SIMULASI EKOSISTEM AKUATIK
Aquatic Ecosystem Simulation
Modeling
Don DeAngelis
U.S. Geological Survey, Florida Integrated Science
Centers
Miami, Florida
Interdisciplinary Modeling for Aquatic Ecosystems
Lake Tahoe, July 17-22, 2005
SUMBER: www.ag.unr.edu/.../Aquatic_Ecosystem_Mo...University of Nevada, Reno
Ecosystem Modeling and Hydrologic
Modeling
Like hydrologic modeling, ecosystem modeling is
primarily a matter of keeping track of a balance of flows.
However, it is important to recognize some basic
differences in practice.
Ecosystem modeling is not as far along in having easily
used ‘off the shelf’ generic models (though some exist).
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Ecosystem Modeling and Hydrologic Modeling
Some problems relate to the uniqueness of individual ecosystems:
Each ecosystem consists of its own suite of species populations, environmental
conditions, spatial complexity, disturbance history, etc., that are unknown. In
particular, the existence of thousands of poorly-understood species (with even less
well known interactions) in an aquatic system always has the capacity to provide
surprises.
Others relate to problems of complexity
Population interactions (predator-prey, competition, mutualism) are highly nonlinear
(density dependent). We are not close to comprehending mathematically the
complex dynamics that result from highly non-linear, multi-variable systems.
Together these create difficulty for predictive ecosystem modeling.
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Aquatic Ecosystem Modeling
My main points are:
Each ecosystem has to be considered carefully on its own and models
crafted to fit that particular system.
Every ‘ecosystem model’ will be limited to describing only certain aspects
of the ecosystem.
Understanding output of dynamic model output (as opposed to the simpler
description of ‘static’ model flows) requires deep understanding of the
behavior of nonlinear mathematics.
We can learn a great deal from ecosystem models (including some off the
shelf models), but we cannot make many predictions with a high degree of
confidence.
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Ecosystem Modeling
What I will do is: Discuss
1.
2.
3.
4.
5.
What we mean by the structure of ecosystems.
How the static fluxes of energy can be described.
How material fluxes can be added to this.
How basic processes are modeled
How complex dynamic phenomena arise from
nonlinearities in population (or functional group)
interactions
6. Uncertainties
7. Resources
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Energy
Flows
Ecosystem Structure
The overall generic
conceptual model of
an ecosystem
Nutrient
Flows
Atmosphere
Storage
Herbivore
s
Carnivore
s
Production
Bacteria
Available
Nutrient
Detritus
DO
M
Biota
Lithosphere
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How do we describe the structure of flows of
energy and matter in ecosystems?
This is done using methods of compartmental modeling,
describing the changes in compartment sizes in terms
of the inflows to and outflows.
Compartment sizes are the state variables of the
system.
This leaves us with the difficult question of deciding
what are the set of compartments.
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STRUKTUR EKOSISTEM
Deciding on what compartments to
include in a model
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This is perhaps the simplest conceptual model
that we can imagine
Heterotrophs
Energy
Autotroph
Detritus
Autotrophs
Nutrient Input
Heterotroph
Detritus
Available
Nutrient
Nutrient Flux
Energy Flux
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But usually we need more disaggregated trophic structure, to
include the food chain
Second-order
Carnivores
First-order
Carnivores
Herbivores
Autotrophs
Detritus
Available
Nutrient
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A further conceptualization includes both ‘herbivore’
and ‘detrital’ food chains
Second-order
Carnivores
Another proposed
representation would
be to divide the
ecosystem into
separate herbivore
and decomposer food
chains, in view of the
fact that these often
are distinct, though
they may share higher
trophic levels.
First-order
Carnivores
First-order
Carnivores
Herbivores
Microbivore
s
Decomposers
Macro Microbial
Autotrophs
Detritus
Available
Nutrient
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A more disaggregated
aquatic system, which
starts to have relevance to
a specific class of
ecosystem types
(Carpenter and Kitchell
1986).
Piscivore
Invertebrate
Planktivore
Vertebrate
Planktivore
Now we have a ‘food web’.
