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

##### 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 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 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.};

- 1 year 9 months ago

#### Exponential Growth

##### Andrew E Long

This simple model demonstrates exponential growth or decay in a population.

A comparable Mathematica file is at

http://www.nku.edu/~longa/classes/2018spring/mat375/mathematica/ExponentialGrowth-and-DecayModel.nb

A comparable Mathematica file is at

http://www.nku.edu/~longa/classes/2018spring/mat375/mathematica/ExponentialGrowth-and-DecayModel.nb

- 1 year 10 months ago

#### A Sleek, non-dimensionalized SIR (Susceptible, Infected, Recovered) model

##### Andrew E Long

This is an example of an SIR (Susceptible, Infected, Recovered) model that has been re-parameterized down to the bare minimum, to illustrated the dynamics possible with the fewest number of parameters.

We're rescaled this SIR model, so that time is given in infection rate-appropriate time units, "rates" are now ratios of rates (with infectivity rate in the denominator), and populations are considered proportions (unfortunately InsightMaker doesn't function properly if I give them all values from 0 to 1, which sum to 1 -- so, at the moment, I give them values that sum to 100, and consider the results percentages).

The new display includes the asymptotics: the three sub-populations will tend to fixed values as time goes to infinity; the infected population goes to zero if the recovery rate is greater than the infectivity rate -- i.e., the disease dies out.

Note the use of a "ghost" stock (for Total Population), which I think is a pretty cool idea. It cuts down on the number of arcs in the model graph.

A comparable model in Mathematica is available at

http://www.nku.edu/~longa/classes/2018spring/mat375/mathematica/SIRModel-rescaled.nb

We're rescaled this SIR model, so that time is given in infection rate-appropriate time units, "rates" are now ratios of rates (with infectivity rate in the denominator), and populations are considered proportions (unfortunately InsightMaker doesn't function properly if I give them all values from 0 to 1, which sum to 1 -- so, at the moment, I give them values that sum to 100, and consider the results percentages).

The new display includes the asymptotics: the three sub-populations will tend to fixed values as time goes to infinity; the infected population goes to zero if the recovery rate is greater than the infectivity rate -- i.e., the disease dies out.

Note the use of a "ghost" stock (for Total Population), which I think is a pretty cool idea. It cuts down on the number of arcs in the model graph.

A comparable model in Mathematica is available at

http://www.nku.edu/~longa/classes/2018spring/mat375/mathematica/SIRModel-rescaled.nb

- 1 year 9 months ago

#### A Simple SIR (Susceptible, Infected, Recovered) without infection

##### Andrew E Long

This is a simple example of (part of a) simple SIR (Susceptible, Infected, Recovered) model, suggested by De Vries, et al. in A Course in Mathematical Biology.

They wanted to illustrate the comparative behavior of differential equations and discrete difference equations. We know that differential equations are generally solved numerically by discretizing them, so that the comparison is a little bit rigged....

A comparable model in Mathematica is available at

http://www.nku.edu/~longa/classes/2018spring/mat375/mathematica/SIRModel-w-discrete-version.nb

They wanted to illustrate the comparative behavior of differential equations and discrete difference equations. We know that differential equations are generally solved numerically by discretizing them, so that the comparison is a little bit rigged....

A comparable model in Mathematica is available at

http://www.nku.edu/~longa/classes/2018spring/mat375/mathematica/SIRModel-w-discrete-version.nb

- 1 year 9 months ago

#### English Mother/Daughter Birth Weights

##### Andrew E Long

This is an introductory example from Olinick's book An Introduction to Mathematical Models in the Social and Life Sciences.

"A recent study focused on the relationship between the birth weights of English women and the birth weights of their daughters. The weights were split into three categories: low (below 6 pounds), average (between 6 and 8 pounds), and high (above 8 pounds). Among women whose own birth weights were low, 50 percent of the daughters had low birth weights, 45 percent had average weights, and 5 percent had high weights. Women with average birth weights had daughters with average weights half of the time, while the half was split evenly between low and high categories. Women with high birth weights had female babies with high weights 40 percent of the time, with low and average weights each occuring 30 percent of the time." p. 274-275.

For the Markov chain, you should make sure that you're taking time steps of length 1 in the settings, and Euler. RK-4 effectively looks beyond a single previous step, so it has a sort of memory!

Thanks Mike! Interesting examples, as always....

Andy Long

Next up: an SIR.

"A recent study focused on the relationship between the birth weights of English women and the birth weights of their daughters. The weights were split into three categories: low (below 6 pounds), average (between 6 and 8 pounds), and high (above 8 pounds). Among women whose own birth weights were low, 50 percent of the daughters had low birth weights, 45 percent had average weights, and 5 percent had high weights. Women with average birth weights had daughters with average weights half of the time, while the half was split evenly between low and high categories. Women with high birth weights had female babies with high weights 40 percent of the time, with low and average weights each occuring 30 percent of the time." p. 274-275.

