Introduction to the Modeling and Analysis of Complex Systems

13.6. REACTION-DIFFUSION SYSTEMS 263Exercise 13.17 Conduct simulations of the Turing pattern formation with severaldifferent parameter settings, and discuss how the parameter variations (especiallyfor the diffusion constants) affect the resulting dynamics.Exercise 13.18 Discretize the Keller-Segel slime mold aggregation model(Eqs. (13.27) and (13.28)) (although this model is not a reaction-diffusion system,this is the perfect time for you to work on this exercise because you can utilizeCode 13.8). Implement its simulation code in Python, and conduct simulationswith µ = 10 −4 , D = 10 −4 , f = 1, and k = 1, while varying χ as a control parameterranging from 0 to 10 −3 . Use a = 1 and c = 0 as initial conditions everywhere, withsmall random perturbations added to them.Belousov-Zhabotinsky reaction The Belousov-Zhabotinsky reaction, or BZ reactionfor short, is a family of oscillatory chemical reactions first discovered by Russian chemistBoris Belousov in the 1950s and then later analyzed by Russian-American chemist AnatolZhabotinsky in the 1960s. One of the common variations of this reaction is essentiallyan oxidation of malonic acid (CH 2 (COOH) 2 ) by an acidified bromate solution, yet this processshows nonlinear oscillatory behavior for a substantial length of time before eventuallyreaching chemical equilibrium. The actual chemical mechanism is quite complex, involvingabout 30 different chemicals. Moreover, if this chemical solution is put into a shallowpetri dish, the chemical oscillation starts in different phases at different locations. Interplaybetween the reaction and the diffusion of the chemicals over the space will result inthe self-organization of dynamic traveling waves (Fig. 13.18), just like those seen in theexcitable media CA model in Section 11.5.A simplified mathematical model called the “Oregonator” was among the first to describethe dynamics of the BZ reaction in a simple form [50]. It was originally proposedas a non-spatial model with three state variables, but the model was later simplified tohave just two variables and then extended to spatial domains [51]. Here are the simplified“Oregonator” equations:ɛ ∂uu − q= u(1 − u) −∂t u + q fv + D u∇ 2 u (13.58)∂v∂t = u − v + D v∇ 2 v (13.59)