Marshall Mayhew & James Franzo
ISAT 420- Dr. Frysinger
September 19, 2015
Exercise 3- Problem 5.6
When on is observing an ecosystem of any sort, it is nearly impossible to be able to recognized
every interaction and relationships that formulate the essence and quality of the ecosystem. With a
substantial background in the natural science and keen senses, one might be able to depict the majority of
them, but the average observer will fall short of having a full understanding of why the system behaves
how it does. One of the more common interactions that one may overlook in their observations are the
various matter cycles that are integrated all throughout every system. It is understandable for why one may
not recognize these interactions because the human eye cannot see the flow of nutrients in its different
states; however, these matter cycles and their efficiency to rotate key nutrients around the ecosystem are
essential to the health and prosperity of ecosystems.
Then, how does one recognize and observe these matter cycles. Each matter cycle has their own
mechanisms and ways they fit in each ecosystem, which depends on what type of matter cycle you are
interested in and the infrastructure of the ecosystem. Once, one is well knowledge on those two aspects of
the scope that they are observing upon, they then can formulate a quality perspective to observe their
presence and importance through various indicators in all forms of matter. As mentioned before, there are
several different types of matter cycle, such as nitrogen and carbon, which flow dynamically throughout
ecosystems in various states of matter and provide the necessities for living organisms. Among the diverse
range of matter cycles, we are going to focus on the phosphorus cycle to understand the importance of
matter cycles in ecosystems and the dynamic role they play in ecosystems through a systems modeling
approach.
The phosphorus cycle is considerably unique among the other matter cycles for various different
reasons, especially in aquatic ecosystems, which is the ecosystem that we will focus on and use to model
and analyze. More specifically, the aquatic ecosystem is a lake with an inflow from an industrial facility
that is located on the lake; in addition, there are no outflows from the lake. Common to all matter cycles,
phosphorus has various forms it can be found in such as an organic living state, a dead organic state, and
an inorganic state. These three states will ultimately be subjected to our three main stocks for our model.
Unlike other nutrients, phosphorus does not exist in a gaseous state within its matter cycle, which why it is
commonly found cycling in aquatic ecosystems and aquatic producers significantly using it for energy;
thus, beginning the producer-consumer-decomposer chain. Unfortunately, this industrial plant and its
inflow of waste that contains inorganic phosphorus forces the phosphorus matter cycle for this lake
ecosystem out of its conserved, natural balance.
In order for us to approach this analysis, we need to define the system in which we are modeling.
Through previous modeling exercises of aquatic systems, we understand how dynamic and delicate the
steady-state balance is; we also understand the vast conglomerate of relationships that could be included in
our analysis that would give a holistic outlook on the effects that each recognized role would dynamically
play throughout the balance. However, we have chosen to set the scope of our model specifically on
phosphorus, the reservoirs that represents the various states in which phosphorus is found in, and how the
nutrient cycles throughout the lake’s ecosystem. We are not including any other stocks for our model
because our goal is to obtain indications on how phosphorus behaves in freshwater aquatics. Though, there
are stocks outside of this nutrient that have influential relationships to different phosphorus stocks, we
believe that those influences will mask the true behaviors of phosphorus’s matter cycle. For the boundaries
that our system is found within, includes a 10,000,000 L lake. In our analysis, we will include organic and
inorganic inhabitants and their relationship to this nutrient cycle but for this behavioral study of the
phosphorus cycle, there will not be anything else in the lake but the freshwater and the nutrient phosphorus.
