simulasi ekosistem akuatik aquatic ecosystem simulation modeling

SIMULASI EKOSISTEM AKUATIK

Aquatic Ecosystem Simulation Modeling

Don DeAngelisU.S. Geological Survey, Florida Integrated Science

CentersMiami, Florida

Interdisciplinary Modeling for Aquatic EcosystemsLake 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).


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.


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.


Ecosystem Modeling

What I will do is: Discuss

1. What we mean by the structure of ecosystems.2. How the static fluxes of energy can be described.3. How material fluxes can be added to this.4. How basic processes are modeled5. How complex dynamic phenomena arise from

nonlinearities in population (or functional group) interactions

6. Uncertainties7. Resources


Storage

Production

Herbivores

Carnivores

Detritus

Biota

AvailableNutrient

Atmosphere

DOM

Lithosphere

Bacteria

EnergyFlows

NutrientFlows

Ecosystem Structure

The overall generic conceptual model of an ecosystem


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.


STRUKTUR EKOSISTEM

Deciding on what compartments to include in a model


Heterotrophs

Autotrophs Heterotroph Detritus

Available Nutrient

Autotroph Detritus

Energy

Nutrient Input

Nutrient Flux

Energy Flux

This is perhaps the simplest conceptual model that we can imagine


But usually we need more disaggregated trophic structure, to include the food chain

Second-order Carnivores

First-order Carnivores

Herbivores

Autotrophs Detritus

Available Nutrient


A further conceptualization includes both ‘herbivore’ and ‘detrital’ food chains

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.

Second-order Carnivores


Herbivores

Autotrophs Detritus

Available Nutrient

Decomposers Macro Microbial

Microbivores



Piscivore

InvertebratePlanktivore

VertebratePlanktivore

SmallCrustaceanZooplankton

LargeCrustaceanZooplankton

Rotifer

Nannoplankton Edible NetPhytoplankton

InediblePhytoplankton

PO4, NH4

A more disaggregated aquatic system, which starts to have relevance to a specific class of ecosystem types (Carpenter and Kitchell 1986).

Now we have a ‘food web’.


A model for a specific aquatic ecosystem (stream) from Meyer and Poepperl (CJFAS 2005)


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.


Ecosystem Bioenergetics

Describing the static energy flows


How do we determine the fluxes of energy through an ecosystem?

We start with the bioenergetics model of a single population (or functional group).

(BA means biomass accumulation.)

BIOMASS BA (Yield)

R, Respiration

M, Natural Mortality

Consumption of prey, PM

NA, Egestion (to decomposers)

PM, Predation Mortality

GP, Assimilation of ingested prey

Population, Functional Group, or Trophic Level


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)


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.


Trophic Level 3

R3

ASSIM3

Trophic Level 2

R2

ASSIM2

Trophic Level 1

R1 , respiration

Photosynthesis (ASSIM1)

PM2

PM1

NA2

NA1 , loss to decomposers

Lindeman’s conceptualization of compartments of an ecosystem as being a series of trophic levels


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


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


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 + PM3

PM1 = NA1 + R2 + PM2

Photosynthesis1 = R1 + PM1


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



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'.


A model for a specific aquatic ecosystem (stream) from Meyer and Poepperl (CJFAS 2005)



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?).


Ecosystem Material Fluxes

Describing the static nutrient flows


Now we must explicitly take into account a pool or pools of nutrient, perhaps in different states

H e te ro tro p h s

A u to tro p h s H e te ro tro p h D e tr itu s

A v a ila b le N u tr ie n t

A u to tro p h D e tr i tu s

E n e rg y

N u tr ie n t In p u t

N u tr ie n t F lu x

E n e rg y F lu x

A d s o rb e d N u tr ie n ts


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.


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).


Ecosystem Processes

Describing the processes that drive flows


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..



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.



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.



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.



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 oxidation-reduction conditions, etc.


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.


Effects of Nonlinearities

Emergence of complex phenomena in ecosystem models


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.


