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.

​ This model is used in a world studies extended essay research. The research question is: In what ways would the water desalination method used in Singapore benefit   water-stressed and economically less developed countries, using Palestine as a case study.

This model is a classic simulation of the production cycle in the ocean, including the effects of the thermocline in switching off advection of dissolved nutrients and detritus to the surface layer.

It illustrates a number of interesting features including the coupling of three state variables in a closed cycle, the use of time to control the duration of advection, and the modulus function for cycling annual temperature data over multiple years.

The model state variables are expressed in nitrogen units (mg N m-3), and the calibration is based on:

Baliño, B.M. 1996. Eutrophication of the North Sea, 1980-1990: An evaluation of anthropogenic nutrient inputs using a 2D phytoplankton production model. Dr. scient. thesis, University of Bergen.
 
Fransz, H.G. & Verhagen, J.H.G. 1985. Modelling Research on the Production Cycle of Phytoplankton in the Southern Bight of the Northn Sea in Relation to Riverborne Nutrient Loads. Netherlands Journal of Sea Research 19 (3/4): 241-250.

This model was first implemented in PowerSim some years ago by one of my M.Sc. students, who then went on to become a Buddhist monk. Although this is a very Zen model, as far as I'm aware, the two facts are unrelated.

European Masters in System Dynamics 2016
New University of Lisbon, Portugal

 Model to represent oyster individual growth by simulating feeding and metabolism. Builds on the core model in three ways: (i) partitions metabolic costs into feeding and fasting catabolism; (ii) adds allometry to clearance rate; (iii) adds temperature dependence to clearance rate.

Here is my systems flow chart for a moose population!

Simple model to illustrate an annual cycle for phytoplankton biomass in temperate waters.
Potential primary production uses Steele's equation and a Michaelis-Menten (or Monod) function for nutrient limitation. Respiratory losses are only a function of biomass.

The World3 model is a detailed simulation of human population growth from 1900 into the future. It includes many environmental and demographic factors.

THIS MODEL BY GUY LAKEMAN, FROM METRICS OBTAINED USING A MORE COMPREHENSIVE VENSIM SOFTWARE MODEL, SHOWS CURRENT CONDITIONS CREATED BY THE LATEST WEATHER EXTREMES AND LOSS OF ARABLE LAND BY THE  ALBEDO EFECT MELTING THE POLAR CAPS TOGETHER WITH NORTHERN JETSTREAM SHIFT NORTHWARDS, AND A NECESSITY TO ACT BEFORE THERE IS HUGE SUFFERING.

HANDY Model of Societal Collapse from Ecological Economics Paper 

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This model is a classic simulation of the production cycle in the ocean, including the effects of the thermocline in switching off advection of dissolved nutrients and detritus to the surface layer.

It illustrates a number of interesting features including the coupling of three state variables in a closed cycle, the use of time to control the duration of advection, and the modulus function for cycling annual temperature data over multiple years.

The model state variables are expressed in nitrogen units (mg N m-3), and the calibration is based on:

Baliño, B.M. 1996. Eutrophication of the North Sea, 1980-1990: An evaluation of anthropogenic nutrient inputs using a 2D phytoplankton production model. Dr. scient. thesis, University of Bergen.
 
Fransz, H.G. & Verhagen, J.H.G. 1985. Modelling Research on the Production Cycle of Phytoplankton in the Southern Bight of the Northn Sea in Relation to Riverborne Nutrient Loads. Netherlands Journal of Sea Research 19 (3/4): 241-250.

This model was first implemented in PowerSim some years ago by one of my M.Sc. students, who then went on to become a Buddhist monk. Although this is a very Zen model, as far as I'm aware, the two facts are unrelated.

This Insight Maker model shows how eagles, as predators, and rats, as prey, interact over time using a predator-prey system. Rats reproduce and grow in number, but their population decreases when eagles hunt them. Eagles increase in number when they eat rats, but their population also declines naturally. The model helps visualize how both populations rise and fall depending on each other.

