Introduction to the Modeling and Analysis of Complex Systems

280 CHAPTER 14. CONTINUOUS FIELD MODELS II: ANALYSISthere are per unit of length), which is called a wave number in physics. The phase offsetφ doesn’t really make any difference in this analysis, but we include it anyway for thesake of generality. Note that, by adopting a particular shape of the perturbation above,we have decoupled spatial structure and temporal dynamics in Eq. (14.60) 1 . Now theonly dynamical variables are ∆a(t) and ∆c(t), which are the amplitudes of the sine waveshapedperturbation added to the homogeneous equilibrium state.By plugging Eq. (14.60) into Eqs. (14.57) and (14.59), we obtain the following:sin(ωx + φ) ∂∆a∂tsin(ωx + φ) ∂∆c∂t= −µω 2 sin(ωx + φ)∆a + χa eq ω 2 sin(ωx + φ)∆c− χω 2 cos 2 (ωx + φ)∆a∆c + χω 2 sin(ωx + φ)∆a∆c (14.61)= −Dω 2 sin(ωx + φ)∆c + f sin(ωx + φ)∆a − k sin(ωx + φ)∆c(14.62)Here, we see the product of two amplitudes (∆a∆c) in the last two terms of Eq. (14.61),which is “infinitely smaller than infinitely small,” so we can safely ignore them to linearizethe equations. Note that one of them is actually the remnant of the product of the twofirst-order spatial derivatives which we had no clue as to how to deal with. We should beglad to see it exiting from the stage!After ignoring those terms, every single term in the equations equally contains sin(ωx+φ), so we can divide the entire equations by sin(ωx + φ) to obtain the following linearordinary differential equations:d∆a= −µω 2 ∆a + χa eq ω 2 ∆cdt(14.63)d∆c= −Dω 2 ∆c + f∆a − k∆cdt(14.64)Or, using a linear algebra notation:( ) (d ∆a −µω2χa=eq ω 2dt ∆c f −Dω 2 − k) ( ∆a∆c)(14.65)We are finally able to convert the spatial dynamics of the original Keller-Segel model(only around its homogeneous equilibrium state) into a very simple, non-spatial linear1 In mathematical terms, this is an example of separation of variables—breaking down the original equationsinto a set of simpler components each of which has fewer independent variables than the originalones. PDEs for which separation of variables is possible are called separable PDEs. Our original Keller-Segel model equations are not separable PDEs, but we are trying to separate variables anyway by focusingon the dynamics around the model’s homogeneous equilibrium state and using linear approximation.