For the Markov chain, you should make sure that you're taking time steps of length 1 in the settings, and Euler. RK-4 effectively looks beyond a single previous step, so it has a sort of memory!

Thanks Mike! Interesting examples, as always....

Andy Long

Next up: an SIR.

- 1 year 8 months ago

#### Flakes no more!

##### Andrew E Long

This is an introductory example from Olinick's book An Introduction to Mathematical Models in the Social and Life Sciences.

Next up: and SIR, and his interesting model of female birth weights.

Next up: and SIR, and his interesting model of female birth weights.

- 1 year 8 months ago

#### Cannibalistic and Chaotic Flour Beetles

##### Andrew E Long

This is an example from Cushing's book An Introduction to Structured Population Dynamics.

The parameters initially included reproduce the bifurcation results on p. 39 of Cushing's manuscript.

The tuning parameter is b, the birthrate.

p. 37: The LPA flour beetle model.

The bifurcation diagram for parameter b is on page 39;

The bifurcation diagram for mu adult is on p. 59;

The bifurcation diagram for C pa is on p. 60.

Andy Long

The parameters initially included reproduce the bifurcation results on p. 39 of Cushing's manuscript.

The tuning parameter is b, the birthrate.

p. 37: The LPA flour beetle model.

The bifurcation diagram for parameter b is on page 39;

The bifurcation diagram for mu adult is on p. 59;

The bifurcation diagram for C pa is on p. 60.

Andy Long

- 1 year 8 months ago

#### A Simple SIR (Susceptible, Infected, Recovered) Example

##### Andrew E Long

This is a first example of a simple SIR (Susceptible, Infected, Recovered) model.

There are three pools of individuals: those who are infected (without them, no disease!), the pool of those who are at risk (susceptible), and the recovered -- who may lose their immunity and become susceptible again.

A comparable model in Mathematica is available at

http://www.nku.edu/~longa/classes/2018spring/mat375/mathematica/SIRModel.nb

There are three pools of individuals: those who are infected (without them, no disease!), the pool of those who are at risk (susceptible), and the recovered -- who may lose their immunity and become susceptible again.

A comparable model in Mathematica is available at

http://www.nku.edu/~longa/classes/2018spring/mat375/mathematica/SIRModel.nb

- 1 year 9 months ago

#### MAT 375 Midterm file: Model of Isle Royale: Predator Prey Interactions

##### Andrew E Long

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

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

- 1 year 8 months ago

#### SIR (poisoning would be better) Markov Model

##### Andrew E Long

This is an example I thought of after reading Olinick's book An Introduction to Mathematical Models in the Social and Life Sciences.

It's an SIR-type model, but one where the equilibrium (ws,wi,wr) is always the same, even as the weights in the transition matrix change.

Actually it might be better to think of this as a poisoning model: the rate of infection is constant, and independent of the existence of an infected population. That's more like disease due to an environmental effect (e.g. lead-poisoning from smelters, or mercury poisoning from the burning of coal). So infected would mean "effected", and "recovered" might be "treated" -- and ultimately released, to be exposed again.

This shows that the equilibrium does not determine the transition probabilities: two different transition matrices can have the same ultimate equilibrium.

There is a constraint on the infection rate that I haven't figured out how to build in:

InfectionRate < Min[1,wi/ws, wr/ws]

I can allow InfectionRate to vary up to 1 if I take

ws < wi

and

ws < wr

However if you violate that, you'll get interesting solutions with negative values of populations. The dynamics are pretty interesting in that case, however! If you want to see them, you'll have to remove the constraints that I put on the parameters in the Recover and LossOfImmunity parameters.

Thanks Mike! Interesting examples, as always....

Andy Long

It's an SIR-type model, but one where the equilibrium (ws,wi,wr) is always the same, even as the weights in the transition matrix change.

Actually it might be better to think of this as a poisoning model: the rate of infection is constant, and independent of the existence of an infected population. That's more like disease due to an environmental effect (e.g. lead-poisoning from smelters, or mercury poisoning from the burning of coal). So infected would mean "effected", and "recovered" might be "treated" -- and ultimately released, to be exposed again.

This shows that the equilibrium does not determine the transition probabilities: two different transition matrices can have the same ultimate equilibrium.

There is a constraint on the infection rate that I haven't figured out how to build in:

InfectionRate < Min[1,wi/ws, wr/ws]

I can allow InfectionRate to vary up to 1 if I take

ws < wi

and

ws < wr

However if you violate that, you'll get interesting solutions with negative values of populations. The dynamics are pretty interesting in that case, however! If you want to see them, you'll have to remove the constraints that I put on the parameters in the Recover and LossOfImmunity parameters.