Rotifer
Nannoplankton
Small
Crustacean
Zooplankton
Edible Net
Phytoplankton
Large
Crustacean
Zooplankton
Inedible
Phytoplankton
PO4, NH4
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A model for a specific aquatic ecosystem (stream)
from Meyer and Poepperl (CJFAS 2005)
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Ecosystems Structure
Unfortunately, very few ecosystem models have been
developed for ecosystems with this high a degree of
resolution, and they each represent collective research
done over decades.
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Ecosystem Bioenergetics
Describing the static energy flows
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PM, Predation Mortality
How do we
determine the
fluxes of energy
through an
ecosystem?
R, Respiration
Population,
Functional Group, or
Trophic Level
We start with the
bioenergetics
model of a single
population (or
functional group).
(BA means
biomass
accumulation.)
M, Natural
Mortality
BIOMASS
BA
(Yield)
GP, Assimilation of ingested prey
NA, Egestion
(to decomposers)
Consumption of prey, PM
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How do we determine the static flows of energy
through the ecosystem?
Now we must also keep track of the gains and losses of each of these trophic levels.
We can write an equation for the rate of change in each trophic level;
d(Biomassn)/dt = Instantaneous gross production (or assimilation of biomass
consumed from lower trophic level)
- Losses (to respiration, natural mortality, predation)
= Gross Production - Respiration - Natural Mortality – Predation Mortality
=
GP
- R
-
NM
- PM
'Biomass' and 'energy' or ‘carbon’ are often used interchangeably, since biomass of
organisms (e.g., in grams) can be converted to energy (e.g., in calories) by a simple
conversion.
(Integration over time gives us biomass changes, BA, shown in preceding slide)
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How do we determine the static flows of energy
through the ecosystem?
Raymond Lindeman (1941) applied this approach to aquatic ecosystems; Cedar
Bog Lake and Lake Mendota.
Problem: It is really difficult to estimate most of the fluxes of the the individual
functional groups.
The genius of Raymond Lindeman was to make useful simplifications.
1.
First, he used a simple trophic level conceptualization of the aquatic
ecosystem; I.e., aggregation of autotrophs, herbivores, primary and
secondarycarnivores.
2.
Second, he ignored natural mortality and assumed that all mortality was due
to predation, also that the highest trophic level suffered no predation.
3.
Third, he assumed a steady state (no net biomass accumulation in any
trophic level through time) on a time scale from year to year.
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Lindeman’s
conceptualization
of compartments of
an ecosystem as
being a series of
trophic levels
Trophic
Level 3
R3
ASSIM3
NA2
PM2
Trophic
Level 2
R2
ASSIM2
NA1 , loss to decomposers
PM1
R1 , respiration
Trophic
Level 1
Photosynthesis (ASSIM1)
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Solving for energy flux through four trophic levels
We can write equations:
dBiomass1/dt =
Photosynthesis1 - R1 - PM1
dBiomass2/dt =
a1 (PM1 - NA1 ) - R2 - PM2
dBiomass3/dt =
a2 (PM2 - NA2) - R3 - PM3
dBiomass4/dt =
a3 (PM3 - NA3) - R4
ai = assimilation by level i
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Lindeman’s Solution
A ‘steady state’ occurs when the sizes of the various compartments do not change
through time (at least when considered on a gross enough time scale); that it, there
was no net accumulation; i.e., BA = 0. In that case one could assume that
dBiomass1/dt = dBiomass2/dt
= dBiomass3/dt =
dBiomass4/dt = 0
Thus, for the four trophic levels, he had a set of equations for energy balance
Trophic level 1 (Autotrophs)
Photosynthesis1 = R1 + PM1
Trophic level 2 (Herbivores)
PM1 - NA1 = R2 + PM2
Trophic level 3 (Carnivores)
PM2 - NA2 = R3 + PM3
Trophic level 4 (Second order carnivores)
PM3 - NA3 = R4
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Lindeman’s Solution
Lindeman was able to obtain estimates of the fraction of consumed energy
that is lost to egestion, NA, and the fraction of assimilated energy that is
respired, R. That enabled him to determine the unknowns, PMi ‘s and
Photosynthesis1
PM3 = NA3 + R4
PM2 = NA2 + R3
PM1 = NA1 +
+ PM3
R2 + PM2
Photosynthesis1 = R1 + PM1
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How do we extend this static energy flow modeling
to food webs?
More detailed food web studies allow one to model ecosystems at
a higher degree of resolution, applying the same type of energetic
balance equations to food webs.