The only flux that is included from outside of the system boundaries of the lake is the inflow of phosphorus
into the water from the industrial facility. There are not any other fluxes because the only outflow that
phosphorus could have from the lake is a water outflow or the harvesting of aquatic consumers and
producers by humans or other predators because phosphorus cannot leave the water in a gaseous state. The
time frame that we will use to analyze our model is over a twenty year period. There are some assumptions
that we need to make in order for our model to maintain its scope and not gain too much complexity, which
would mask the goal of the systems approach. As mentioned above, the model will just focus phosphorus
stocks in the lake and not any other organic/inorganic stocks or atmospheric flows and stocks. Also, without
the presence of producer or consumer stocks, the rate constants that subsequently relate the stocks for each
flow are based off of ratios for a conserved, steady-state phosphorus cycle that is presented in the text and
not based off of the lifecycles of the conglomerated stocks. Another assumption is that our model does not
have factors that incorporate seasonal effects, which are a strong influence on the phosphorus cycle, because
we have a twenty year time frame; thus, assuming that our spatial resolution locates our lake in a tropical
climate. In addition, we must assume that the different rate constants respectively incorporate driving
processes, such as solar energy and gravity, in their ratios within the flow relationships between reservoirs,
even though they are not recognized within the model.
Represented in Figure 1 below is a diagram of our system. From this system we were able to narrow
down the stocks and flows that met the representation characteristics of the goal of our analysis and,
ultimately, defining the scope and flux(es) for our model. As Figure x represents, the phosphorus cycle is
integrated within a very dynamic system, which takes a lot of physical and chemical coherence to maintain
a steady state.
Figure 1- Shows our completed Systems Diagram with the Stocks as boxes and the flows as arrows with
red text describing the nature of the flow. Not all of these stocks were included in the running model, but
it is still important to note the relationships between all of them.
Once we strip the system down to our model’s scope, we found our model, Figure 2, consisting of
three different stocks and four flows that related those stocks to one another. In, addition we created a
model, Figure 3, that represents the phosphorous cycle in the lake without the inorganic matter flow from
the industrial facility, creating a representation of a conserved phosphorus matter cycle.
Relationships in matter cycles are very important because they help define the matter cycle system
and how it is going to be have; therefore, when one is evaluating a relationship between two states (stocks),
they must be very attentive to the role that the relationship plays for the system, the nature of the flow
between states (indicators), and the quantitative exchange of nutrients. There are three different types of
flow processes that represents relationships throughout the Producer-Consumer-Decomposer Chain; those
flow types are called a Donor-Controlled Flow, a Receptor-Controlled Flow, and a combination of both
called a Donor- and Receptor-Controlled Flow. Each of these flows have their own characteristics and own
way to be mathematically represented.
In Figure 2, we have the model that represents the conserved phosphorus cycle in the lake (aquatic)
ecosystem. The three stocks that are represented in the model are “Living Organic Phosphorus,” “Dead
Organic Phosphorus,” and “Inorganic Phosphorus”. To represent the natural cycle of phosphorus and its
quantity exchange from state to state, we placed a flow from “Living Organic Phosphorus” to “Dead
Organic Phosphorus”; we placed a flow from “Dead Organic Phosphorus” to “Inorganic Phosphorus”; and
we placed a flow from “Inorganic Phosphorus” to “Living Organic Phosphorus.” After connecting all the
reservoirs in our model with the flows needed to properly delineate the natural cyclical movement of the
phosphorus cycle, we defined each of them.
Figure 2- Shows the Original Stella Model given to us in the textbook before we modified it. This is a
steady state system of the Phosphorus cycle. The Graph on the right shows the 3 stocks and it is clear
that they are not changing hence “Steady State”.
The flow that connects “Living Organic Phosphorus” to “Dead Organic Phosphorus,” is a donorcontrolled flow because the “Living Organic Phosphorus” stock is pushing out units to the “Dead Organic
Phosphorus” stocks through the process called “DEATH”. “DEATH” controls the quantity of phosphorus
that is transfers through the rate constant and the current state of the “Living Organic Phosphorus” stock.
The rate constant we used was 0.25 years-1, which is the same rate constant that they presented in the book.
