The body of research and studies generated on the Fryingpan River between the 1940s and the present supports the development of a conceptual model of ecosystem responses to hydrological regime behavior and streamflow management activities. This conceptual model should encourage conversations about system behavior and collective understanding among stakeholders regarding connections between specific hydrological regime characteristics affected by management of Ruedi Reservoir and the ecological or biological variables important to local communities. For the sake of simplicity, the model includes mostly unidirectional relationships—feedback loops are exploded to reveal intermediate connections between variables. This approach increases the number of variables represented in the system, perhaps increasing its complexity at first glance. However, the primary benefit to the end user is that the model becomes more readable and explicit in its representation of system behavior. 

 

The conceptual model presented here likely differs by degrees from those held by the various investigators who considered Fryingpan River processes over the previous 80 years. However, it affectively aggregates the ideas main presented by each of those individuals. This model focuses on hydrological and biological variables and does not incorporate the entire diversity of human uses and needs for water from the Fryingpan River (e.g. hydropower production for the City of Aspen, revenue generated in the Town of Basalt by angling activities, etc.).  Rather it attempts to illustrate how the conditional state of important ecosystem characteristics might respond to reservoir management activities that impact typical spring flows, peak flow timing and magnitude, summer flows, fall flows, and winter flows. 

This model implements the one-dimensional version of the advection-dispersion equation for an estuary. The equation is:

dS/dt = (1/A)d(QS)/dx - (1/A)d(EA)/dx(dS/dx) (Eq. 1)

Where S: salinity (or any other constituent such as chlorophyll or dissolved oxygen), (e.g. kg m-3); t: time (s); A: cross-sectional area (m2); Q: river flow (m3 s-1); x: length of box (m); E: dispersion coefficient (m2 s-1).

For a given length delta x, Adx = V, the box volume. For a set value of Q, the equation becomes:

VdS/dt = QdS - (d(EA)/dx) dS (Eq. 2)

EA/x, i.e. (m2 X m2) / (m s) = E(b), the bulk dispersion coefficient, units in m3 s-1, i.e. a flow, equivalent to Q

At steady state, dS/dt = 0, therefore we can rewrite Eq. 2 for one estuarine box as:

Q(Sr-Se)=E(b)r,e(Sr-Se)-E(b)e,s(Se-Ss) (Eq. 3)

Where Sr: river salinity (=0), Se: mean estuary salinity; Ss: mean ocean salinity

E(b)r,e: dispersion coefficient between river and estuary, and E(b)e,s: dispersion coefficient between the estuary and ocean.

By definition the value of E(b)r,e is zero, otherwise we are not at the head (upstream limit of salt intrusion) of the estuary. Likewise Sr is zero, otherwise we're not in the river. Therefore:

QSe=E(b)e,s(Se-Ss) (Eq. 4)

At steady state

E(b)e,s = QSe/(Se-Ss) (Eq 5)

The longitudinal dispersion simulates the turbulent mixiing of water in the estuary during flood and ebb, which supplies salt water to the estuary on the flood tide, and make the sea a little more brackish on the ebb.

You can use the slider to turn off dispersion (set to zero), and see that if the tidal wave did not mix with the estuary water due to turbulence, the estuary would quickly become a freshwater system.

Simple Model of the Food Chain

Eastern oyster growth model calibrated for Long Island Sound
Developed and implemented by Joao G. Ferreira and Camille Saurel; growth data from Eva Galimany, Gary Wickfors, and Julie Rose; driver data from Julie Rose and Suzanne Bricker; Culture practice from the REServ team and Tessa Getchis. This model is a workbench for combining ecological and economic components for REServ. Economic component added by Trina Wellman.

This is a one box model for an idealized farm with one million oysters seeded (one hectare @ a stocking density of 100 oysters per square meter)

1. Run WinShell individual growth model for one year with Long Island Sound growth drivers;

2. Determine the scope for growth (in dry tissue weight per day) for oysters centered on the five weight classes)
 
3. Apply a classic population dynamics equation:

dn(s,t)/dt = -d[n(s,t)g(s,t)]/ds - u(s)n(s,t)

s: Weight (g)
t: Time
n: Number of individuals of weight s
g: Scope for growth (g day-1)
u: Mortality rate (day-1)

4. Set mortality at 30% per year, slider allows scenarios from 30% to 80% per year

5. Determine harvestable biomass, i.e. weight class 5, 40-50 g (roughly three inches length)

Summary of UNEP ecosystems services CBA 2011 article by Wegner and Pascual

Modeling the change in concentration of O2 in a lake with a continuous loading of BOD, modeled as a CSTR

Bathtub SFD

In Chile, 60% of its population are exposed to levels of Particulate Matter (PM) above international standards. Air Pollution is causing 4,000 premature deaths per year, including health costs over US$8 billion.

