A PRIMER ON ECOLOGICAL MODELLING IN LIMNOLOGY
Carlos Ruberto Fragoso Jr1; Tiago Finkler Ferreira1; David da Motta Marques1
1
Federal University of Rio Grande do Sul, Hydraulic Research Institute, CP 15029, Porto
Alegre, RS, Brazil. crubertofj@hotmail.com, tiagofferreira@hotmail.com, dmm@iph.ufrgs.br
LIMNOLOGY AND ECOLOGICAL MODELLING
The shared management of natural resources, based on specific knowledge, is the best
form of promoting environmental conservation. In aquatic ecosystems, the conservation of water
resources depends on specific understanding and management of limnological variables.
However the wide range of physical, chemical and biological processes and factors, and their
interactions, makes the quantitative analysis of the aquatic ecosystems very difficult. In addition,
management of aquatic ecosystems is, by their characteristic, a field of multidisciplinary action,
where there is a great number of alternatives on planning and forecasting, considering their uses,
availability and preservation (Tucci, 1998).Thus, the diversity of methodological approaches
available to quantify processes are essential to acquire higher understanding of the dynamics in
natural systems and promoting tools for accurate decision making. One of these approaches is
the mathematical ecological modelling applied to limnology.
Models focused on ecological themes are thinking pad at the limnologist disposal. It
allows the formulation of questions and obtaining answers considering the pre-established
outline conditions, it allows the formulation of hypotheses and their test, as well as the
development and consolidation or the refutation of theories. A model is the representation of
some object or system, in a language easily accessed and used, with the means of understanding
and searching for answers according to different inputs. In order to better represent reality, a
model must simulate the highest possible number of processes that occur in nature. The larger
the number of involved interactions the more complex the systems, and, consequently, the more
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2
challenging and necessary are the models. In Limnology, a model is a tool developed in order to
assist the understanding of an aquatic ecosystem, considering the involved drivers, direct or
indirect, and in different scales, including the antropic (e.g. social, economical) and natural (e.g.
climatic, chemicals, biological, hydrographic basin, hydrology, hydrodynamic) factors, and their
interactions and processes. Ecological models can be applied on the Limnology, e.g., for: (a)
evaluation responses of trophic cascading interactions before changes in nutrient concentrations
(Janse, 2005; Jeppesen et al., 2002; Jakobsen et al., 2004; Scheffer et al., 1993; Moss, 1990;
Moss et al., 1996; Perrow et al., 1994, Ferreira et al., 2008,); (b) prediction of phytoplankton
blooming (Fragoso Jr, 2005, Lucas et al. 1999a, 1999b; (c) determination of the trophic state in
aquatic ecosystems (Kishia et al., 2007); (d) investigation of ecological concepts in limnology
such as the theory of alternative stable states (Van Nes et al., 1999; Ferreira et al., 2008; Fragoso
Jr. et al., 2007); (e) assesment of biomanipulation impacts (e.g. fishing or alteration of trophic
interactions) on the system (Carpenter & Kitchell, 1993; Hansson et al., 1998; Meijer et al.,
1994; Shapiro et al., 1975; Shapiro & Wright, 1984; Van Donk et al., 1990); (f) estimative of
gross primary production, community respiration and net production of the ecosystem in terms of
carbon (Mukherjee, et al, 2002; Sandberg, 2007); (g) evaluation of pollution levels.
A model must not be considered as an objective, but as a tool for reaching a determined
goal. It may be utilized with the means of prediction; understanding processes; filling the
variables of interest within a period without survey; and the generation of hypotheses, which
may be tested experimentally or in situ. It is worthy to emphasize that the modelling should be
utilized in partnership with experimental, laboratorial and monitoring works, otherwise its
potentiality of application may become fulfilled.
There is a wide range of software with restrict access or publicly available (e.g. SWMM,
2004; AQUATOX, 2004; Chapra and Pelletier, 2003), which may be acquired for analyzing a
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3
determined ecological issue. However, these models manuals, many times, do not disclose an
appropriate description of the applied methodology, neither a description of their limitations and
advantages. Therefore, this chapter aims at providing conditions of analyzing existent models,
supplying a conceptual base for the construction of simple ecological models to the application
in environmental problems associated with aquatic ecosystems or in scientific investigations.