The “DEATH” Flow can be mathematically represented as, 𝐷𝐸𝐴𝑇𝐻 = 𝑎 ∗ Living_Organic_State. In
addition, the flow that connects “Dead Organic Phosphorus” to “Inorganic Phosphorus” is a donorcontrolled stock too because, though, the bacteria is the active agent for the process, the amount of
phosphorus that will flow to the into “Inorganic Phosphorus” stock is controlled by the immensity of the
“Dead Organic Phosphorus” stock and the decomposition rate constant, b, of the bacteria. The rate constant
we used to represent the bacteria’s decomposition rate is 0.05 years-1, which is the same rate used by Deaton
just as the other three rate constants are. This donor-controlled relationship is exhibited as DECOMP in
Figure 2; it can be mathematically represented as 𝐷𝐸𝐶𝑂𝑀𝑃 = 𝑏 ∗ 𝐷𝑒𝑎𝑑_𝑂𝑟𝑔𝑎𝑛𝑖𝑐 _𝑃ℎ𝑜𝑠𝑝ℎ𝑜𝑟𝑢𝑠.
The last flow between “Inorganic Phosphorus” to “Living Organic Phosphorus” is a unique flow
due to the dependency on information from both stocks that will ultimately determine the quantity of
phosphorus that will be transferred to the “Living Organic Phosphorus” stock; the information that the flow
is dependent upon includes the amount of available nutrients in the “Inorganic Phosphorus” reservoir and
the amount of organisms in the “Living Organic Phosphorus” reservoir that can reap the inorganic matter.
Thus, making it a donor- & receptor-controlled flow noted as “UPTAKE”. In this flow, “UPTAKE” is
defined with one rate constant that helps determine the amount of inorganic matter that is converted to
organic matter through producers and consumers. We set the rate constant in the “UPTAKE” flow to 2.5
years-1*mole-1; the mathematical representation of the flow is represented as 𝑈𝑃𝑇𝐴𝐾𝐸 = 𝑐 ∗
𝐼𝑛𝑜𝑟𝑔𝑎𝑛𝑖𝑐_𝑃ℎ𝑜𝑠𝑝ℎ𝑜𝑟𝑢𝑠 ∗ 𝐿𝑖𝑣𝑖𝑛𝑔_𝑂𝑟𝑔𝑎𝑛𝑖𝑐_𝑃ℎ𝑜𝑠𝑝ℎ𝑜𝑟𝑢𝑠.
The initial stock levels that we started with are also the same ones that Deaton used in their
conserved phosphorus model representations. The initial stock level for “Living Organic Phosphorus” was
0.2 moles of Phosphorus; the initial stock level for “Dead Organic Phosphorus” was 1.0 moles of
Phosphorus; and the initial stock level for “Inorganic Phosphorus” was 1.0 moles of Phosphorus. We
decided to use moles as the unit for our stock because of how vast the lake is and the mole unit gave us a
proficient perspective on the concentration of phosphorus in the lake, which is important for analyzing the
data with the potential effects it could have on components throughout the lakes ecosystem, outside of the
scope.