Thanks Mike! Interesting examples, as always....

Andy Long

- 1 year 8 months ago

#### SIR Model

##### Clay Frink

- 1 year 9 months ago

#### Day 22: Isle Royale: Predator/Prey Model for Moose and Wolves

##### Jacob Englert

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.};

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.};

- 1 year 9 months ago

#### SIR Markov

##### Jacob Englert

- 1 year 8 months ago

#### Driehaus SIR Markov Chain

##### Rachel Driehaus

This is an introductory example from Olinick's book An Introduction to Mathematical Models in the Social and Life Sciences.

"A recent study focused on the relationship between the birth weights of English women and the birth weights of their daughters. The weights were split into three categories: low (below 6 pounds), average (between 6 and 8 pounds), and high (above 8 pounds). Among women whose own birth weights were low, 50 percent of the daughters had low birth weights, 45 percent had average weights, and 5 percent had high weights. Women with average birth weights had daughters with average weights half of the time, while the half was split evenly between low and high categories. Women with high birth weights had female babies with high weights 40 percent of the time, with low and average weights each occuring 30 percent of the time." p. 274-275.

For the Markov chain, you should make sure that you're taking time steps of length 1 in the settings, and Euler. RK-4 effectively looks beyond a single previous step, so it has a sort of memory!

Thanks Mike! Interesting examples, as always....

Andy Long

Next up: an SIR.

"A recent study focused on the relationship between the birth weights of English women and the birth weights of their daughters. The weights were split into three categories: low (below 6 pounds), average (between 6 and 8 pounds), and high (above 8 pounds). Among women whose own birth weights were low, 50 percent of the daughters had low birth weights, 45 percent had average weights, and 5 percent had high weights. Women with average birth weights had daughters with average weights half of the time, while the half was split evenly between low and high categories. Women with high birth weights had female babies with high weights 40 percent of the time, with low and average weights each occuring 30 percent of the time." p. 274-275.

For the Markov chain, you should make sure that you're taking time steps of length 1 in the settings, and Euler. RK-4 effectively looks beyond a single previous step, so it has a sort of memory!

Thanks Mike! Interesting examples, as always....

Andy Long

Next up: an SIR.

- 1 year 8 months ago

#### Markov Chain SIR Model 3/26

##### Sally Dufek

This is an introductory example from Olinick's book An Introduction to Mathematical Models in the Social and Life Sciences.

"A recent study focused on the relationship between the birth weights of English women and the birth weights of their daughters. The weights were split into three categories: low (below 6 pounds), average (between 6 and 8 pounds), and high (above 8 pounds). Among women whose own birth weights were low, 50 percent of the daughters had low birth weights, 45 percent had average weights, and 5 percent had high weights. Women with average birth weights had daughters with average weights half of the time, while the half was split evenly between low and high categories. Women with high birth weights had female babies with high weights 40 percent of the time, with low and average weights each occuring 30 percent of the time." p. 274-275.

For the Markov chain, you should make sure that you're taking time steps of length 1 in the settings, and Euler. RK-4 effectively looks beyond a single previous step, so it has a sort of memory!

Thanks Mike! Interesting examples, as always....

Andy Long

Next up: an SIR.

"A recent study focused on the relationship between the birth weights of English women and the birth weights of their daughters. The weights were split into three categories: low (below 6 pounds), average (between 6 and 8 pounds), and high (above 8 pounds). Among women whose own birth weights were low, 50 percent of the daughters had low birth weights, 45 percent had average weights, and 5 percent had high weights. Women with average birth weights had daughters with average weights half of the time, while the half was split evenly between low and high categories. Women with high birth weights had female babies with high weights 40 percent of the time, with low and average weights each occuring 30 percent of the time." p. 274-275.

For the Markov chain, you should make sure that you're taking time steps of length 1 in the settings, and Euler. RK-4 effectively looks beyond a single previous step, so it has a sort of memory!

Thanks Mike! Interesting examples, as always....

Andy Long

Next up: an SIR.

- 1 year 8 months ago

#### Driehaus Flakes no more!

##### Rachel Driehaus

This is an introductory example from Olinick's book An Introduction to Mathematical Models in the Social and Life Sciences.

Next up: and SIR, and his interesting model of female birth weights.

Next up: and SIR, and his interesting model of female birth weights.

- 1 year 8 months ago

#### Logistic Growth

##### Andrew E Long

This simple model demonstrates logistic growth.The differential equation looks like

y'(t)=by(t)(1-y(t)/K)

where K is the carrying capacity of the quantity y.

y'(t)=by(t) - b/K*y(t)^2

so the loss term is of the form b/K.