Pure Bottom-Up Control in Typical Four-Level Food Chain

Carnivore 1

Herbivore

Primary Producer

Carnivore 2 size limited by Carnivore 1 availability - thus competition is strong

Carnivore 1 feeding limited by herbivore availability

Herbivore feeding is limited by primary productivity

Herbivore size limited by primary producers - thus competition is strong

Primary producer size limited by resources - thus competition is strong

Carnivore 2

Carnivore 2 feeding limited by Carnivore 1 availability

Carnivore 1 size limited by herbivores - thus competition is strong

Trophic Cascades

The earlier point of view was that the direction of effects in food chains was solely from bottom to top.

However, the nonlinear nature of interactions between trophic levels leads to a more complex situation.


Mixed Top-Down and Bottom-Up Control in Typical

Four-Level Food Chain

Carnivore 1

Herbivore

Primary Producer

Carnivore 2 size limited by Carnivore 1 availability - thus competition is strong

Weak negative effect of carnivores on herbivores

Carnivore 1 feeding not limited by herbivore availability, as herbivores are abundant.

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 size determined by predation - thus competition is weak

Carnivore 2

Carnivore 2 feeding limited by Carnivore 1 availability

Strong negative effect by Carnivore 2 on Carnivore 1

Carnivore 1 size limited by Carnivore 2 - thus competition is weak

Figure 6

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.


Piscivore

Invertebrate Planktivore

Vertebrate Planktivore

Small Crustacean Zooplankton

Large Crustacean Zooplankton

Rotifer

Nannoplankton Edible Net Phytoplankton

Inedible Phytoplankton

PO4, NH4

These trophic cascades occur in nature. The ability of predatory fish to control prey populations is well-documented. 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).


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.)


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.


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

-


Uncertainties

Error propagation in complex ecosystem models


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.

.


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.


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.


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.


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.


Heterotrophs

Autotrophs Heterotroph Detritus

Available Nutrient

Autotroph Detritus

Energy

Nutrient Input

Nutrient Flux

Energy Flux


The equations take the form

hhdecplantheter

h

ppdecplantplant

p

heterheterheterheterplantheterheter

plantplantheterheterplantheterplantplant

plant

plantplanthhdecppdec

BkBddtdB

BkBddtdB

BdBBBFdtdB

BdBBBFBNFdtdB

BNFBkBkdtdN

det,,

det,

det,,

det,

det,,det,,

),(

),()(

)(


Let’s assume that nutrient is completely recycled and there are no inputs or outputs of nutrient. The equations then can be written:

hpheterplanttotal

ppdecplantplant

p

heterheterheterheterplantheterheter

plantplantheterheterplantheterplantplant

plant

plantplanthhdecppdec

BBBBNN

BkBddtdB

BdBBFdtdB

BdBBBFBNFdtdB

BNFBkBkdtdN

det,det,

det,,

det,

det,,det,,

),(

),()(

)(


and let’s also assume

plantha

planta

heterplantheter

half

uplant

BkkBk

BBF

NkNkNF

1),(

)(


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:


Equilibrium Values of Model Variables as Function of Total Nutrient

0

0.5

1

1.5

2

2.5

3

0 2 4 6 8 10

TOTAL NUTRIENT

CO

MPA

RTM

ENT

SIZE

S AT

EQ

UIL

IBR

IUM

0.1*Bplant

Bheter

N


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.


Equilibrium Values as Function of Solar RadiationTotal Nutrient in System Fixed.

0

1

2

3

4

5

6

7

0 0.05 0.1 0.15 0.2 0.25 0.3

CO

MPA

RTM

ENT

SIZE

S

SOLAR RADIATION

N

0.1*Bplant

Bheter


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.


Ntotal = 2.0

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

1 12 23 34 45 56 67 78 89 100 111 122 133 144 155 166 177 188

CO

MPA

RTM

ENT

SIZE

S

TIME

0.1*Bplant

N

Bheter


Ntotal = 6.0

0

1

2

3

4

5

6

1 12 23 34 45 56 67 78 89 100 111 122 133 144 155 166 177

CO

MPA

RTM

ENT

SIZE

S

TIME

0.1*Bplant

N

Bheter


Ntotal = 9.0

0

5

10

15

20

25

30

35

1 14 27 40 53 66 79 92 105 118 131 144 157 170 183 196 209 222 235 248

CO

MPA

RTM

ENT

SIZE

S

TIME

Bplant

N Bheter


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.


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