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# Ecology

#### Dystopia: simple energy system model

Basic idea is to model demand with endogenous growth (but "satiation" becomes possible - eventually - at some notional "sufficiency" level); and supply then attempts to track demand with some time lag (~5-50 years - characteristic of commissioning/decommissioning large scale energy infrastructure). But supply also produces pollution, which accoumulates. We can specify a notional constraint/limit; approaching this should trumpdemand and forces supply to zero. In this version we'll only have one source. so no substitution is possible. We expect to see a fairly sudden supply crash. Of course, "demand" will still carry merrily on its way up anyway, but the interpretation of the consequently growing supply shortfall will be left to the eye of the beholder. NB: this version doesn't automatically succeed in limiting P to P_max. It forces dS/dt to zero as A*(P/ P_max) reaches 1; and then as that value exceeds 1, dS/dt is forced negative. But this dynamics has no way to "undo" any overshoot of P over P_max (which would require S itself to become negative: "negative emissions"). Need to manual find/choose a big enough value of A to limit P effectively.
• 2 years 5 months ago

#### Clone of Final Midterm Student version of A More Realistic Model of Isle Royale: Predator Prey Interactions

Allison Zembrodt's Model
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.

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

equations I used in kill rate :

power model - 12*0.1251361120909615*([Moose]/[Wolves])^.44491970277839954*[Wolves]

Kill rate sqrt = 12*(0.0933207+.0873463*([Moose]/[Wolves])^.5)*[Wolves]

Holling Type III - ((0.986198*([Moose]/[Wolves])^2)/ (601.468 +([Moose]/[Wolves])^2))*[Wolves]*12

linear - 12*[Wolves]*(.400271+.00560299([Moose]/[Wolves]))

• 2 years 10 months ago

#### Ecological

Showing a more detailed energy flow model (some areas/details have been omitted in order to "simplify")
• 3 years 11 months ago

#### Predator and Prey model

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

Experiment with adjusting the initial number of moose and wolves on the island.
• 1 year 6 months ago

#### Clone of Spring and fall bloom

Introduction
Simple model of the spring bloom in coastal temperate coastal waters. Nitrogen is assumed to be the limiting nutrient, so the model is based on N only. The model represents one liter of water. Dissolved inorganic nitrogen (DIN) accumulates in the water column during winter and has reached 250 µmol/L on March 1st where the model starts. At this time the light intensity have just reached the level necessary to initiate the bloom.

Model setup
N uptake: Michaelis Menten kinetics with a maximum growth rate that doubles the population each day. Km=5µM.

Grazing: Michaelis Menten kinetics with a maximum daily uptake equal to the N in the population. Km=50µM.

Sloppy eating: 60% of the grazing is wasted to PON

Death: 5% of the zooplankton dies each day

Mineralization: 1% of the PON is mineralized to DIN each day

Results
For the first 6 days the phytoplankton grows exponentially and depletes the DIN pool. The peak in phytoplankton is followed by a delayed peak in zooplankton due to its slower growth rate. Slowly the zooplankton graze down the spring bloom and the nitrogen is transformed to the pool of particulate dead organic nitrogen (PON). While this happens the phytoplankton is kept low by the still high zooplankton which allow the DIN pool to increase from day 25 to day 55. Eventually the phytoplankton escapes the top down control and we see a secondary bloom based on regenerated DIN.
• 5 years 11 months ago

#### Clone of Predator-Prey Model ("Lotka'Volterra")

Dynamic simulation modelers are particularly interested in understanding and being able to distinguish between the behavior of stocks and flows that result from internal interactions and those that result from external forces acting on a system.  For some time modelers have been particularly interested in internal interactions that result in stable oscillations in the absence of any external forces acting on a system.  The model in this last scenario was independently developed by Alfred Lotka (1924) and Vito Volterra (1926).  Lotka was interested in understanding internal dynamics that might explain oscillations in moth and butterfly populations and the parasitoids that attack them.  Volterra was interested in explaining an increase in coastal populations of predatory fish and a decrease in their prey that was observed during World War I when human fishing pressures on the predator species declined.  Both discovered that a relatively simple model is capable of producing the cyclical behaviors they observed.  Since that time, several researchers have been able to reproduce the modeling dynamics in simple experimental systems consisting of only predators and prey.  It is now generally recognized that the model world that Lotka and Volterra produced is too simple to explain the complexity of most and predator-prey dynamics in nature.  And yet, the model significantly advanced our understanding of the critical role of feedback in predator-prey interactions and in feeding relationships that result in community dynamics.The Lotka–Volterra model makes a number of assumptions about the environment and evolution of the predator and prey populations:
1. The prey population finds ample food at all times.2. The food supply of the predator population depends entirely on the size of the prey population.3. The rate of change of population is proportional to its size.4. During the process, the environment does not change in favour of one species and genetic adaptation is inconsequential.5. Predators have limitless appetite.As differential equations are used, the solution is deterministic and continuous. This, in turn, implies that the generations of both the predator and prey are continually overlapping.[23]
Prey
When multiplied out, the prey equation becomesdx/dt = αx - βxy The prey are assumed to have an unlimited food supply, and to reproduce exponentially unless subject to predation; this exponential growth is represented in the equation above by the term αx. The rate of predation upon the prey is assumed to be proportional to the rate at which the predators and the prey meet; this is represented above by βxy. If either x or y is zero then there can be no predation.

With these two terms the equation above can be interpreted as: the change in the prey's numbers is given by its own growth minus the rate at which it is preyed upon.

Predators

The predator equation becomes

dy/dt =  -

In this equation, {\displaystyle \displaystyle \delta xy} represents the growth of the predator population. (Note the similarity to the predation rate; however, a different constant is used as the rate at which the predator population grows is not necessarily equal to the rate at which it consumes the prey). {\displaystyle \displaystyle \gamma y} represents the loss rate of the predators due to either natural death or emigration; it leads to an exponential decay in the absence of prey.

Hence the equation expresses the change in the predator population as growth fueled by the food supply, minus natural death.

• 1 year 6 months ago

#### Rachel Driehaus Midterm MAT 375

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.

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

• 2 years 10 months ago

#### wolf ~ boom and bust

This model is to be used with Mr. Roderick's AP biology activity on population growth. See steveroderick.net for a copy of the activity worksheet.
Use the sliders below to quickly change the initial values of components of the model.
• 5 years 8 months ago