The differential equations that we used for the each of the stocks and their inflow & outflows
include:



𝐿𝑖𝑣𝑖𝑛𝑔_𝑂𝑟𝑔𝑎𝑛𝑖𝑐_𝑃ℎ𝑜𝑠𝑝ℎ𝑜𝑟𝑢𝑠(𝑡)=Living_Organic_Phosphorus(t-dt)+(UPTAKE-DEATH)*dt
o Inflow UPTAKE Flow
o Outflow DEATH Flow
𝐷𝑒𝑎𝑑_𝑂𝑟𝑔𝑎𝑛𝑖𝑐_𝑃ℎ𝑜𝑠𝑝ℎ𝑜𝑟𝑢𝑠(𝑡)=Dead_Organic_Phosphorus(t-dt)+(DEATH-DECOMP)*dt
o Inflow DEATH Flow
o Outflow DECOMP Flow
𝐼𝑛𝑜𝑟𝑔𝑎𝑛𝑖𝑐_𝑃ℎ𝑜𝑠𝑝ℎ𝑜𝑟𝑢𝑠(𝑡)=Inorganic_Phosphorus(t-dt)+(DECOMP-UPTAKE)*dt
o Inflow DECOMP Flow
o Outflow UPTAKE Flow
Once our model of a conserved phosphorus cycle was set up in STELLA®, we ran the model for a
twenty year period and the model proved to represent a steady-state system, where each phosphorus stocks
held constant concentration levels and there were no increase or decreases overtime. With no concentration
increases or decreases over time, the derivative of each stock is zero. Therefore, proving the steady-state
relationship,
which
𝑑𝐷𝑒𝑎𝑑_𝑂𝑟𝑔𝑎𝑛𝑖𝑐_𝑃ℎ𝑜𝑠𝑝ℎ𝑜𝑟𝑢𝑠
𝑑𝑡
is
=
mathematically
𝑑𝐼𝑛𝑜𝑟𝑔𝑎𝑛𝑖𝑐_𝑃ℎ𝑜𝑠𝑝ℎ𝑜𝑟𝑢𝑠
𝑑𝑡
represented
as
𝑑𝐿𝑖𝑣𝑖𝑛𝑔_𝑂𝑟𝑔𝑎𝑛𝑖𝑐_𝑃ℎ𝑜𝑠𝑝ℎ𝑜𝑟𝑢𝑠
𝑑𝑡
=
=0.
Now, the model we just presented does not include the industrial facility that is located on the edge
of the lake and is included in the scope of our system’s definition for which we are providing an analysis
upon through exploring what effects it would have on the ecosystem and its phosphorus cycle.
When incorporating the industrial facility into the lake’s aquatic ecosystem, there are many effects
that the effluent byproducts from the facility have on the dynamic system present. However, the facility’s
discharge has one predominate contribution to the phosphorus cycle, within the discharge phosphorus
matter is found in only an inorganic state. As the inorganic matter flow enters the lake’s ecosystem, the
original conserved phosphorus matter that we observed earlier is corrupted and, consequently, becomes a
non-conserved system with the inflow of matter from outside of the natural cycle and making the total
amount of phosphorus in the cycle no longer constant. In addition, harboring this influx of phosphorus from
the effluent issues a loss of the natural balance of the system, which we found in the conserved phosphorus
cycle (Figure 2). Due to dynamic infrastructure of the system we are studying and through the specific
scope in which we begin our observations with, one knows the ripple effect that this unbalance can cause
throughout the whole ecosystem, especially if the system cannot activate a resilient process to reobtain a
steady-state.
In order to compensate for this influx and understand the new behaviors that the system will
produce, we must further develop the model to represent its new, non-conserved state. We mentioned above
that addition of matter from the industrial facility cause a ripple effect throughout the whole system;
however, in order to gain a further understanding constituent of matter cycles in ecosystems, we decided to
continue to use the same scope that we used for the conserved system with the addition of phosphorus
matter inflow because we felt that the simplicity would expose clearer indicators within the phosphorus
reservoirs and their flows.
The compensations required to represent all the active contributors within our system was simple.
Since we determined that we would all aspects from the conserved phosphorus cycle, we only need to
correctly place the inflow of phosphorus from the industrial facility; define its flow type and mathematical
representation; and determine the parameters and rate constants. We knew that the only state of phosphorus
that was entering the matter cycle was in the inorganic state; therefore, we were respectively represent the
inorganic phosphorus of the industrial facility flowing into the natural “Inorganic Phosphorus in Lake.”
