Introduction to the Modeling and Analysis of Complex Systems

13.4. MODELING SPATIAL MOVEMENT 243Plugging J = −α∇c into Eq. (13.14), we obtain∂c∂t= −∇ · (−α∇c) + s = ∇ · (α∇c) + s. (13.16)If α is a homogeneous constant that doesn’t depend on spatial locations, then we cantake it out of the parentheses:∂c∂t = α∇2 c + s (13.17)Now we see the Laplacian coming out! This is called the diffusion equation, one ofthe most fundamental PDEs that has been applied to many spatio-temporal dynamicsin physics, biology, ecology, engineering, social science, marketing science, and manyother disciplines. This equation is also called the heat equation or Fick’s second law ofdiffusion. α is called the diffusion constant, which determines how fast the diffusion takesplace.Exercise 13.10 We could still consider an alternative smoothing model in whichthe flux is given by −c(α∇c), which makes the following model equation:∂c∂t= α∇ · (c∇c) + s (13.18)Explain what kind of behavior this equation is modeling. Discuss the differencebetween this model and the diffusion equation (Eq. (13.17)).Exercise 13.11 Develop a PDE model that describes a circular motion of particlesaround a certain point in space.Exercise 13.12each other.Develop a PDE model that describes the attraction of particles toWe can develop more complex continuous field models by combining spatial movementand local dynamics together, and also by including more than one state variable.Let’s try developing a PDE-based model of interactions between two state variables: populationdistribution (p for “people”) and economic activity (m for “money”). Their local(non-spatial) dynamics are assumed as follows: