Follow the steps below and do your work in the provided GeoGebra windows. [br][i][br]NOTE to VSC students taking this class with me[/i]: be sure you are accessing this through a GeoGebra Classroom Link! The URL of this page should have the word “classroom” in it. If not, then go back to Canvas and be sure to access this page from the Project 6 assignment. Also: I strongly recommend you login to GeoGebra.org with a free account so your work is saved, and you can come back later to review or modify it. [br][i][br]For external readers[/i]: these "project" activities are meant to be taken as part of my course, so these sections of the GeoGebra book my not be as intelligible as others. My apologies.

The primary goal of this project is to learn about nonlinear [b]Lotka-Volterra (Predator-Prey) Models[/b], a classical application of non linear first order systems of differential equations.

The Lotka-Volterra (or Predator-Prey) Models are a system of first order non-linear differential equations that model populations of two species, [math]x[/math], the prey, and [math]y[/math], a predator of the prey.[br][br]As summarized nicely in [url=https://mc-stan.org/users/documentation/case-studies/lotka-volterra-predator-prey.html]this article[/url]:[br][br][quote]The Lotka-Volterra equations (Lotka 1925; Volterra 1926, 1927) are based on the assumptions that the predator population intrinsically shrinks, the prey population intrinsically grows, a larger prey population leads to a larger predator population, and a larger predator population leads to a smaller prey population. [br][br][br]More specifically, the rate of growth of the prey population is proportional to the size of the prey population, leading to exponential growth if unchecked. The prey population simultaneously shrinks at a rate proportional to the size of the product of the prey and predator populations. For the predator species, the direction of growth is reversed. The predator population shrinks at a rate proportional to its size and grows at a rate proportional to the product of its size and the prey population’s size.[/quote]The Lotka-Volterra equations are: [br][br][math]\frac{dx}{dt}=\alpha\cdot x-\beta\cdot x\cdot y[/math][br][math]\frac{dy}{dt}=\delta\cdot x\cdot y-\gamma\cdot y[/math][br][br]When using these equations to model species' populations, the coefficients [math]\alpha,\beta,\delta,[/math] and [math]\gamma[/math] are always real positive numbers. [br][br]The traditional interpretation of the coefficients of the Lotka-Volterra equations are:[br][br]Prey coefficients:[br][math]\alpha[/math]: the natural reproduction rate of the prey[br][math]\beta[/math]: the death rate per interaction of the prey and predator[br][br]Predator coefficients:[br][math]\delta[/math]: the efficiency of the predators at turning the prey into baby predators (i.e. the rate of new baby predators per interaction of the predator and prey)[br][math]\gamma[/math]: the natural death rate of predator in the absence of food (prey)[br][br]Higher dimensional versions of the Lotka-Volterra equations have been used to model a wide variety of additional interaction phenomena. They are widely praised for their simplicity to setup along with their oftentimes chaotic response to initial conditions. (In this context, "chaos" refers to the phenomenon of a differential equation in which a small change in an initial condition leads to a substantial change in the specific solution). See the [url=https://www.geogebra.org/m/cxgtwkqa#material/tsbftxud]next activity of the book[/url] for a quick overview and a link to additional resources.[br][br]You can explore the phase portrait perspective of specific solutions of the Lotka-Volterra equations in the applet below. In the applet you can drag the green dot to different initial conditions, and the specific solution of the system of equations is estimated out to [math]t=3[/math]. You can also adjust the coefficients.

The next applet is the traditional perspective of the specific solutions. [br][br]This applet shows the specific solutions for the prey, [math]x(t)[/math], and the predators, [math]y(t)[/math], plotted against time [math]t[/math]. In this applet you can view the solutions over a longer period of time by adjusting [code]TimeFrame[/code] (the previous applet was capped at [math]t=3[/math]).[br][br][i]Note[/i]: Due to a bug in the code somewhere, you may need to "jiggle" the point [code]InitialCondition[/code] to get this applet to perform. Note that the point [code]InitialCondition[/code] is setting BOTH the initial condition on [code]x[/code] and [code]y[/code]. You can also make changes to the [code]InitialCondition[/code] in the left hand input box.

In Landmark Meadow in Stowe, which we saw [url=https://www.geogebra.org/m/cxgtwkqa#material/jumcyzry]earlier[/url], the following parameters govern the interaction of the mouse (prey) and owl (predator) populations:[br][br]Mouse (prey) coefficients:[br][math]\alpha[/math]: the natural reproduction rate of the mice (prey) is [b]0.055[/b] mice per day.[br][math]\beta[/math]: the death rate (by consumption by the owls) per interaction of the owls and mice is [b]0.004[/b] mice per day (per mouse, per owl). So, for instance, if there are 500 mice, then each owl will consume 2=0.004*500 mice. [br][br]Owl (predator) coefficients:[br][math]\delta[/math]: the efficiency of the owls at turning the mice into baby owls (i.e. the rate of new baby owls per interaction of the owls and mice) is [b]0.001[/b] owls per day (per owl per mouse). In reality, this is probably a little lower, but we can't set the constants less than 1 thousandth in this app.[br][math]\gamma[/math]: the natural death rate of owls in the absence of mice is [b]0.3[/b] owls per owl per day. Another way of understanding this coefficient, is that it is saying that if the owls have no food, 30% would die each day. In reality, this is probably a little too quick. [br][br]Suppose that there are initially 500 mice and 13 owls occupying the meadow, and a Lotka-Volterra model is run for 60 days.

If you're interested in learning more about the application of 2 dimensional Lotka-Volterra Equations, [url=https://mc-stan.org/users/documentation/case-studies/lotka-volterra-predator-prey.html]this article[/url] gives an excellent overview, and an interesting use case of the equations modeling populations of hare and lynx. It also does a good job of giving insight into how statistical methods can be brought to bear on the challenge of identifying appropriate coefficients to utilize these equations. It also discusses some adjustments to the model.