After placing the flow, we contemplated on whether or not there should be a stock for the inorganic
phosphorus from the industrial facility; ultimately, we decided not to include one because it is only the flow
that is penetrating the systems boundaries and flow of inorganic phosphorus was not going to be dependent
upon how much matter the industrial facility had but from the natural forces that drive matter cycles such
as gravity, runoff, and weather. Thus, making the amount of matter in the “Inorganic Phosphorus in Lake”
and the rate constant of matter cycle’s natural driving forces the determining factors for the influx of
inorganic phosphorus quantity. With those two determining factors, we defined the nature of the influx to
be a receptor-controlled flow, as the matter was being pulled into the cycle. The development from the
conserved phosphorus cycle to the representation of a non-conserved phosphorus cycle is represented in
Figure 3, also in Figure 4 and 5 too, where the addition of inorganic phosphorus from the industrial facility
is represented as “INFLOW” and the cycle’s natural driving forces as the rate constant “r.” The flow can
be mathematically described as 𝐼𝑁𝐹𝐿𝑂𝑊 = 𝑟 ∗ 𝐼𝑛𝑜𝑟𝑔𝑎𝑛𝑖𝑐_𝑃ℎ𝑜𝑠𝑝ℎ𝑜𝑟𝑢𝑠_𝑖𝑛_𝐿𝑎𝑘𝑒.
Figure 3 – Shows the model after it has been modified to include an inflow of inorganic phosphorus from
an industrial source. This make the system non-steady state. The graph on the right shows a steady
increase in dead and living organic phosphorus while the inorganic phosphorus rapidly increases before
rapidly declining and leveling off.
As stated above we knew that the addition of inorganic phosphorus from the industrial plant to the
matter cycle would offset the natural balance found in the conserved model; however, we did not know in
what ways the balance would be offset and how the different areas of the model would behave. The ultimate
goal for representing the phosphorus matter cycle system through a model and analyzing the model and
observing its behaviors with specified parameters was to find a steady-state solution for the model. In order
to do so, we ran simulations on our original non-conserved phosphorus cycle model and made a series of
calibrations.
In Figure 3, we have our original non-conserved phosphorus cycle model, all the parameters from
the conserved phosphorus cycle model, Figure 2, were kept the same but we still had to set the rate constant
for the “INFLOW” flow. After reevaluating the effects of the original rate constant parameters in Figure 2
on their flow constraints, we determined the initial “INFLOW”’s rate constant to be 0.25 years-1. When we
simulated the model with the same 20 year period time frame and the results can be found in the graph in
Figure 3, which represents the stock levels in our model. We observed initially that the “Inorganic
Phosphorus in Lake” stock sky rockets with addition of immediate inflow of inorganic phosphorus from
the industrial facility to the decomposition inflow the stock normally had experienced. After about two and
a half years, the “Inorganic Phosphorus in Lake” stock imitates a negative parabolic trend with a sudden
decrease in matter within its reservoir because the donor- & receptor-controlled flow to the “Living Organic
Phosphorus” stock catches up to the “Inorganic Phosphorus in Lake” stock increase. This reaction indicated
a lagged inverse relationship between the two stocks and due to the extreme increase of the “Inorganic
Phosphorus in Lake” stock, the “Living Organic Phosphorus” stock had an extreme gain to its reservoir
from overturned of inorganic matter. Over the 20 year time period, the “Dead Organic Phosphorus” stock
continuously increased its stock’s matter quantity because of the death rate constant in the “DEATH” flow,
but the “Dead Organic Phosphorus” stock could not maintain any balance due to the rate constant of
“DEATH” being a lot greater that the decomposition rate constant in the “DECOMP” flow, which provides
an exponential inflow with the “Living Organic Phosphorus” stock increase and a low-level linear outflow
with “DECOMP” to “Inorganic Phosphorus in Lake” stock. With those conditions, the model is far from
steady state. We concluded that the “UPTAKE” rate constant needed to be lowered slightly to moderate the
“Living Organic Phosphorus” stock increase; the “DECOMP” rate constant needed to be increased to match
the inflow fixed from the “DEATH” relationship; and to delay the influx of inorganic phosphorus from
“INFLOW” for a small fraction at the beginning of the time period.
Figure 4 represents the changes we made from our original non-conserved model in order to obtain
a steady-state solution. In the model, the parameters we changed were the rate constant for “DECOMP”,
the “UPTAKE” rate constant, and we altered the rate constant for “INFLOW” in terms of time and ratio.
