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# Ecology

#### Clone of Isle Royale: Predator/Prey Model for Moose and Wolves

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.};

• 2 years 6 months ago

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

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 6 months ago

#### Clone of Prey&Predator

​Physical meaning of the equationsThe 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.

• 2 years 10 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 7 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 7 months ago

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

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 6 months ago

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

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 6 months ago

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

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 6 months ago

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

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 6 months ago