Introduction to the Modeling and Analysis of Complex Systems

13.4. MODELING SPATIAL MOVEMENT 245be represented by including a linear term of m with a negative coefficient in the equationfor m, so that:∂m∂t= αp − βm. (13.21)Again, β is a positive constant that determines the decay rate of the economy. Thisequation correctly shows exponential decay if p = 0, which agrees with the assumption.So far, all the assumptions implemented are about local, non-spatial dynamics. Thereforewe didn’t see any spatial derivatives.Now, Assumption 4 says we should let people and money diffuse over space. Thisis about spatial movement. It can be modeled using the diffusion equation we discussedabove (Eq. (13.17)). There is no need to add source/sink terms, so we just add Laplacianterms to both equations:∂p∂t = D p∇ 2 p (13.22)∂m∂t = D m∇ 2 m + αp − βm (13.23)Here, D p and D m are the positive diffusion constants of people and money, respectively.Finally, Assumption 5 says people can sense the “smell” of money and move towardareas where there is more money. This is where we can use the transport equation. Inthis case, we can use Eq. (13.15), because all the people at a certain location would besensing the same “smell” and thus be moving toward the same direction on average. Wecan represent this movement in the following transport term−∇ · (p γ∇m), (13.24)where the gradient of m is used to obtain the average velocity of people’s movement (withyet another positive constant γ). Adding this term to the equation for p represents people’smovement toward money.So, the completed model equations look like this:∂p∂t = D p∇ 2 p − γ∇ · (p∇m) (13.25)∂m∂t = D m∇ 2 m + αp − βm (13.26)How does this model compare to yours?Interestingly, a mathematical model that was essentially identical to our equationsabove was proposed nearly half a century ago by two physicists/applied mathematicians,