Introduction to the Modeling and Analysis of Complex Systems

14.3. LINEAR STABILITY ANALYSIS OF CONTINUOUS FIELD MODELS 279we will study. As you may have done this in Exercise 14.3, any a eq and c eq that satisfyfa eq = kc eq (14.51)can be a homogeneous equilibrium state of this model. Here we denote the equilibriumstate as((a eq , c eq ) = a eq , f )k a eq . (14.52)Then we introduce small perturbations into this homogeneous equilibrium state, as follows:( ) ( ) ( )a(x, t) aeq ∆a(x, t)⇒ +. (14.53)c(x, t)∆c(x, t)fk a eqHere, we assume that the space is just one-dimensional for simplicity (therefore no y’sabove). By applying these variable replacements to the Keller-Segel model, we obtain∂∆a= µ ∂2∂t ∂x (a 2 eq + ∆a) − χ ∂ ((a eq + ∆a) ∂ ( f∂x∂x k a eq + ∆c))(14.54)= µ ∂2 ∆a− χ ∂ ()∂∆ca∂x 2eq∂x ∂x + ∆a∂∆c(14.55)∂x= µ ∂2 ∆a ∂ 2 ∆c− χa∂x 2 eq∂x − χ ∂ (∆a ∂∆c )(14.56)2 ∂x ∂x= µ ∂2 ∆a ∂ 2 ∆c− χa∂x 2 eq∂x − χ∂∆a ∂∆c2 ∂x ∂x − ∆cχ∆a∂2 ∂x , (14.57)( )( 2 )∂∆c= D ∂2 f f∂t ∂x 2 k a eq + ∆c + f (a eq + ∆a) − kk a eq + ∆c(14.58)= D ∂2 ∆c+ f∆a − k∆c. (14.59)∂x2 In the equations above, both the second-order and first-order spatial derivatives remain.We can’t find an eigenfunction that eliminates both simultaneously, so let’s adopt sinewaves, i.e., the eigenfunction for the second-order spatial derivatives that appear moreoften in the equations above, and see how the product of two first-order spatial derivativesin Eq. (14.57) responds to it. Hence, we will assume( ∆a(x, t)∆c(x, t))= sin(ωx + φ)( ∆a(t)∆c(t)), (14.60)where ω and φ are parameters that determine the spatial frequency and phase offsetof the perturbation, respectively. ω/2π will give a spatial frequency (= how many waves