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 office hours for this project!

- Finite Element Method for the 1D Wave Equation

The wave equation is partial differential 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

∂2φ

∂t2

= ∆φ = XNi=1

∂2

∂x2i φ

The version with one spatial dimension simplifies to

∂2

∂t2φ(t, x) = ∂2

∂x2φ(t, x)

Although this is a fairly simple PDE with many known features and solution techniques, it is still

fundamentally an infinite dimensional problem since φ lies inside a function space.

Our goal will be to create a simplified, finite dimensional, model by approximating the spatial

dimension by a finite 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 (simplified) case of what is called the “Finite Element Method”.

Figure 1. Crude drawing of a collection of masses and springs, with two fixed walls

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 fixed 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 reflect off of the walls.

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

• What happens when the spring constants are different 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:

x0 = ax x bxy

y0 = cy + dxy

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 effects.

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 differ.

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

2.2. Suggested Explorations.

• What if a predator had a predator?

• Explore the effects 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-defies-math-in-keeping-ecosystems-stable-20180926/

i.e. An ecosystem with two species whose self-reinforcing benefit to each other drive all other

species to go extinct?

American University, Washington, DC 20016

Email address: knapp@american.edu