One needs information on:
•Compartment sizes
•Process rates (consumption, assimilation, respiration, etc.)
•Diet composition
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How do we extend this static energy flow modeling
to food webs?
Example:
Meyer and Poepperl developed a diagram of their food web. This includes
'Aufwuchs' or periphyton as the primary producer and base for herbivore,
and also detritus as the base for the decomposer part of the web, although
the herbivore and decomposer parts mesh at higher trophic levels.
It is impossible to try to account for every species population in an
ecosystem, so these are grouped into 'guilds (often also called 'functional
groups'), or 'groups of species having similar ecological resource
requirements, foraging strategies, and predators'.
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A model for a specific aquatic ecosystem (stream)
from Meyer and Poepperl (CJFAS 2005)
SUMBER: www.ag.unr.edu/.../Aquatic_Ecosystem_Mo...University of Nevada, Reno
How do we extend this static energy flow modeling
to food webs?
Meyer and Poepperl were able to find empirical estimates of many of
the process rates through studies on a particular stream over many
years.
They also compiled data on 'who eats whom', or more precisely, how
much of the diet of a consumer is made up of various prey.
These data do not directly tell us all of the flows. Meyer and Poepperl
use the above input data, and then perform a mass-balance network
analysis to find the 'matrix of flows' and other outputs, analogous to
Lindeman’s approach, but much more sophisticated. This can also
involve linear optimization techniques that construct the full set of
flows that is completely balanced and matches the known flows as
best as possible (e.g., Christensen and Pauly 1992, their ECOPATH,
Diffendorfer, Richards, and DeAngelis, Ecological Modelling,1999?).
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Ecosystem Material Fluxes
Describing the static nutrient flows
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Now we must explicitly take into account a pool
or pools of nutrient, perhaps in different states
Heterotrophs
Energy
Autotroph
Detritus
Autotrophs
Nutrient Input
Heterotroph
Detritus
Available
Nutrient
Nutrient Flux
Adsorbed
Nutrients
Energy Flux
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How do we extend such static models t mass
(nutrients) flow?
Stoichiometry
Integrating nutrient cycling into models also requires inclusion of
concepts of stoichiometry. Organisms tend to require internal
concentrations of nutrients in particular ratios. For example, the
ratios of C to N to P tend to be around 106:16:1 in algae (Redfield
ratio).
But different species tend to have different ratios.
Consider predator-prey interaction in which we are taking into
account nutrients; nitrogen (N) and phosphorus (P), in which the
predator and prey have different N:P ratios.
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How do we extend such static models to mass
(nutrients) flow?
It is necessary to make sure that there is mass balance of both
nutrients through the ratio of nutrients released as waste
(Nx: Py)predator + (Na:Pb)prey
(Nx:Py)predator + (Nc:Pd)waste
Stoichiometry has implications for ecological phenomena because
growth rates and life history strategies of organisms are linked to
their N:P ratios (Sterner and Elser, Ecological Stoichiometry 2003).
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Ecosystem Processes
Describing the processes that drive flows
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What are the ecosystem processes that drive fluxes of
matter and energy?
Fluxes of energy and nutrients through ecosystems depend on processes
of energy and material conversion.
The flows of energy and nutrients in ecosystems are governed by
processes; primarily photosynthesis, respiration, consumption (herbivory
and carnivory), and decomposition. (But there are actually many more,
including spatial movement, that must often be taken into account.)
These are all complex and depend on densities of organisms, their
physiology and behaviors, and environmental factors..
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What are the ecosystem processes that drive fluxes of
matter and energy?
Primary production (phytoplankton, periphyton, aquatic macrophytes)
The amount of photosynthesis is a given area is generally proportional to the
biomass density, Biomass1. However, self-shading can occur, which requires a
non-linear dependence.
Photosynthesis will be limited by either available light or nutrients, such as
phosphorus or nitrogen, and the rate is proportional to a Michaelis-Menten
factor, μN/(k + N), where N is nutrient concentration.
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What are the ecosystem processes that drive fluxes of
matter and energy?
