I'm currently taking a Partial Differential Equations class, and for one assignment we have to come up with a model for a theoretical zombie outbreak. Well anyways, this is what I have gathered thus far:
- I am defining my u(r,z,t) to be the population density of humans, where r=radius, z=zombies, and t=time.
- There will be a continuous flow in and out of humans out of the boundary.
- I am letting my boundary be a circular region, suppose a 35 meter radius.
- The population density of both zombies and humans is dependent on the radius, r, of the region. For example if you have 100 zombies in a particular radius with 50 humans, if you increase the radius then the population density decreases.
I think I may have my boundary condition where Du/Dr(35,z,t)= flux, since the normal derivative will always be the radius.
My Initial condition is u(r,z,0)= u0
Now, the PDE is where I am having trouble, I can't figure out what Du/Dt is (the rate of change of human population density with respect to time).I tried modeling it similar to the heat equation, but that doesn't work since I only have one spatial dimension in r, and no theta. As r changes as does the total density (zombies and humans) and therefore human density.