Introduction to the Modeling and Analysis of Complex Systems

14.4. LINEAR STABILITY ANALYSIS OF REACTION-DIFFUSION SYSTEMS 291There are a few more useful predictions we can make about spontaneous patternformation in reaction-diffusion systems. Let’s continue to use the Turing model discussedabove as an example. We can calculate the actual eigenvalues of the coefficient matrix,as follows:∣ 1 − 10−4 ω 2 − λ−12 −1.5 − 6 × 10 −4 ω 2 − λ ∣ = 0 (14.114)(1 − 10 −4 ω 2 − λ)(−1.5 − 6 × 10 −4 ω 2 − λ) − (−2) = 0 (14.115)λ 2 + (0.5 + 7 × 10 −4 ω 2 )λ + (1 − 10 −4 ω 2 )(−1.5 − 6 × 10 −4 ω 2 ) + 2 = 0 (14.116)λ = 1 (− (0.5 + 7 × 10 −4 ω 2 )2± √ )(0.5 + 7 × 10 −4 ω 2 ) 2 − 4(1 − 10 −4 ω 2 )(−1.5 − 6 × 10 −4 ω 2 ) − 8 (14.117)= 1 (− (0.5 + 7 × 10 −4 ω 2 ) ± √ )2.5 × 102−7 w 4 + 2.5 × 10 −3 w 2 − 1.75 (14.118)Out of these two eigenvalues, the one that could have a positive real part is the one withthe “+” sign (let’s call it λ + ).Here, what we are going to do is to calculate the value of ω that attains the largest realpart of λ + . This is a meaningful question, because the largest real part of eigenvaluescorresponds to the dominant eigenfunction (sin(ωx + φ)) that grows fastest, which shouldbe the most visible spatial pattern arising in the system’s state. If we find out such a valueof ω, then 2π/ω gives us the length scale of the dominant eigenfunction.We can get an answer to this question by analyzing where the extremum of λ + occurs.To make analysis simpler, we let z = ω 2 again and use z as an independent variable, asfollows:dλ +dz = 1 2()− 7 × 10 −4 5 × 10 −7 z + 2.5 × 10 −3+2 √ = 0 (14.119)2.5 × 10 −7 z 2 + 2.5 × 10 −3 z − 1.757 × 10 −4 (2 √ 2.5 × 10 −7 z 2 + 2.5 × 10 −3 z − 1.75) = 5 × 10 −7 z + 2.5 × 10 −3 (14.120)1.96 × 10 −6 (2.5 × 10 −7 z 2 + 2.5 × 10 −3 z − 1.75)= 2.5 × 10 −13 z 2 + 2.5 × 10 −9 z + 6.25 × 10 −6 (14.121)(. . . blah blah blah . . .)2.4 × 10 −13 z 2 + 2.4 × 10 −9 z − 9.68 × 10 −6 = 0 (14.122)z = 3082.9, −13082.9 (14.123)Phew. Exhausted. Anyway, since z = ω 2 , the value of ω that corresponds to the dominant