We can write an expression for photosynthesis as
Photosynthesis = min[(μN/(k + N), f1(APAR, Biomass)]
*f2(Temperature)*Biomass1
where min[ . , . ] means that whichever factor, nutrient or available light
(available photosynthetically active radiation, or APAR), is more limiting,
that is smaller, will control the photosynthetic rate. The function f1 for the
dependence of photosynthesis on APAR will depend on the physiology of
the autotrophs, which may saturate at high light levels. Temperature also
modulates the rate.
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What are the ecosystem processes that drive fluxes of
matter and energy?
Heterotrophy and the functional response:
The rate of consumption of prey biomass by the consumer is usually
modeled a linear function of consumer biomass (Biomassn) and a
saturating function of prey biomass (Biomassn-1);
Consumption = a Biomassn-1 Biomassn /(1 + h Biomassn-1)
where a and h are constant parameters. The factor in the above equation
multiplying Biomassn on the right hand side is called the functional
response. So the loss of biomass from a prey compartment will increase
linearly with consumer biomass, but not with prey biomass. High levels of
prey biomass (1 < h Biomassn-1 ), will saturate the consumer’s ability to
consume it.
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What are the ecosystem processes that drive fluxes of
matter and energy?
Decomposition and nutrient recycling processes:
Decomposition may be highly complex because different materials
decompose at different rates, depending on the type of biomass, and
whether it is in the water column or sediment.
The array of processes that affect nutrient recycling can be highly complex,
as in the case of nitrogen, which is first broken down through biomass
decomposition into ammonia, hydrolyzes to ammonium ions, and may
undergo nitrification to nitrate ions, and then denitrification to gaseous
compounds that can escape into the atmosphere. This can be quantified
in equations, but that is often very complex, depending on oxidationreduction conditions, etc.
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Question: What are the ecosystem processes that
drive fluxes of matter and energy?
Other processes:
Other processes that affect the above processes and may need to
be modeling in certain situations include evapotranspiration, water
movement and changes in depth, nutrient inputs and leaching,
nutrient adsorption, sedimentation, removal of certain nutrients by
complexing, organism migrations, and many more.
All processes can be modeled in various levels of detail.
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Effects of Nonlinearities
Emergence of complex phenomena in
ecosystem models
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What are the implications of nonlinearities for
dynamics, structure, and flows?
Nonlinearities occur in ecosystem models because of the density
dependencies in various processes.
The nonlinearities in the population interactions drastically affect
the way ecosystems function.
This includes oscillatory behavior and chaos, as well as trophic
cascades and mathematical catastrophes.
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Pure Bottom-Up Control in Typical Four-Level Food Chain
Carnivore 2
Trophic Cascades
Carnivore 2 size limited by
Carnivore 1 availability - thus
competition is strong
Carnivore 2 feeding
limited by Carnivore 1
availability
The earlier point of view
was that the direction of
effects in food chains was
solely from bottom to top.
Carnivore 1
Carnivore 1 size limited by
herbivores - thus competition is
strong
Carnivore 1 feeding
limited by herbivore
availability
However, the nonlinear
nature of interactions
between trophic levels
leads to a more complex
situation.
Herbivore
Herbivore size limited by
primary producers - thus
competition is strong
Herbivore feeding is
limited by primary
productivity
Primary Producer
Primary producer size
limited by resources - thus
competition is strong
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Mixed Top-Down and Bottom-Up Control in Typical
Four-Level Food Chain
Carnivore 2 size limited by
Carnivore 1 availability - thus
competition is strong
Carnivore 2
Because of the nonlinear
interactions, trophic
cascades propagate down
food chains starting from
the highest carnivores, the
piscivores. An increase in
piscivore biomass causes
a decrease in planktivore
biomass, increasing
herbivore biomass (as well
as allowing the herbivore
community to shift towards
larger zooplankters),
decreasing phytoplankton
biomass.
Carnivore 2 feeding
limited by Carnivore 1
availability
Strong negative effect by
Carnivore 2 on Carnivore 1
Carnivore 1
Carnivore 1 feeding not
limited by herbivore
availability, as herbivores
are abundant.
Carnivore 1 size limited by
Carnivore 2 - thus competition
is weak
Weak negative effect of
carnivores on herbivores
Herbivore
Herbivore feeding is
limited by primary
productivity
Herbivore size not limited by
carnivores - thus competition is
strong
Strong negative effect of
herbivory on autotrophs
Primary Producer
Primary producer size
determined by predation thus competition is weak
Figure 6
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These trophic
cascades occur in
nature. The ability
of predatory fish to
control prey
populations is welldocumented. This
can cause
suppression of the
forage species,
which affects
species composition
and size structure of
the zooplankton
community, and in
turn influence the
phytoplankton
community
(Carpenter and
others).