Modeling forest succession in a northeast deciduous forest.

Simple model to illustrate Michaelis-Menten equation for nutrient uptake by phytoplankton.

The equation is:

P = Ppot S / (Ks + S)

Where:

P: Nutrient-limited production (e.g. d-1, or mg C m-2 d-1)
Ppot: Potential production (same units as P)
S: Nutrient concentation (e.g. umol N L-1)
Ks: Half saturation constant for nutrient (same units as S)

The model contains no state variables, just illustrates the rate of production, by making the value of S equal to the timestep (in days). Move the slider to the left for more pronounced hyperbolic response, to the right for linear response.

This model provides a dynamic simulation of the Sverdrup (1953) paper on the vernal blooming of phytoplankton.

The model simulates the dynamics of the mixed layer over the year, and illustrates how it's depth variation leads to conditions that trigger the spring bloom. In order for the bloom to occur, production of algae in the water column must exceed respiration.

This can only occur if vertical mixing cannot transport algae into deeper, darker water, for long periods, where they are unable to grow.

Sverdrup, H.U., 1953. On conditions for the vernal blooming of phytoplankton. J. Cons. Perm. Int. Exp. Mer, 18: 287-295

HANDY Model of Societal Collapse from Ecological Economics Paper 

This model describes the flow of energy from generation to consumption for neighborhoods in the metro Atlanta area. It also calculates the cost of energy production and the number of years it will take to recover that cost.

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.

​ This model is used in a world studies extended essay research. The research question is: In what ways would the water desalination method used in Singapore benefit   water-stressed and economically less developed countries, using Palestine as a case study.

This model is based off Meadows economic capital with reinforcing growth loop constrained by a renewable resource model.

Simple model to illustrate oyster growth based on primary production of Phytoplankton as a state variable, forced by light and nutrients, running for a yearly period.

Phytoplankton growth based on on Steele's and Michaelis-Menten equations), where: 

Primary Production=(([Pmax]*[I]/[Iopt]*exp(1-[I]/[Iopt])*[S])/([Ks]+[S]))

Pmax: Maximum production (d-1)
I: Light energy at depth of interest (uE m-2 s-1)
Iopt: Light energy at which Pmax occurs (uE m-2 s-1)
S: Nutrient concentration (umol N L-1)
Ks: Half saturation constant for nutrient (umol N L-1).

Further developments:
- Nutrients as state variable in cycle with detritus from phytoplankton and oyster biomass.
- Light limited by the concentration of phytoplankton.
- Temperature effect on phytoplankton and Oyster growth.

Modèle simple pour illustrer la croissance des huîtres sur la base de la production primaire de phytoplancton comme une variable d'état, forcé par la lumière et les éléments nutritifs, en cours d'exécution pour une période annuelle.

La croissance du phytoplancton sur la base de Steele et équations de Michaelis-Menten), où:

Production primaire = (([Pmax] * [I] / [Iopt] * exp (1 - [I] / [Iopt]) * [S]) / ([K] + [S]))

Pmax: production maximale (d-1)
I: L'énergie lumineuse en profondeur de l'intérêt (Ue m-2 s-1)
Iopt: L'énergie lumineuse à laquelle se produit Pmax (Ue m-2 s-1)
S: concentration des éléments nutritifs (N umol L-1)
KS: Demi constants de saturation en nutriments (N umol L-1).

D'autres développements:
- Les éléments nutritifs comme variable d'état dans le cycle de détritus de phytoplancton et d'huîtres de la biomasse.
- Lumière limitée par la concentration de phytoplancton.
- Effet de la température sur le phytoplancton et la croissance des huîtres.

Life and Death

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.

This model describes the flow of energy from generation to consumption for neighborhoods in the metro Atlanta area. It also calculates the cost of energy production and the number of years it will take to recover that cost.