The application of models for scientific questions is almost compulsory if we want to
understand a complex system as an aquatic ecosystem. It is not simple to investigate all the
components and their interactions in the ecosystem without using models as a synthesis tool. The
use of modelling as a tool for understanding the proprieties of the system has advantages and has
been revealing gaps in our knowledge. Perhaps the main contribution provided by a model may
be the establishment of research priorities, which may reveal proprieties of the system from
scientific hypotheses generated by the model itself. Therefore, the models by simulating the
interactions in the aquatic ecosystem not only generate results that may be compared with in situ
or experimental observations, but also may serve as a thinking base for important scientific
questions.
History of the development of models
The evolution of modelling may be divided in four great phases (Figure 1). These phases
are related to the social interest and the computational capacity offered in the period. The first
works on modelling appeared around the 20’s with the urban wastewater problem. The pioneer
modelling work was done by Streeter and Phelps (1925) in Ohio River. This work and the
subsequent ones were focused on the evaluation of the dissolved oxygen levels in rivers and
estuaries. Still without the availability of computers, these applications were limited to linear
solutions, with simple geometry and considering a steady state.
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Reaeration
1925 -1960 ( Streeter -Phelps )
Problems: untreated and primary effluent
Pollutants: BOD/DO
Systems: rivers/estuaries (1D)
Kinetics: linear
Solu tions: analytical
DO
BOD
P
R
DO sed
1960 -1970 (computerization)
Problems: untreated and primary effluent
Pollutants: BOD/DO
Systems: rivers/estuaries (1D / 2D)
Kinetics: linear
Solu tions: analytical and numerical
Fish
1970 -1977 (Biology)
Problems: eutrophication
Pollutants: nutrients
Systems: rivers/lakes/estuaries (1D / 2D / 3D)
Kinetics: nonlinear
Solution: numerical
Zoo
Phyto
NO 3
NH 3
PO4
Porg
Norg
1977 - Present (Toxics)
Problems: toxics
Pollutants: organics and metals
- interactions/
Systems: water-sediment
food-web interactios (1D / 2D / 3D)
Kinetics: nonlinear
Solutions: numerical and analytical
Solids
Toxics
Biota
water
sediment
Solids
Pore
water
Benthos
Figure 1. Four periods of development of the models for limnology (Adapted from Chapra,
1997).
In the 60’s the computers appeared as a widely available tool and this leading to a greater
advance of the models and their applications. The oxygen was still the focus, but the computers
allowed the solution of more complicated problems, as complex geometries, more detailing of
the kinetic reactions and non-steady simulations (dynamic simulations).
In the 70’s, another outstanding phase occurred stimulated by the environmental
consciousness from the period. The dissolved oxygen problems associated to point sewage
sources gave rise to the awareness of eutrophication problems in aquatic ecosystems. In this
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5
period, the first models that were developed would represent the dynamic of the aquatic food
chain, such as the models by Chen (1970) and Di Toro et al. (1971). With the additional
computational advances, non-linear and retroactive solutions could be applied in systems with
complex domains.
The most recent phase of models development turned to issues involving toxic,
pathogenic and trace-metal substances, which represent a great threat for the human health and
the aquatic ecosystem. This issue has also been effectively marked by generated political
debates. However, the past problems still endure theses days, once the computational progress
has provided solutions closer to reality.
Ecological models
An ecological model considers, in its conceptual structure, part or fraction of the
processes related to the biota of the ecosystem. In aquatic ecosystems, the ecological model tries
to simulate the processes inherent of the aquatic trophic interactions in order to evaluate the
organisms’ dynamics (Figure 2). The abiotic and biotic components in the aquatic environment
have different processes of development and, therefore, different approximations to be included
in the modelling. Empiric or deterministic functions may approximate these processes. The
mathematics representation of processes, such as, the primary, secondary production and other
eco-physiologic traits, are available in the literature, although many of them depict the reality of
temperate ecosystems. A few models can distinguish classes of groups with phytoplankton,
macrophyte and fish, and, therefore, generalize the main processes of great groups as a state
variable for all. However, currently, there have already been models capable of distinguishing
classes of phytoplankton (i.e. cyanobacteria, green algae, diatoms, etc), macrophyte (i.e.
A primer on Ecological Modelling in Limnology
6
submerged, emergent and floating) and fish (i.e. piscivorous, omnivorous and planktivorous),
considering their stages of life (juvenile and adult) (e.g. Janse, 2005; Fragoso Jr. 2007).
