Introduction to the Modeling and Analysis of Complex Systems

14.4. LINEAR STABILITY ANALYSIS OF REACTION-DIFFUSION SYSTEMS 289can take a positive value for some z > 0. g(z) can be rewritten asg(z) = −D u D v(z − aD v + dD u2D u D v) 2+ (aD v + dD u ) 24D u D v− det(A). (14.104)There are two potential scenarios in which this polynomial can be positive for some z > 0,as shown in Fig. 14.5.g(z)g(z)0z0zFigure 14.5: Two possible scenarios in which g(z) can take a positive value for somez > 0. Left: When the peak exists on the positive side of z. Right: When the peakexists on the negative side of z.If the peak exists on the positive side of z (aD v + dD u > 0; Fig. 14.5 left), the onlycondition is that the peak should stick out above the z-axis, i.e.(aD v + dD u ) 24D u D v− det(A) > 0. (14.105)Or, if the peak exists on the negative side z (aD v + dD u < 0; Fig. 14.5 right), the conditionis that the intercept of g(z) should be positive, i.e.,g(0) = − det(A) > 0, (14.106)but this can’t be true if the original non-spatial model is stable. Therefore, the only possibilityfor diffusion to destabilize the otherwise stable system is the first case, whosecondition can be simplified toaD v + dD u > 2 √ D u D v det(A). (14.107)