This is a two-stock (ocean and atmosphere) climate model simulating the behavior of the earth climate from time zero. The initial conditions of the stocks are also set zero, so it demonstrates how long the earth takes to reach the temperature suitable for life.

Westley, F. R., O. Tjornbo, L. Schultz, P. Olsson, C. Folke, B. Crona and Ö. Bodin. 2013. A theory of transformative agency in linked social-ecological systems. Ecology and Society 18(3): 27. link

This diagram provides a stylised description of important feedbacks within a shallow-lake system.

This model illustrates predator prey interactions using real-life data of wolf and moose populations on the Isle Royale.

We incorporate logistic growth into the moose dynamics, and we replace the death flow of the moose with a kill rate modeled from the kill rate data found on the Isle Royale website.

I start with these parameters:
Wolf Death Rate = 0.15
Wolf Birth Rate = 0.0187963
Moose Birth Rate = 0.4
Carrying Capacity = 2000
Initial Moose: 563
Initial Wolves: 20

I used RK-4 with step-size 0.1, from 1959 for 60 years.

The moose birth flow is logistic, MBR*M*(1-M/K)
Moose death flow is Kill Rate (in Moose/Year)
Wolf birth flow is WBR*Kill Rate (in Wolves/Year)
Wolf death flow is WDR*W

Harvested fishery with endogenous investment and ship deployment policy. Ch 9 p345-360 John Morecroft (2007) Strategic Modelling and Business Dynamics. See simpler models at IM-2990 and IM-2991

Very simple model demonstrating growth of phytoplankton using Steele's equation for potential production and Michaelis-Menten equation for nutrient limitation.

Both light and nutrients (e.g. nitrogen) are modelled as forcing functions, and the model is "over-calibrated" for stability.

The phytoplankton model approximately reproduces the spring-summer diatom bloom and the (smaller) late summer dinoflagellate bloom.
 
Oyster growth is modelled only as a throughput from algae. Further developments would include filtration as a function of oyster biomass, oyster mortality, and other adjustments.

In Chile, 60% of its population are exposed to levels of Particulate Matter (PM) above international standards. Air Pollution is causing 4,000 premature deaths per year, including health costs over US$8 billion.

This is step 2 in making a climate model based on our insights of how trees actively contribute to the cooling capcacity of the Earth.​

In this step we divide the incoming energy from the sun to the land and to the oceaan.

This model implements the equations proposed by Ketchum in 1954. The rationale behind the concept is that only phytoplankton that grows above a certain rate will not be flushed out of an estuary.

For biological processes:

Pt  =  Po exp(kt)

Where Pt is the phytoplankton biomass at time t, Po is the initial biomass, and k is the growth rate.

For physical processes:

Pm  =  Po (1-r)^m

Where Pm is the phytoplankton biomass after m tidal cycles, and r is the exchange ratio (proportion of estuary water which does not return each tidal cycle).

By substitution, and replacing t by m in the first equation, we get:

Pm = Poexp(km).(1-r)^m

For phytoplankton to exist in an estuary, Pm = Po (at least), i.e. 1 / (1-r)^m = exp(km)
ln(1) - m.ln(1-r) = km
-m.ln(1-r) = km
k = -ln(1-r)

Ketchum (1954) Relation between circulation and planktonic populations in estuaries. Ecology 35: 191-200.

In 2005, Ferreira and co-workers showed that this balance has direct implications on biodiversity of estuarine phytoplankton, and discussed how this could be relevant for water management, in particular for the EU Water Framework Directive 60/2000/EC (Ecological Modelling, 187(4) 513-523).

This model describes phosphorus cycling in a dune-lake system in the Northland region of New Zealand. It is based on stock and flow diagrams where each orange oval represents an input, while each blue box represents a stock. Each arrow represents a flow. Flows involve a loss from the stock at which they start and add to the stock at which they end.