For the rate constant in the “DECOMP” flow, we raised the ratio to 0.1 years-1 to help increase outflow
from the “Dead Organic Phosphorus” reservoir and compensate the heavy inflow with “DEATH” from the
“Living Organic Phosphorus”. For the rate constant in the “UPTAKE” flow, we slightly lowered the ratio
to 2.0 (years-1)(moles-1) to help lower the amount of phosphorus that the living organisms were taking up
while not undercompensating for the high volume of matter that the “Inorganic Phosphorus in Lake” is
receiving. For the rate constant of “INFLOW”, we wrote an IF-THEN-ELSE statement that would change
the rate constant level depending on the current time with in the model’s simulation. The time-rate constant
changes included that from years 0 to 2 rate constant=0.15, from years 2 to 5 rate constant= 0, from
years 5 to 15 rate constant= 0.25, and from years 15 to 25 rate constant= 0.20. This equation was
representing in STELLA as IF (Time<=2) THEN 0.15 ELSE (IF (2<Time<=5) THEN 0 ELSE(IF
(5<Time<=15) THEN 0.25 ELSE(IF (Time>15) THEN 0.20 ELSE 0))). When we ran the simulation, we
observed that initially there was the same inverse trend between the “Inorganic Phosphorus in Lake” and
the “Living Organic Phosphorus”; however, the delay of the trend that was observed in Figure 3 was not
present in this new infrastructure due to the initial reduction of inorganic phosphorus influx for the first two
years. Meanwhile, the “Dead Organic Phosphorus” stock has a large decrease of matter in its reservoir until
it leveled off with the decreased quantity of matter in the “Living Organic Phosphorus” stock. After the
first two years once the inorganic phosphorus influx was deactivated within the “INFLOW” flow into the
matter cycle, all three stocks leveled-off with similar negative and positive exponential trends into stocks
resembling constant behaviors in each of the respective stocks. Each stock obtained a constant stock level,
which means the rate of each of the flows were consistent as well, after about six years into the model and
4 years after the inorganic phosphorus was cut off, thus, resembling a model with steady-state behavior.
The behaviors we observed represents that of what we would see from this sort of matter cycle
system within an aquatic ecosystem if it were to experience a large nutrient influx and set the ecosystem
into an unbalanced process of eutrophication, which can have a large effect all over the ecosystem and
ultimately an increase of BOD and a decrease in DO. If eutrophication processes are not controlled, then
their effects could greatly, especially with the decrease in DO, can kill off many different species that are
living in the ecosystem and depend on the steady-state of the phosphorus cycle. We then tried to evaluate
what possible solutions were realistic for a system that our model represented besides immediately stopping
the influx of inorganic phosphorus from the industrial facility. With no gaseous outflow was possible with
phosphorus, we brainstormed if any other outflow of phosphorus was possible. We then came to the
conclusion that there were not any possible outflows for the lake and its ecosystem, unless there were to be
an anthropological harvest of living and dead organisms for the release of phosphorus from the system.
However, that was not a realistic solution. Therefore, we then looked into natural processes that were
applicable to our model and a solution for eutrophication. We then revaluated the changes that we made in
our non-conserved phosphorus cycles model and its parameters to obtain a steady-state model. We found
that as the balanced was regained in the infrastructure of the phosphorus cycle from the changes found in
Figure 4 of our model, it resembled the natural process of hypertrophication, which is the natural response
of resiliency that contributors apart of the ecosystem do when they detect oscillations of nutrient levels and
unbalance.
Figure 4 – Shows the Model in Figure 3 with some different parameters. “A” was kept the same, but r, c
and b were decreased slightly with “b” being decreased the most. The Graph shows a lagging inverse
trend between Inorganic Phosphorus and Living Organic Phosphorus as well as their derivative and the
direct smoothing trend of Living Phosporus to Dead Organic Phosphorus.