Piscivore
Invertebrate
Planktivore
Rotifer
Nannoplankton
Vertebrate
Planktivore
Small
Crustacean
Zooplankton
Edible Net
Phytoplankton
Large
Crustacean
Zooplankton
Inedible
Phytoplankton
PO4, NH4
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Mathematical catastrophes in ecosystems
Top-down effects can also lead to what are known as mathematical
"catastrophes" in ecological systems. Such “catastrophes” are sudden
changes that can occur in an ecosystem as the result of slow, gradual
changes in an environmental parameter. These catastrophes involve shifts
in which there is a change from one state of an ecosystem to another,
usually with a change in dominance of the species community. These are
of interest mathematically as well as ecologically, because they involve
certain types of nonlinearities. (see especially, papers and book by Marten
Scheffer.)
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Consider a shallow lake that is relatively clear and has a lot of aquatic macrophytes
on the bottom. Suppose there is a slow buildup of nutrients in the lake. For many
years there is no discernible change in the lake’s biotic community. But suddenly,
one summer, there is a big phytoplankton bloom. The bloom is there next year too,
and soon the aquatic macrophytes die off.
To make matters worse, once the shallow lake has "tipped" from a clear,
"macrophyte-dominated" lake to a turbid "algal-dominated" lake, it is not easy to
change it back. Even if you are able to drastically reduce the nutrient input, the lake
may not change back to its former clear self.
The above scenario has occurred very often in lakes. Theoretical ecologists have
examined this phenomenon in terms of "mathematical catastrophe" theory, which is
describes how a system may undergo major changes due to small changes in some
environmental parameter.
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The reason that such a dramatic change in the ecosystem can occur is that there
are "self-reinforcing" processes occurring within a lake system (or any other
ecosystem) that tend to maintain it in a stable state. But if you push the system too
far, those self-reinforcing processes break down and turn against the original
system.
Aquatic
Macrophytes
Phytoplankton
-
+
+
-
Zooplankton
Nutrients
Planktivorous
Fish
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Uncertainties
Error propagation in complex ecosystem
models
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Yodzis’s Result
Peter Yodzis (Ecology 1988) studied the propagation of uncertainty
in large nonlinear food web models (> 12 or so species or
functional groups).
He found that even if parameter values are known to within 15% or
so, the propagation of uncertainty is such that, for a particular
choice of parameters within the range of uncertainty, a perturbation
in one functional group is equally likely to affect another given
functional group positively or negatively.
.
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Incompleteness of Ecosystem Models
Modeling means choosing what to include in a model system.
However, it is impossible to know a priori which components and
processes in an ecosystem are likely to be important.
Hence, ecosystem models always are missing components and
processes that may at some time be important in the modeled real
system.
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Lack of Analytic Understanding
Mathematical ecology has not supplied an understanding of the
behavior of complex ecological systems; that is, nonlinear systems
with more than two or three species.
Thus, analytic solutions cannot help us evaluate the output of
model simulations.
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Resources
Packaged modeling platforms such as AQUATOX and
EcoSim exist.
These should by all means be exercised as a part of
experiencing the complexity of ecosystem behavior.
However, one should be aware of assumptions in these
models, including ad hoc assumptions to maintain
stability.
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My view is that one should not trust the output of such
packaged models unless one has a knowledge of the
assumptions in the models and a deep understanding of
nonlinear differential equation models.
For example, one should be able to understand the
output of a model such as the following.
