PROJECT 1

                  ADAM C. KNAPP

               Monday, February 18

For the following project, everyone should choose exactly one of the suggested projects in the sections below. (If you have your own idea for a project which covers similar material, feel free to suggest it.)

In either case, you should create a document which discusses the model you created and the choices that were involved. It should include any data or graphs that you generate and any Matlab code, in an appendix. Be sure to thoroughly explore your model for interesting behavior!

              I do expect you to come to offiffiffice hours for this project!

       1. Finite Element Method for the 1D Wave Equation

The wave equation is partial difffferential equation (PDE) which describes how any of a variety of waves (pressure, light, etc.) travel through a medium or space. A general form, where φ is a function of time, t, and a collection of spatial coordinates, x1, . . . , xN, is

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Although this is a fairly simple PDE with many known features and solution techniques, it is still fundamentally an infifinite dimensional problem since φ lies inside a function space.

Our goal will be to create a simplifified, fifinite dimensional, model by approximating the spatial dimension by a fifinite number of points. Essentially, our model is that of a linkage of springs (governed by Hooke’s law) as shown in Figure 1.

This is a (simplifified) case of what is called the “Finite Element Method”.

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1.1. Minimum Tasks.

• Find a collection of parameters and variables to describe the system. Write down any relations between the variables. For example, in Figure 1 I have added two fifixed walls which lead to a constraint on the total lengths of the springs.

• Write down the forces acting on a mass and translate this to a 1st order ODE on phase space.

• Implement the ODE in Matlab.

• Examine a collection of initial values.

• Plot solutions as positions of masses.

1.2. Suggested Explorations.

• See if you can get waves to reflflect offff of the walls.

• What happens when the masses are difffferent on the left and right?

• What happens when the spring constants are difffferent on the left and right?

                     2. Multi-species predator-prey

In this project you will modify the Lotka-Volterra equations to model several species. Recall the equations from class:

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with parameters a, b, c, d > 0. Here x gives the prey population and y the predator population. The parameters a, c give self-growth rates while b, d give species interaction effffects.

You will add several more species, each a predator or prey and add interaction terms

2.1. Minimum Tasks.

• Update the model to have one predator species and two prey species. Explore what happens when the interaction terms between the predator and the two prey species diffffer.

• Similarly, explore the effffects of two predator species on one prey species.

2.2. Suggested Explorations.

• What if a predator had a predator?

• Explore the effffects of a “W” shaped ecology. That is, two prey species and three predators. One predator can consume both prey species while the other two can only consume one. Do changes in population for the predators at the ends of the W propagate through the system? In what way?

• Can you create the unstable system described in the following article?

https://www.quantamagazine.org/how-nature-defifies-math-in-keeping-ecosystems-stable-20180926/

i.e. An ecosystem with two species whose self-reinforcing benefifit to each other drive all other species to go extinct?

American University, Washington, DC 20016

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