​Physical meaning of the equations

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.

​Physical meaning of the equations

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.

The body of research and studies generated on the Fryingpan River between the 1940s and the present supports the development of a conceptual model of ecosystem responses to hydrological regime behavior and streamflow management activities. This conceptual model should encourage conversations about system behavior and collective understanding among stakeholders regarding connections between specific hydrological regime characteristics affected by management of Ruedi Reservoir and the ecological or biological variables important to local communities. For the sake of simplicity, the model includes mostly unidirectional relationships—feedback loops are exploded to reveal intermediate connections between variables. This approach increases the number of variables represented in the system, perhaps increasing its complexity at first glance. However, the primary benefit to the end user is that the model becomes more readable and explicit in its representation of system behavior. 

 

The conceptual model presented here likely differs by degrees from those held by the various investigators who considered Fryingpan River processes over the previous 80 years. However, it affectively aggregates the ideas main presented by each of those individuals. This model focuses on hydrological and biological variables and does not incorporate the entire diversity of human uses and needs for water from the Fryingpan River (e.g. hydropower production for the City of Aspen, revenue generated in the Town of Basalt by angling activities, etc.).  Rather it attempts to illustrate how the conditional state of important ecosystem characteristics might respond to reservoir management activities that impact typical spring flows, peak flow timing and magnitude, summer flows, fall flows, and winter flows. 

​Physical meaning of the equations

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.

​Physical meaning of the equations

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.

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

Overview

This model simulates logging and mountain biking competition in Derby, Tasmania. The Simulation is referenced to simulate Derby mountain biking with logging.

 

Model Work

The tourism industry is represented on the model's left side, and the logging industry is on the right side. Interactions between these two industries generate tax revenues. Logging and tourism have different growth rates regarding people working/consuming. The initial values of these two industries in the model are not fixed but increase yearly due to inflation or economic growth.

 

Detail Insights

From the perspective of tourism, as the number of tourists keeps growing, the number of people who choose to ride in Derby City also gradually increases. And the people who ride rate the ride. The negative feedback feeds back into the cycling population. Similarly, positive cycling reviews lead to more customer visits. And all the customers will create a revenue through tourism, and a certain proportion of the income will become tourism tax.

From a logging perspective, it is very similar to the tourism industry. As the number of people working in the industry is forecast to increase, the industry's overall size is predicted to grow. And as the industry's size continues to rise, the taxes on the logging industry will also continue to rise. Since logging is an industry, the tax contribution will be more significant than the tourism excise tax.

 

This model assumption is illustrated below:

1. The amount of tax reflects the level of industrial development.

2. The goal of reducing carbon emissions lets us always pay attention to the environmental damage caused by the logging industry.

3. The government's regulatory goal is to increase overall income while ensuring the environment.

4. Logging will lead to environmental damage, which will decrease the number of tourists.

 

This model is based on tourism tax revenue versus logging tax revenue. Tourism tax revenue is more incredible than logging tax revenue, indicating a better environment. As a result of government policy, the logging industry will be heavily developed in the short term. Growth in the logging industry will increase by 40%. A growth rate of 0.8 and 0.6 of the original is obtained when logging taxes are 2 and 4 times higher than tourism taxes.

 

Furthermore, tourism tax and logging tax also act on the positive rate, which is the probability that customers give a positive evaluation. The over-development of the logging industry will lead to the destruction of environmental resources and further affect the tourism industry. The logging tax will also affect the tourism Ride Rate, which is the probability that all tourism customers will choose Derby city.

 

This model more accurately reflects logging and tourism's natural growth and ties the two industries together environmentally. Two ways of development are evident in the two industries. Compared to tourism, logging shows an upward spiral influenced by government policies. Government attitudes also affect tourism revenue, but more by the logging industry. 

​Physical meaning of the equations

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.

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

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

Experiment with adjusting the moose birth-rate to simulate Over-shoot followed by environmental recovery

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.};

A model which simulates the competition between logging versus adventure tourism (mountain bike ridding) in Derby Tasmania.  Simulation borrowed from the Easter Island simulation.

​Physical meaning of the equations

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.

​Physical meaning of the equations

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.

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

Experiment with adjusting the moose birth-rate to simulate Over-shoot followed by environmental recovery

This insight models the intertwining relationships of sea grass, Chelonia mydas (green sea turtles), and Galeocerdo cuvier (tiger sharks). Each of these populations are connected to one another through a food chain interaction, making all of the populations dependent upon each other.

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.};

http://www.nku.edu/~longa/classes/mat375/mathematica/Lotka-Volterra.nb

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.

Simple BIDE model