Lastly we have the non-conserved phosphorus matter cycle that is represented in the model of
Figure 5. In this model, there is one true difference in comparison to the model shown in Figure 4; the
difference is that we changed the donor- & receptor-controlled flow between “Inorganic Phosphorus in
Lake” and “Living Organic Phosphorus” from a multiplicative flow equation to an additive flow equation.
We made this change because in the previous model’s set up, we found that the “Living Organic
Phosphorus” stock was receiving a large amount of nutrients and not oscillating down while the “Inorganic
Phosphorus in Lake” stock was experiencing the opposite trend. Though, ultimately both obtained levels
of steady-state, we thought that the incorporation of the additive donor- & receptor-controlled flow would
help keep the flows within a safe medium because of the required second rate constant. The additive donor& receptor-controlled flow can be mathematically represented as 𝑈𝑃𝑇𝐴𝐾𝐸 = (𝑐2 ∗
𝐼𝑛𝑜𝑟𝑔𝑎𝑛𝑖𝑐_𝑃ℎ𝑜𝑠𝑝ℎ𝑜𝑟𝑢𝑠_𝑖𝑛_𝐿𝑎𝑘𝑒) + (𝑐 ∗ 𝐿𝑖𝑣𝑖𝑛𝑔_𝑂𝑟𝑔𝑎𝑖𝑛𝑖𝑐_𝑃ℎ𝑜𝑠𝑝ℎ𝑜𝑟𝑢𝑠). The additional rate constant
that is required is represented by “c2” and it is applied to the outflow of the “Inorganic Phosphorus in Lake”
stock, quantified as a ratio with the units (years-1)(moles-1), just like the original rate constant. The additional
rate constant was set to 0.2 (years-1)(moles-1) in the model and all the parameters from the previous model
remained the same. After simulating the model with the same twenty year period, we observed similar
trends that represented in the models from Figure 3. The trends produced included inverse rates between
the living organic phosphorus and the inorganic phosphorus; in addition, the “Dead Organic Phosphorus”
stocks and the “Living Organic Phosphorus” stock had the same quantity level of matter at the end of the
simulation. We then tried to algebraically determine a rate constant for the “Inorganic Phosphorus in Lake”
stock’s outflow that would level out the inverse trend; however the results we received showed that the
additive flow equation did not improve the model for the behavioral medium that we desired.
Figure 5 – Shows the modified model with an added Additive donor receptor controlled relationship (c2).
This is not a steady state system, as you can see in the graph on the right, a steady increase in inorganic
phosphorus is present, while there is a decrease in both dead organic phosphorus and living organic
phosphorus.
For each of the different models we analyzed we had an evaluation process that assisted our approach.
The steps that were included in our evaluation process included:
1. Did the phosphorus cycle system represent a Conserved Matter Cycle or a Non-Conserved Matter
Cycle?
2. Determine the stocks of the phosphorus cycle
3. Determine the flow types for each flow that connected the phosphorus stocks within the matter
cycle
 i.e. the nature of the flow equation
4. Develop and layout (diagram) phosphorus’ stocks and flows
5. Associate mathematical relationships to stocks and flows
6. Determine the rate constants for the specific flows in the model
7. Observe interactions and explore the behaviors of the relationships
8. Did the model represent a steady-state system?
9. Evaluate approached for steady-state solutions for the model
Through those steps in the evaluation process, we were able to gain sufficient knowledge of the phosphorus
matter cycle and the dynamic system it is. The phosphorus cycle, itself, can be simply broken down, but it
will also always prove to be very dynamic, especially with the large effects it can cause from the slight
adjustment to its balance.
Bibliography
1. Deaton, M., & Winebrake, J. (2000). Dynamic modeling of environmental systems. New
York: Springer.
2. https://www.boundless.com/microbiology/textbooks/boundless-microbiology-textbook/microbial3.
ecology-16/nutrient-cycles-195/the-phosphorus-cycle-983-489/
http://bcn.boulder.co.us/basin/data/NEW/info/TP.html