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Heterotrophs
Energy
Autotroph
Detritus
Autotrophs
Nutrient Input
Heterotroph
Detritus
Available
Nutrient
Nutrient Flux
Energy Flux
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The equations take the form
dN

 k B  k B  F ( N ) B
dt

dB
 F ( N )B  F (B , B )B  d
dt
dB
 F ( B , B ) B  d B
dt
dB
 d B k B
dt
dB
 d B k B
dt
dec , p
det, p
dec ,h
det,h
plant
plant
plant
plant
plant
heter
plant
heter
heter
plant
B
plant
heter
heter
plant
heter
heter
heter
heter
det, p
plant
plant
dec , p
heter
plant
dec ,h
det, p
det,h
det,h
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Let’s assume that nutrient is completely recycled and there are no inputs
or outputs of nutrient. The equations then can be written:
dN

 k dec, p Bdet,p  k dec,h Bdet,h  Fplant ( N ) B plant
dt

dBplant
 Fplant ( N ) B plant  Fheter ( B plant , Bheter ) Bheter  d plant B plant
dt
dBheter
 Fheter ( B plant , Bheter ) heter  d heter Bheter
dt
dBdet,p
 d plant B plant  k dec, p Bdet,p
dt




N total  N  B plant  Bheter  Bdet,p  Bdet,h
SUMBER: www.ag.unr.edu/.../Aquatic_Ecosystem_Mo...University of Nevada, Reno
and let’s also assume
k N
F (N ) 
k N
u
plant
half
kB
F (B , B ) 
1 k k B
a
heter
plant
plant
heter
a
h
plant
SUMBER: www.ag.unr.edu/.../Aquatic_Ecosystem_Mo...University of Nevada, Reno
Evalutation of Model
The model can be evaluated for sets of specific parameters;
e.g.
ku = 0.05
khalf = 0.5
ka = 0.02
kh = 1.0
dplant = 0.02
dheter = 0.005
kdec,p = 0.01
kdec,h = 0.1
 = 0.05
 = 0.02
If the total nutrient in the system, Ntotal, is allowed to vary,
then the nutrient, plant biomass, and heterotroph biomass
compartments behave as follows:
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Equilibrium Values of Model Variables as Function of Total
Nutrient
COMPARTMENT SIZES AT
EQUILIBRIUM
3
2.5
N
2
0.1*Bplant
1.5
1
Bheter
0.5
0
0
2
4
6
8
10
TOTAL NUTRIENT
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Analysis of Model
It is also possible to examine the effects of other
parameters. For example, suppose the total nutrient in the
system, Ntotal, is held constant.
Let ku vary, representing changing rate of photosynthesis
due to changes in solar radiation. Then the changes in the
main variables are as in the following figure.
SUMBER: www.ag.unr.edu/.../Aquatic_Ecosystem_Mo...University of Nevada, Reno
Equilibrium Values as Function of Solar Radiation
Total Nutrient in System Fixed.
7
COMPARTMENT SIZES
6
N
5
4
3
0.1*Bplant
2
Bheter
1
0
0
0.05
0.1
0.15
0.2
0.25
0.3
SOLAR RADIATION
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Analysis of Model
One can also study the temporal
dynamics of the model for any given
value of Ntotal by solving the set of
differential equations plus constraint
on nutrients.
Here, all other parameter values are
fixed.
SUMBER: www.ag.unr.edu/.../Aquatic_Ecosystem_Mo...University of Nevada, Reno
Ntotal = 2.0
4.5
COMPARTMENT SIZES
4
3.5
0.1*Bplant
3
2.5
2
1.5
1
N
0.5
Bheter
0
1
12
23
34
45
56
67
78
89
100
111
122
133
144
155
166
177
188
TIME
SUMBER: www.ag.unr.edu/.../Aquatic_Ecosystem_Mo...University of Nevada, Reno
Ntotal = 6.0
COMPARTMENT SIZES
6
5
0.1*Bplant
4
3
2
N
1
Bheter
0
1
12
23
34
45
56
67
78
89
100
111
122
133
144
155
166
177
TIME
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Ntotal = 9.0
35
COMPARTMENT SIZES
30
Bplant
25
20
15
10
N
5
Bheter
0
1
14
27
40
53
66
79
92
105
118
131
144
157
170
183
196
209
222
235
248
TIME
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The simple model here is about at the limits of what can
easily be analyzed mathematically.
The model ignores the complexities of multiple
limiting nutrients, of oxygen availability, carbon
dioxide levels, etc., as it was developed for idealized
ecosystems.
One should have an understanding of the behavior of
such simple models before moving on to the more
elaborate packaged aquatic ecosystem models.
SUMBER: www.ag.unr.edu/.../Aquatic_Ecosystem_Mo...University of Nevada, Reno