In order to depict the high level of functional diversity of the aquatic organisms, the
ecological models must include the main processes of each group within modules form that
contain a group of differential equations representative of ecological functions and metabolic
coefficients related to the biological processes. These coefficients are found in situ or
experimentally such as (a) respiration rates, primary and secondary production; (b) limit capacity
of support (carrying capacity) of biomass or species density per area or water volume, (c)
nutrients assimilation (i.e. phosphorous, nitrogen and silicate) by primary producers, (d)
competition by nutrients available in the water column, (e) absorption of photosinthetically
active radiation (PAR), growing rates, reproduction and mortality; (f) excretion, biomass loss
and decomposition. Currently, with the computational available capacity, the mathematical
approaches tend to include, with a greater level of detail, all the elements of the aquatic food-web
(i.e. communities, complete cycle of phosphorous, nitrogen, silica, carbon and their interactions
among the organisms), which are essential for the evaluation of stocks (e.g. plankton
compartments, aquatic macrophyte, fish, and benthos), and system processes and patterns..
A primer on Ecological Modelling in Limnology
7
Figure 2. Simplification of the aquatic food web. The thickness of the arrows indicates the
strength of interaction. (Adapted from Carpenter & Kitchell, 1993).
MODELLING ELEMENTS
A mathematical modelling consists, basically, of four components aiming to represent a
specific phenomenon of interest: (a) governing functions or external variables; (b) state
variables; (c) mathematical equations; and (d) parameters (Figure 3). These components assist
the translation of a determined phenomenon existent in the nature towards a mathematical
language. On this section, we will thoroughly describe each component of the mathematical
modelling and their inter-relations.
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Phenomenon of interest
Parameters
Driving forces
or
external variables
Equations
State
variables
Figure 3. Elements of modelling and their inter-relations for explaining a determined
phenomenon.
Phenomenon of interest
The phenomena are patterns found in nature that may be observed or noticed (e.g.
precipitation, rivers flow, eutrophication, and alteration of the aquatic trophic structure promoted
by some natural or anthropogenic stressors). Typically, the phenomena are described from the
pre-established assumptions related to homogeneity, uniformity and universality of the
proprieties and their main components, which include temporal and spatial relations descibing
the phenomenon. Nevertheless, in order to model the phenomena with the necessary level of
realism, these rigid assumptions have been simplified and approximated in such form so that the
model may be able to represent (Couclelis, 1997):
• The space as a non-homogeneous entity as well in its proprieties as in its structure.
• The surroundings as non-estationary relations.
• The rules of transition as non-universal rules.
• The variation of time as a regular or irregular process.
• The system as an open environment to external influences.
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For implementing dynamic spatially ecosystems with the characteristics mentioned
above, some basic principles relative to the main elements from these systems must be
considered. Among these principles, it must be emphasized: (a) the way of temporal and spatial
representation, (b) the structure of the model itself to be utilized for the representation of spatial
phenomenon and (c) the computational approach for implementing these principles in an
integrated and consistent form. On the next sections, we will discuss on the elements of
mathematical modelling utilized for the representation of an interest phenomenon.
Driving forces or external variables
Functions or variables of nature are the ones that influence the state of the aquatic
ecosystem. In the management context, the problem may be formulated in the following way: if
some phenomena are variable, which one will influence the state of the ecosystem? In this sense,
the model is used for predicting the change in the ecosystem when external variables are altered
in time and space. For example, a nutrient input, climate changes, inflow or outflow in the
system may be considered as external variables or driving forces.
State variable
The state variables describe, as the name indicates, the state of the ecosystem. The
selection of state variables is crucial for the model structure, but generally, in most cases, this
selection is trivial. We may, for example, choose for modelling the state of eutrophication in a
lake, where the concentration of phytoplankton and nutrients as state variables is intuitive. The
states variables are related to the external variables and may be considered as a model outcome.
Depending on the purpose of the application of a model, it will contain more state variables than
is necessary, since a state variable may explain others. For instance, in models of eutrophication,
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the zooplankton could be included as a state variable, which promote a grazing control on the
phytoplankton concentration.
Mathematical equations
The physical, chemical, physical and biological processes (e.g. nitrification, primary
production, mortality, interactions, sediment nutrients release) are represented in the model
through mathematical equations. These equations are the relations between the external variables
and state variables. The same process may be found in different aquatic ecosystems, suggesting
that the same equation may be used in different models. The relations for each process may be
found in the literature or may be obtained directly from field and experimental work (e.g.
Jorgensen, 1986; Sheffer, 1998; Chapra, 1997). A determined process can be made of
innumerable mathematical equations, and, thereby, the modeler has to decide which equation
better represent that process with the smallest number of simplifications.
Parameters
The parameter is a value that characterizes a process in the ecosystem, and may be
considered constant for a particular system or for one part of the system, indicating that a
parameter can also be variable in time and space. In ecological models the parameters have a
scientific definition, as, for instance, the maximum growth rate of phytoplankton or the grazing
rate of zooplankton. The complexity of a model has been represented by the quantity of applied
parameters. Simple models have a lower quantity of parameters, while complex models have a
larger number. Groups of known or suggested values for specific parameters can be found in the
literature. However, the majority of parameters are subjected to adjustments with the means of
approximating the maximum outcome of the model to the values observed in the field.
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THE MODELLING STEP BY STEP
As a research method modeling has a methodological orientation to be followed. In this
sense, different schemes were elaborated aiming to describe the phases concerning a process of
Mathematical Modelling. There have been several methods with different number of phases,
however, it is important that each method contemplate the specific objectives of the problem.
Different objectives need different spatial and temporal scales. A general scheme has been
compounded in eight phases (Figure 4). From these phases, some have been very generic and
may be particularly treated by each modeler, although others have been considered normative
(i.e. pattern) and have deserved a greater detailing, and are described briefly as follows.
Definition of the problem
Given a real situation, the problem to be studied must be identified. The problem is
nothing more than an observed phenomenon that we want to represent mathematically.
Subsequently, the elements of modelling, necessary for its solution, have to be specified.
There are several problems in Limnology to which a mathematical model might
contribute, somehow, in the understanding of processes, prediction, generation of hypotheses, or
even in the behavior of variables of interest for periods without observation. (Figure 5).
A primer on Ecological Modelling in Limnology
Definition of the problem
Conceptual model
12
Selection of
complexity
Association of the problem with time,
space and sub-systems
Quality of data?
Data acquisition
Simplification and formulation of
hypotheses (conceptual)
Deduction of the mathematical
model (equations)
Resolution of the mathematical problem
Demanded
Revision
Calibration and validation of the model
Application
Figure 4. An approximation of the modelling procedure (Adapted from Jørgensen, 1986).
Eutrophication
Cyanobacteria
blooms
Net ecosystem
production
Biomanipulation
Themes
in
Limnology
Trophic
interactions
Carbon cycle
Fishing
Water management
Alternative
steady states
Figure 5. Themes in Limnology that can be evaluated using mathematical models.
Simplification and formulation of hypotheses
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In this phase, the modelling elements are examined and selected in order to preserve the
characteristics of the problem. For that, it is necessary to simplify the reality and define the the
driving functions, processes and state variables representative of the phenomenon of interest.
Additionally, the parsimony principle must be applied. This principle advocates the adequate
representation of a process and/or system behavior by a model with the possible smaller number
of variables and/or parameters. For example, in estuaries where the tides action rules the
hydrodynamic of the system, and, consequently, the pollutants transportation, the effect of the
wind may be discarded or simplified. In deep lakes or reservoirs, the vertical processes are more
important than the horizontal processes. On the other hand, in shallow lakes, where stratification
of the water column does not occur, the horizontal processes are more important. In summary,
this is a decisive phase for modelling, in which the modeler defines the conceptual model. This
conceptual model is a simplified scheme, utilizing blocks and arrows, which shows the involved
state variables, the processes and the interactions among the variables of interest. For example,
for exploring some basic proprieties of interactions between the phytoplankton and zooplankton,
the simple model of prey-predator of Lotka-Volterra is generally utilized (Scheffer, 1998). This
example will be adopted for the elaboration of its conceptual scheme. This model considers the
phytoplankton and zooplankton biomass as variables of interest (blocks) and the processes of
interactions among the organisms as flows (arrows) (Figure 6).
The processes or interactions among the variables of interest are mathematically
represented by equations. It is up to the modeler to choose the approximation that will be
adopted in each process. For example, the production of phytoplankton biomass is a biological
process that depends on several factors, such as the distribution of the light through the water
column, temperature of water and nutrients availability. The modeler could opt for choosing an
equation that would involve all these factors, a combinations of them or, simply, choosing a
A primer on Ecological Modelling in Limnology
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constant rate of daily production that would reduce all the driving factors for the phytoplankton
production in only one coefficient (Figure 7). The larger the number of parameters and external
variables involved in the calculation of a process the better is the approximation with the reality
and greater is the difficulty of their estimation in field or experimentally works. The
representation of a process by a constant value may be a gross simplification of reality, however
it is a way to condensate all the process through only one parameter (constant rate). Thereby, the
understanding and control of such process is facilitated.
