My take on simulating the peak oil production

​ 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.

Simple mass balance model for lakes, based on the Vollenweider equation:

dMw/dt = Min - sMw - Mout

The model was first used in the 1960s to determine the phosphorus concentration in lakes and reservoirs, for eutrophication assessment.

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.

M.Sc. in Environmental Engineering SIMA 2018
New University of Lisbon, Portugal

 Model to represent oyster individual growth by simulating feeding and metabolism. Model (i) partitions metabolic costs into feeding and fasting catabolism; (ii) adds allometry to clearance rate; (iii) adds temperature dependence to clearance rate; (iv) illustrates how clearance rate per gram is used if we multiply by the oyster biomass

Clone of: 

Simulation of MTBF with controls


The inverse curve is the trust time
On the right the increase in failures brings its inverse which is loss of trust and move into suspicion and lack of confidence.
This can be seen in strategic social applications with those who put economy before providing the priorities of the basic living infrastructures for all.

This applies to policies and strategic decisions as well as physical equipment.
A) Equipment wears out through friction and preventive maintenance can increase the useful lifetime, 
B) Policies/working practices/guidelines have to be updated to reflect changes in the external environment and eventually be replaced when for instance a population rises too large (constitutional changes are required to keep pace with evolution, e.g. the concepts of the ancient Greeks, 3000 years ago, who based their thoughts on a small population cannot be applied in 2013 except where populations can be contained into productive working communities with balanced profit and loss centers to ensure sustainability)

Early Life
Useful Life

Wearout

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

This model depicts a very simplified series of interactions between water quality inspectors and cannabis cultivators in northern California.

The Logistic Map is a polynomial mapping (equivalently, recurrence relation) of degree 2, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations. The map was popularized in a seminal 1976 paper by the biologist Robert May, in part as a discrete-time demographic model analogous to the logistic equation first created by Pierre François Verhulst. 

Mathematically, the logistic map is written

where:

 is a number between zero and one, and represents the ratio of existing population to the maximum possible population at year n, and hence x0 represents the initial ratio of population to max. population (at year 0)r is a positive number, and represents a combined rate for reproduction and starvation. To generate a bifurcation diagram, set 'r base' to 2 and 'r ramp' to 1
To demonstrate sensitivity to initial conditions, try two runs with 'r base' set to 3 and 'Initial X' of 0.5 and 0.501, then look at first ~20 time steps

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 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.

Two households with PV systems and Electric Vehicles, sharing a battery and connected to the grid. What are the advantages?

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.

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.

This model illustrates predator prey interactions using real-life data of wolf and moose populations on the Isle Royale. It was "cloned" from a model that InsightMaker provides to its users, at
https://insightmaker.com/insight/2068/Isle-Royale-Predator-Prey-Interactions
Thanks Scott Fortmann-Roe.

I've created a Mathematica file that replicates the model, at
http://www.nku.edu/~longa/classes/2018spring/mat375/mathematica/Moose-n-Wolf-InsightMaker.nb

It allows one to experiment with adjusting the initial number of moose and wolves on the island.

I used steepest descent in Mathematica to optimize the parameters, with my objective data being the ratio of wolves to moose. You can try my (admittedly) kludgy code, at
http://www.nku.edu/~longa/classes/2018spring/mat375/mathematica/Moose-n-Wolf-InsightMaker-BestFit.nb

{WolfBirthRateFactorStart,
WolfDeathRateStart,
MooseBirthRateStart,
MooseDeathRateFactorStart,
moStart,
woStart} =
{0.000267409,
0.239821,
0.269755,
0.0113679,
591,
23.};

Examining the ecosystem of the sea turtle and how that influences its population as an endangered species.

Simple mass balance model for lakes, based on the Vollenweider equation:

dMw/dt = Min - sMw - Mout

The model was first used in the 1960s to determine the phosphorus concentration in lakes and reservoirs, for eutrophication assessment.

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.

Simple mass balance model for lakes, based on the Vollenweider equation:

dMw/dt = Min - sMw - Mout

The model was first used in the 1960s to determine the phosphorus concentration in lakes and reservoirs, for eutrophication assessment.

Simple mass balance model for lakes based on the Vollenweider equation:

dMw/dt = Min - sMw + pMs - Mout

The model was first used in the 1960s to determine the phosphorus concentration in lakes and reservoirs for eutrophication assessment.

This version considers mercury, and adds diagenesis, using an extra state variable (mercury in the sediment), and incorporates desorption processes that release mercury trapped in the sediment back to the water column.

The temporal dynamics of the model simulate the typical development of pollution in time.

1. Low loading, low Hg concentration in lake
2. High loading, increasing Hg concentration in lake
3. Desorption rate is low, Hg in sediment increases
4. Measures implemented for source control, loading reduces
5. Hg in lake gradually decreases, but below a certain point, desorption increases, and lake Hg concentration does not improve
6. Recovery only occurs when the secondary load in the sediment is strongly reduced.

​ 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 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.