Introduction to the Modeling and Analysis of Complex Systems

256 CHAPTER 13. CONTINUOUS FIELD MODELS I: MODELINGlations, a escaping from diffusing b:∂a∂t = −µ a∇ · (a(−∇b)) (13.44)∂b∂t = µ b∇ 2 b (13.45)Here µ a and µ b are the parameters that determine the mobility of the two species. Theseequations can be expanded and then discretized for a 2-D space, as follows (you shouldalso try doing this all by yourself!):⎛∂a∂t = −µ ⎜a ⎝∂∂x ∂∂y⎞⎟⎠T ⎛⎜⎝−a ∂b∂x−a ∂b∂y⎞( ( ∂= µ a a ∂b )+ ∂ (a ∂b ))∂x ∂x ∂y ∂y( ∂a ∂b= µ a∂x ∂x + a ∂2 b∂x + ∂a )∂b2 ∂y ∂y + a ∂2 b∂y( 2 ∂a ∂b= µ a∂x ∂x + ∂a )∂b∂y ∂y + a∇2 b⎟⎠ (13.46)(13.47)(13.48)(13.49)a(x, y, t + ∆t) ≈ a(x, y, t)( a(x + ∆h, y, t) − a(x − ∆h, y, t) b(x + ∆h, y, t) − b(x − ∆h, y, t)+ µ a +2∆h2∆ha(x, y + ∆h, t) − a(x, y − ∆h, t) b(x, y + ∆h, t) − b(x, y − ∆h, t)+ a(x, y, t)×2∆h2∆h)b(x + ∆h, y, t) + b(x − ∆h, y, t) + b(x, y + ∆h, t) + b(x, y − ∆h, t) − 4b(x, y, t)∆t∆h 2a ′ C ≈ a C + µ a(aR − a L2∆hb R − b L2∆h+ a U − a D2∆hb U − b D2∆h(13.50))+ a b R + b L + b U + b D − 4b CC∆t∆h 2(13.51)Note that I used simpler notations in the last equation, where subscripts (C, R, L, U, D)represent states of the central cell as well as its four neighbors, and a ′ C is the next valueof a C . In the meantime, the discretized equation for b is simply given byb ′ C ≈ b C + µ bb R + b L + b U + b D − 4b C∆h 2 ∆t. (13.52)