Figure 6. Scheme representing the prey-predator model of Lotka-Volterra, which the variable of
interest is the phytoplanton and zooplankton biomass.
The challenge of the modeler is always finding the easiest alternative, considering the
complexity of the model (number of involved parameters), to investigate the object of study
without missing the approximation with reality (Figure 8). As mentioned previously, at this time
it is worthy the application of the parsimony principle that is the adequate representation of the
behavior of a system or process, through a model, with a possible smaller number of parameters.
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Production
Light
L 

2,781  f  e  1  e   2
ke H
Constant rate
Temperature

T  Gmax  TT
Nutrients
20
N 
N
kN  N
Figure 7. Possible mathematical representations of the phytoplanktonic biomass production.
Complexity
Aproximation
Optimal number of
parameters
Number of parameters
Figure 8. Model complexity versus solution approximation (difference between real and modeled
solutions). The joint of the two curves represents the optimal number of parameters utilized to
represent a specific process or phenomenon.
Deduction of the mathematical model
In this stage, the conceptual language is replaced by a coherent mathematical language.
That is, the state variables and flows have been written in mathematical terms. For each state
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variable, a flow balance is checked with means of representing the continuity in an infinitesimal
interval of time (dt) (Figure 9).
The equation of continuity for a state variable, A, discarding the flows related to
transportation and chemical reactions, may be written as:
dA
 input  output
dt
(1)
The differentiation of state variable (A) in relation to time (t) represents the balance of the
variable of interest in a certain interval of time or its internal variation in that interval. The
differential has a flow unit and, thus, the balance of the variable of interest must be also in terms
of flows. For example, the prey-predator model mentioned previously could be represented by
the following differential equations:
dF
 rF  g z FZ  production  grazing
dt
(2)
dZ
 e z g z FZ  m z Z  growth  mortality
dt
(3)
The first term on the right side of the Equation 2 represents the quantity of biomass fixed
through the photosynthesis within the time interval. The second term describes the losses of
phytoplankton biomass (negative flow) due to the consumption by zooplankton. The population
of zooplankton converts the ingested food in growing with certain efficiency (ez) and suffers
losses due to respiration and mortality for other organisms. The phytoplankton biomass (F) and
zooplankton (Z) are the state variables of interest in this model.
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Figure 9. Representation of the input and output flows (processes) for a specific state variable
(A). The flows related to transport and chemical reactions could also be included in the model.
It is worth to emphasize that the decision of what factors will be included in each term for
better representing a determined process is responsibility of the modeler. For example, the
primary production is associated with various factors, such as, temperature, light, nutrients, and
concentration of the phytoplankton itself, among others. In this simplified model, it was assumed
that the primary production would only depend on the phytoplankton concentration and other
factors would be neglected. This false supposition may not represent this process at all, but, on
the other hand, only one parameter was used (r) in order to minimize the complexity of the
model (principle of parsimony).
Resolution of the mathematical problem
During this phase mathematics and computational resources are used by the modeler
aiming the solution of the formulated mathematical problem. The mathematical methods utilized
to solve the differential equations may be analytical or numerical. After the equations resolution,
the next step is choosing an appropriate computational language for the implementation of the
differential equations of the model (Figure10). There are several available softwares that refer to
this subject, such as, EXCEL, MAPLE, MATLAB, FORTRAN, C++, DELPHI e TURBO
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PASCAL. The mathematical software choice is directly related to the modeler familiarity with
the program, as well as to the complexity of the problem to be resolved. Some mathematical
programs have taken advantages in relation to others in terms of processing speed and
availability of pre-built functions. The model code is an important part of model development
and may take some time to optimize and run free of bugs.
Figure 10. Computational mathematical programs which process information and generate
results.
Calibration and validation of the model
After the model is formulated , it is necessary to calibrate and validate the model. This is
achieved by adjusting the parameters so that the model’s outcome approximates to the observed
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data. (Figure 11). The calibration of the model may be achieved by try and error or by algorithms
able to calibrate the parameters automatically utilizing objective functions that minimize the
difference between the calculated and observed values. The validation of the model consists of
testing the calibrated parameters in another period with the observed data. In case the model is
not be considered valid, that is, if the model’s outcome was not close to the observed data, the
modeler must reformulate hypothesis and simplifications and restarts the process.
