kinetic, chemical, heat, and
biological energy.
Aggregation
The grouping of organisms into mutually exclusive trophic levels
was made without much reflection simply because standard text books
split up the world that way. Some thought should, therefore, go into
the question of the best suited mechanisms of aggregation.
Compartments
Two attributes, mass and energy, are presently kept track of. It
may well be advisable to describe living matter through more attributes.
A first one that comes to mind for the primary producers is nutrients.
This would allow the decoupling of external concentration from growth,
a decoupling that 4s known to exist in reality through, for example,
the phenomenon of luxury consumption.
Hydrology
The most useful extension in hydrology would be a thorough survey
of existing hydrological model. Depending on the type of model
lL
available, it may be possible to adapt one, or, at the very least,
knowledge of other hydrological models will greatly reduce the develop-
ment cost of a realistic’ hydrology sector. We feel that such a sector
should allow for precipitation and temperature being transformed into
monthly (or weekly) runoff, for ground- and river water exchanges if
that is significant, and for evaporation from lakes. It should also
be easy to simulate withdrawal of water in any given reach, introduc-
tion in other reaches, and regulation of flow.
Geography
Theoretically it is possible to cut the river system up into
smaller and smaller reaches, so that the assumption of uniform
conditions within the reach becomes more and more true. In practice,
however, two problems arise. First, the solution interval must be
not larger than 1/5 or so of the smallest time constant in the model
to ensure reasonable computational accuracy. If reaches become tiny,
retention time becomes extremely short and a model run of 5 to 10
years requires many millions of computations. Computers can handle
this but the cost per run may be judged to be too high.
Second, the assumption of uniformity extends to exogenous inputs
to the reach, It may be difficult, if not impossible, to get the input
resolution to match the fine reach resolution. Taking larger input
units and dividing them by the number of reaches over which they are
spread defeats the whole idea of small reaches.
Data
Much of the data needed for 2 model of the kind described in this
report is difficult to get. One model extension is, therefore, to
12
begin a dialogue with professional data gatherers to work towards a
better match between data needed and data offered. Nodern data gather-
ing has developed to the point of being virtually automatic and con=
tinuous. The promise of a model like ours, which is to provide a
theoretically solid and consistent framework for the analysis of these
floods of data should make it easier to open such a dialogue.
Conclusion
Models are means. The end is to better understand the working of
an entire river system to make more intelligent decisions about a
limited resource. Model extensions should always be designed ‘with
that end in mind and not be allowed to become ends in themselves.
Acknowledgements
The work reported in this paper was conducted at the Resource
Policy Group, Sagveieu 21, Oslo 4, Norway, in collaboration with the
Norwegian Institut of Water Research.. The full paper is available
from the Resource Policy Group.
280
