This model shows how a persistent pollutant such as mercury or DDT can be bioamplified along a trophic chain to levels that result in reduction of top predator populations.

This model simulates the growth of carp in an aquaculture pond, both with respect to production and environmental effects.

 Carp are mainly cultivated in Asia and Europe, and contribute to the world food supply.

Aquaculture currently produces sixty million tonnes of fish and shellfish every year. In 2011, aquaculture production overtook wild fisheries for human consumption.

This paradigm shift last occurred in the Neolithic period, ten thousand years ago, when agriculture displaced hunter-gatherers as a source of human food.

Aquaculture is here to stay, and wild fish capture (fishing) will never again exceed cultivation.

Recreational fishing will remain a human activity, just as hunting still is, after ten thousand years - but it won't be a major source of food from the seas.

The best way to preserve wild fish is not to fish them.

Simulates Ag biogeochemical cycling using data from Rauch and Pacyna (2009). This Insight forms part of the engaged lear​ning exercise for a SESYNC case study about the human relationship with silver as a natural resource throughout history.

The beginning of a systems dynamics model for teaching NRM 320.

A basic model of the short-term carbon cycle.

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.

Thanks to Jacob Englert for the model if-then-else structure.

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

Here is an average representation of Earth today. Enjoy!

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 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.

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

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.

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.

The simple  graph shows two feedback loops that interact to make climate change and its consequences worse, leading to an unexpected (& inescapable?) dilemma. Presently,  there are 413 ppm of CO2 gasses in the atmosphere. Even without any further emission of greenhouse gasses, this high level of CO2 in the atmosphere will ensure constantly worsening climatic consequences because of delays that operate in the climate system. The dilemma is caused by relentlessly worsening of extreme weather events, droughts, forest fires etc., the need for draconian measures to deal with the situation and the opposition to the measures, described by the feedback loop B2.This opposition is rooted in human nature, the psychological defence mechanisms that cause us to repress or even deny unpalatable truths that threaten our basic assumptions and the way we understand life. Together,  the loops B1 & B2 create a vicious reinforcing loop that describes the escalating and worsening situation created by the dilemma.

Please look at Insight No. 238770 that provides background information and also at the information labels attached to majority of the variables in the 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.

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

An attempt to model some of the features in the national water market in Australia.

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.};

Here is my systems flow chart for a moose population!

HANDY Model of Societal Collapse from Ecological Economics Paper 

Bathtub SFD

Simple Model of the Food Chain