Introduction to the Modeling and Analysis of Complex Systems

224 CHAPTER 12. CELLULAR AUTOMATA II: ANALYSIS1.00.80.60.40.20.00.0 0.2 0.4 0.6 0.8 1.0Figure 12.7: Cobweb plot of Eq. (12.13).obtained analytically as follows:p c = Φ(p c ) = p 4 c + 4p 3 c(1 − p c ) + 4p 2 c(1 − p c ) 2 (12.14)0 = p c (p 3 c + 4p 2 c(1 − p c ) + 4p c (1 − p c ) 2 − 1) (12.15)0 = p c (1 − p c )(−p 2 c − p c − 1 + 4p 2 c + 4p c (1 − p c )) (12.16)0 = p c (1 − p c )(−1 + 3p c − p 2 c)) (12.17)p c = 0, 1,3 ± √ 52Among those solutions, p c = (3 − √ 5)/2 ≈ 0.382 is what we are looking for.(12.18)So, the bottom line is, if the tree density in the forest is below 38%, the burned area willremain small, but if it is above 38%, almost the entire forest will be burned down. You cancheck if this prediction made by the renormalization group analysis is accurate or not bycarrying out systematic simulations. You should be surprised that this prediction is prettygood; the percolation indeed occurs for densities above about this threshold!