#### Isle Royale: Predator/Prey Model for Moose and Wolves, with Total Population

##### Andrew E Long

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 added in an adjustment to handle population.

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.};

https://insightmaker.com/insight/2068/Isle-Royale-Predator-Prey-Interactions

Thanks Scott Fortmann-Roe.

I've added in an adjustment to handle population.

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 3 months ago

#### Education Don

##### Don Fernando

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.

Experiment with adjusting the initial number of moose and wolves on the island.

- 5 years 1 month ago

#### Clone of Isle Royale: Predator Prey Interactions

##### Matt Young

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.

Experiment with adjusting the initial number of moose and wolves on the island.

- 4 years 10 months ago

#### Sleek, non-dimensionalized Isle Royale: Predator/Prey Model for Moose and Wolves

##### Andrew E Long

This non-dimensionalized, sleekest most neatest model illustrates predator prey interactions using logistic growth for the moose population, for the wolf and moose populations on Isle Royale.

Thanks Scott Fortmann-Roe for the original model.

I've added in an adjustment to handle population sizes, by dividing by moose carrying capacity.

Time is scaled by the moose birth parameter:

tau=bm*t

There are therefore only three parameters left to account for any dynamics:

beta = bw/bm (relative wolf to moose births)

delta = dm/bm (relative death to birth ratio for moose)

gamma = dw/bm (wolf deaths to moose births)

The equations are thus

dM/dtau = M [ (1-M) - delta W ]

dW/dtau = W [beta M - gamma ]

There is a stable equilibrium pair of population values, relative to the carrying capacity:

M^* = gamma / beta

W^* = (1-gamma / beta) / delta

I have a sleek version with a logistical growth term for the moose, at

http://www.nku.edu/~longa/classes/2018spring/mat375/mathematica/Moose-n-Wolf-InsightMaker-sleek.nb

Thanks Scott Fortmann-Roe for the original model.

I've added in an adjustment to handle population sizes, by dividing by moose carrying capacity.

Time is scaled by the moose birth parameter:

tau=bm*t

There are therefore only three parameters left to account for any dynamics:

beta = bw/bm (relative wolf to moose births)

delta = dm/bm (relative death to birth ratio for moose)

gamma = dw/bm (wolf deaths to moose births)

The equations are thus

dM/dtau = M [ (1-M) - delta W ]

dW/dtau = W [beta M - gamma ]

There is a stable equilibrium pair of population values, relative to the carrying capacity:

M^* = gamma / beta

W^* = (1-gamma / beta) / delta

I have a sleek version with a logistical growth term for the moose, at

http://www.nku.edu/~longa/classes/2018spring/mat375/mathematica/Moose-n-Wolf-InsightMaker-sleek.nb

- 2 years 4 months ago

#### Clone of (3) Copy of "Isle Royale: Predator Prey Interactions"

##### Nora Baumgartner

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.

Experiment with adjusting the initial number of moose and wolves on the island.

- 5 years 7 months ago

#### Clone of Clone of Isle Royale: Predator Prey Interactions

##### Aleksandr

Experiment with adjusting the initial number of moose and wolves on the island.

- 5 years 12 months ago

#### Clone of Isle Royale: Predator Prey Interactions

##### Sarah Supp

Experiment with adjusting the initial number of moose and wolves on the island.

- 3 years 7 months ago

#### Day 22: More Realistic Model of Isle Royale: Predator Prey Interactions

##### Jacob Englert

Experiment with adjusting the initial number of moose and wolves on the island.

- 2 years 3 months ago

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

##### Allison Zembrodt

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.

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

equations I used in kill rate :

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

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 3 months ago

#### Model

##### Victor Wong

- 3 years 3 months ago

#### Predator and Prey model

##### Abbott Van

Experiment with adjusting the initial number of moose and wolves on the island.

- 11 months 4 weeks ago

#### Parker Realistic Isle Royale: Predator Prey Interactions

##### Parker Kain

Experiment with adjusting the initial number of moose and wolves on the island.

- 2 years 4 months ago

#### Rachel Driehaus Midterm MAT 375

##### Rachel Driehaus

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

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

- 2 years 3 months ago

#### A More Realistic Model of Isle Royale: Predator Prey Interactions

##### Alyssa Farmer

Experiment with adjusting the initial number of moose and wolves on the island.

- 2 years 4 months ago

#### Clone of Isle Royale: Predator Prey Interactions

##### Bob Jones

Experiment with adjusting the initial number of moose and wolves on the island.

- 5 years 2 months ago

#### Clone of Jacob Englert MAT 375 Midterm: Model of Isle Royale: Predator Prey Interactions

##### Urasa Kongdechakul

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

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

- 11 months 4 weeks ago

#### Driehaus Isle Royale: Meese and Wolves

##### Rachel Driehaus

Experiment with adjusting the initial number of moose and wolves on the island.

- 2 years 4 months ago

#### Clone of Isle Royale: Predator Prey Interactions

##### Trent J. Lythgoe

Experiment with adjusting the initial number of moose and wolves on the island.

- 5 years 5 months ago

#### Clone of Isle Royale: Predator Prey Interactions

##### Karen Doore

Experiment with adjusting the initial number of moose and wolves on the island.

- 6 years 6 months ago

#### Clone of Isle Royale: Predator Prey Interactions

##### machendiran

Experiment with adjusting the initial number of moose and wolves on the island.

- 7 years 8 months ago

#### Clone of (3) Copy of "Isle Royale: Predator Prey Interactions"

##### Eliecer Alvarado Rodriguez

Experiment with adjusting the initial number of moose and wolves on the island.

- 5 years 7 months ago

#### Clone of Isle Royale: Predator Prey Interactions

##### Michael Afghani

Experiment with adjusting the initial number of moose and wolves on the island.

- 5 years 2 months ago

#### Clone of (3) Copy of "Isle Royale: Predator Prey Interactions"

##### JanH

Experiment with adjusting the initial number of moose and wolves on the island.

- 4 years 7 months ago

#### Clone of Isle Royale: Predator Prey Interactions

##### Nicole M Radziwill

Experiment with adjusting the initial number of moose and wolves on the island.

- 4 years 4 months ago