A comparable Mathematica file is available at

http://www.nku.edu/~longa/classes/2018spring/mat375/mathematica/LogisticGrowth-and-DecayModel.nb

y'(t)=by(t)(1-y(t)/K)

where K is the carrying capacity of the quantity y.

y'(t)=by(t) - b/K*y(t)^2

so the loss term is of the form b/K.

A comparable Mathematica file is available at

http://www.nku.edu/~longa/classes/2018spring/mat375/mathematica/LogisticGrowth-and-DecayModel.nb

- 1 year 10 months ago

#### Driehaus English Mother/Daughter Birth Weights

##### Rachel Driehaus

"A recent study focused on the relationship between the birth weights of English women and the birth weights of their daughters. The weights were split into three categories: low (below 6 pounds), average (between 6 and 8 pounds), and high (above 8 pounds). Among women whose own birth weights were low, 50 percent of the daughters had low birth weights, 45 percent had average weights, and 5 percent had high weights. Women with average birth weights had daughters with average weights half of the time, while the half was split evenly between low and high categories. Women with high birth weights had female babies with high weights 40 percent of the time, with low and average weights each occuring 30 percent of the time." p. 274-275.

For the Markov chain, you should make sure that you're taking time steps of length 1 in the settings, and Euler. RK-4 effectively looks beyond a single previous step, so it has a sort of memory!

Thanks Mike! Interesting examples, as always....

Andy Long

Next up: an SIR.

- 1 year 8 months ago

#### Clone of A Simple Infection-only SIR (Susceptible, Infected, Recovered) Example

##### Sally Dufek

This is a simple example of (part of a) simple SIR (Susceptible, Infected, Recovered) model, suggested by De Vries, et al. in A Course in Mathematical Biology.

They wanted to illustrate the comparative behavior of differential equations and discrete difference equations. We know that differential equations are generally solved numerically by discretizing them, so that the comparison is a little bit rigged....

A comparable model in Mathematica is available at

http://www.nku.edu/~longa/classes/2018spring/mat375/mathematica/SIRModel-w-discrete-version.nb

They wanted to illustrate the comparative behavior of differential equations and discrete difference equations. We know that differential equations are generally solved numerically by discretizing them, so that the comparison is a little bit rigged....

A comparable model in Mathematica is available at

http://www.nku.edu/~longa/classes/2018spring/mat375/mathematica/SIRModel-w-discrete-version.nb

- 1 year 9 months ago

#### Galla example from day 35

##### Connor Edwards

- 1 year 8 months ago

#### Driehaus Galla Example

##### Rachel Driehaus

This is an introductory example from Olinick's book An Introduction to Mathematical Models in the Social and Life Sciences.

Galla with the different classes.

Thanks Mike! Interesting examples, as always....

Andy Long

Next up: an SIR.

Galla with the different classes.

Thanks Mike! Interesting examples, as always....

Andy Long

Next up: an SIR.

- 1 year 8 months ago

#### Galla example

##### Leah Gillespie

"A recent study focused on the relationship between the birth weights of English women and the birth weights of their daughters. The weights were split into three categories: low (below 6 pounds), average (between 6 and 8 pounds), and high (above 8 pounds). Among women whose own birth weights were low, 50 percent of the daughters had low birth weights, 45 percent had average weights, and 5 percent had high weights. Women with average birth weights had daughters with average weights half of the time, while the half was split evenly between low and high categories. Women with high birth weights had female babies with high weights 40 percent of the time, with low and average weights each occuring 30 percent of the time." p. 274-275.

For the Markov chain, you should make sure that you're taking time steps of length 1 in the settings, and Euler. RK-4 effectively looks beyond a single previous step, so it has a sort of memory!

Thanks Mike! Interesting examples, as always....

Andy Long

Next up: an SIR.

- 1 year 8 months ago

#### Galla Example

##### Sally Dufek

This is an introductory example from Olinick's book An Introduction to Mathematical Models in the Social and Life Sciences.

Galla Age Distribution Model.

Thanks Mike! Interesting examples, as always....

Andy Long

Galla Age Distribution Model.

Thanks Mike! Interesting examples, as always....

Andy Long

- 1 year 8 months ago

#### Galla Example

##### Andrew E Long

This is an introductory example from Olinick's book An Introduction to Mathematical Models in the Social and Life Sciences.

Galla age distribution model.

Thanks Mike! Interesting examples, as always....

Andy Long

Next up: an SIR.

Galla age distribution model.

Thanks Mike! Interesting examples, as always....

Andy Long

Next up: an SIR.

- 1 year 8 months ago