Observ
Observated
Calculated
Calcula
A
Calibration period
Validation period
Figure 11. Process of model calibration and validation.
The efficiency of the model estimative is measured through statistical techniques that
evaluate particular characteristics from the calculated series. Examples of these techniques are
presented in Table 1. The coefficient of determination Nash-Sutcliffe (R2) prioritises the
comparison of values with the observed values mean; the root-mean-square error (RMSE) gives
a higher weight to the values with a greater magnitude, and, the root-mean-squared-error
inverted (RMSEI) prevails the adjustment to the small order values.
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These techniques of model efficiency analysis may demonstrate interesting aspects for
the comprehension of restrictions, limitations and advantages of models.
In general, due to the great number of differential equations used in ecological models
(what leads to an increased quantity of parameters), the determination of a set of parameters that
is better in agreement with the observed data may be a challenging task. In order to minimize
this complexity, some abiotic processes and eco-physiological aspects of the aquatic organisms
of the ecosystem could be adjusted separately in specific experiments, what would reduce the
number of involved parameters in the calibration phase. This type of experimental calibration is
denominated parameterization, which is guided to experimental adjustment of global abiotic and
biological process coefficients within the aquatic ecosystem. That is, the parameterization refers
to the adjustment of parameters through the variation estimative of such processes within
established gradients. For example, in order to adjust the eco-physiological parameters related to
state variables of phytoplankton or macrophytes (e.g. primary production, respiration, excretion),
it is necessary to evaluate the variation of these processes within a controlled variation of the
controlling abiotic variables (e.g. temperature, light, pH, etc).
The necessity of parameterization is due to the large number of existent parameters in
complex ecological models for representing, in a more approximate way, possible ecological,
populational and eco-physiological processes and functions.
Table 1. Coefficients used for describing the efficiency of the adjustments of the models.
Coefficients
Coefficient of determination by
Nash-Sutcliffe (R2)
Equation1
R2  1
 YObs t   YCal t 

2
2
 YObs t   YObs t 
A primer on Ecological Modelling in Limnology
Root-Mean-Square-Error (RMSE)
Root-Mean-Square-Error
(RMSEI)
21
RMSE 
Inverted
RMSEI 
 YObs t   YCal t 
N
2
 1
1 


 

 YObs t  YCal t  
N
2
is the observed value, YCal is the value calculated by the model, YObs is the mean values observed and
1
YObs
N
is the total number of the values.
Application of the model
If the model has been considered valid, the same may be utilized in applications with
diverse objectives, such as generating hypotheses, better understanding of a problem, explaining
a phenomenon, analyzing the behavior of state variables, making forecasting and making
decisions based on the outcome results. The last application option is the one that makes possible
the management of situations associated with impacts on aquatic systems (i.e. study of
scenarios).
Considering the prey-predator model, proposed by Equations 2 and 3, the two organisms
interact among themselves, where one of them serves as a primary food source to the other. In
this application, the mathematical modeling was utilized for better understanding the related
processes in the competition between these two organisms. For a specific set of parameters
(ecosystem condition), phytoplankton works as prey and zooplankton as predators (Figure12).
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10
F
Z
6
(mg/L)
Concentration (mg/L)
8
4
2
0
0
200
400
600
Time (days)
800
1000
Figure 12. Simulation of prey versus predator processes in the water and sediment compartments
in a hypothetic lake involving zooplankton (Z) and phytoplankton (F).
With a small initial concentration of zooplankton (predator), the production of
phytoplankton (prey) in the environment starts to increase. In a certain point the prey population
becomes so numerous that the predator population starts to grow. Eventually, the predators’
increase causes the phytoplankton availability decline. This decline leads the zooplankton
population to a decrease due to the lack of food. The process, then, becomes seasonably cyclic.
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of Science and Technology, Washington.
CARPENTER, S.R.; KITCHELL, J. E. 1993. The trophic cascade in lakes. Cambridge Univ.
Press.
CHAPRA, S. 1997. Surface water-quality modeling. McGraw-Hill series in water resources and
environmental engineering. Editora: McGraw-Hill.
CHAPRA, S. C. ; PELLETIER, G. J. 2003. QUAL2K: A Modeling Framework for Simulating
River and Stream Water Quality: Documentation and Users Manual. Civil and
Environmental Engineering Dept., Tufts University, Medford.
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