Introduction to the Modeling and Analysis of Complex Systems

13.4. MODELING SPATIAL MOVEMENT 241Exercise 13.9 Plot the Laplacian of f(x, y) = sin(xy) for −4 ≤ x, y ≤ 4 usingPython. Compare the result with the outputs of the exercises above.13.4 Modeling Spatial MovementNow we will discuss how to write a PDE-based mathematical model for a dynamical processthat involves spatial movement of some stuff. There are many approaches to writingPDEs, but here in this textbook, we will use only one “template,” called the transportequation. Here is the equation:∂c∂t= −∇ · J + s (13.14)Here, c is the system’s state defined as a spatio-temporal function that represents theconcentration of the stuff moving in the space. J is a vector field called the flux of c.The magnitude of the vector J at a particular location represents the number of particlesmoving through a cross-section of the space per area per unit of time at that location.Note that this is exactly the same as the vector field v we discussed when we tried tounderstand the meaning of divergence. Therefore, the first term of Eq. (13.14) is comingdirectly from Eq. (13.12). The second term, s, is often called a source/sink term, whichis a scalar field that represents any increase or decrease of c taking place locally. Thisis often the result of influx or outflux of particles from/to the outside of the system. Ifthe system is closed and the total amount of the stuff is conserved, you don’t need thesource/sink term, and the transport equation becomes identical to Eq. (13.12).There is another way of writing a transport equation. While Eq. (13.14) explicitly describesthe amount of particles’ movement per area per unit of time in the form of fluxJ, another convenient way of describing the movement is to specify the velocity of theparticles, and then assume that all the particles move at that velocity. Specifically:∂c∂t= −∇ · (cw) + s (13.15)Here, w is the vector field that represents the velocity of particles at each location. Sinceall the particles are moving at this velocity, the flux is given by J = cw. Which formulationyou should use, Eq. (13.14) or Eq. (13.15), depends on the nature of the system you aregoing to model.As you can see above, the transport equation is very simple. But it is quite usefulas a starting point when you want to describe various kinds of spatial phenomena. One