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14: Continuous Field Models II - Analysis
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- 14.1: Finding Equilibrium States
- One nice thing about PDE-based continuous ﬁeld models is that, unlike CA models, everything is still written in smooth differential equations so we may be able to conduct systematic mathematical analysis to investigate their dynamics (especially their stability or instability) using the same techniques as those we learned for non-spatial dynamical systems in Chapter 7. The ﬁrst step is, as always, to ﬁnd the system’s equilibrium states.
- 14.2: Variable Rescaling
- Variable rescaling of continuous ﬁeld models comes with yet another bonus variable, i.e., space, which you can rescale to potentially eliminate more parameters from the model. In a 2-D or higher dimensional space, you can, theoretically, have two or more spatial variables to rescale independently. But the space is usually assumed to be isotropic (i.e., there is no difference among the directions of space) in most spatial models, so it may not be practically meaningful to use different rescaling
- 14.3: Linear Stability Analysis of Continuous Field Models
- We can apply the linear stability analysis to continuous ﬁeld models. This allows us to analytically obtain the conditions for which a homogeneous equilibrium state of a spatial system loses its stability and thereby the system spontaneously forms non-homogeneous spatial patterns. Note again that the homogeneous equilibrium state discussed here is no longer a single point, but it is a straight line (or a ﬂat plane) that covers the entire spatial domain.
- 14.4: Linear Stability Analysis of Reaction-Diffusion Systems
- You may have found that the linear stability analysis of continuous ﬁeld models isn’t as easy as that of non-spatial models. For the latter, we have a very convenient tool called the Jacobian matrices, and the stability analysis is just calculating a Jacobian matrix and then investigating its eigenvalues. Everything is so mechanistic and automatic, compared to what we went through in the previous section. You may wonder, aren’t there any easier shortcuts in analyzing the stability of continuous