With the linear analysis we can only determine the small amplitude initial behaviour of u and v about the uniform steady state when the steady state is driven unstable to spatially heterogenous perturbations. These spatially inhomogeneous solutions initially grow exponentially and are clearly not valid for all time. For the class of problems here we can carry out a nonlinear asymptotic analysis and obtain the solutions to O(e) (and in principle to higher orders but the algebra is prohibitive) which are valid for all time. As mentioned the details of the procedure are given by Zhu and Murray (1995) for several pattern formation mechanisms including a diffusion-chemotaxis one. For the more complex chemotaxis mechanism the analysis has been carried out by Tyson (1996). Here we sketch the analytical procedure, namely, a multi-scale asymptotic analysis, for determining small perturbation solutions valid for all time of the system of equations (5.41) and (5.42) the general forms of which are (5.44) and (5.45). We start by writing u = u* + u = u* + (eu1 + e2u2 + eiu3 + •••) v = v* + V = v* + (ev1 + e2v2 + e3v3 +----), (5.66)
where (u*, v*) is the spatially homogeneous steady state which depends on the model parameters and which, as we saw in the last section, can be driven unstable as a parameter passes through the bifurcation value which results in spatially unstable solutions. We scale time by writing
Consider equation (5.44). Substituting the expansions (5.66) and (5.67) into the individual terms, we get d û ^ d Û „2 „2 ^ — = ci—, V2û = V2 û dt d T
tV ■ [ûx(v)Vv] = aV ■ [ (û* + Û)(x* + x*v + 1 x„v„v2 + = û*x*V2v + (û*x*Vv + x*VÛ) ■ Vi + (û* 1 x*vV(v2) + x*V(Ûv)) ■vv + ■
f (Û, v) = f * + ( f** + f**)Û + 2 ( f* + f*)û2 + f* Û v
The first two expressions have linear terms in u,, while the last two expressions have linear, quadratic, cubic and so on with higher-order terms in u and v; f * = 0 by definition of the steady state. So (5.44) transforms into another equation with linear, quadratic and higher-order terms in u and v. Equation (5.45) transforms in an equivalent way. In general then, (5.44) and (5.45) with (5.67) and (5.68) take the form
and where * denotes evaluation at the steady state (u*, v*).
The quantities AH, Q(u) and C(H), represent the linear, quadratic and cubic terms respectively, of the expansion of the chemotaxis and reaction functions about the steady state. The matrices Au and D were determined above in the linear analysis. We are interested in situations where the model parameters have particular values such that X = 0. This occurs when the parameters are defined by (5.60); we call this set a critical parameter set. Basically, it means the parameter set is sitting on the boundary between growing spatially heterogeneous solutions and spatially homogeneous solutions.
If we now perturb one of the model parameters a, say, which can be any one of the parameters in (5.44) and (5.45), about its value in a critical set, the eigenvalue for temporal growth becomes k(a) = k(ac ) +
since X(ac) = 0 by definition of the critical parameter ac. We take the perturbation to be such that dk2 d a
= O, a c a c a c so that the perturbation effect on the solution eXt+lk'x is restricted to a change in the temporal growth rate X. Then, if the change in ac makes Re(X(a)) positive, the pattern mode corresponding to kC is predicted to grow according to linear theory. Depending on the parameters, the result can be a stable or unstable spatially heterogeneous solution. If this growth is sufficiently slow, we can predict whether or not it will develop into a temporally stable pattern and furthermore, what the characteristics of the pattern will be such as spots or stripes. We start by perturbing the steady state model (5.69) about the critical set. To keep the analysis simple we perturb only one parameter, and to keep the analysis general we call the parameter a. Tyson (1996) carried out the analysis with the actual parameters from the model equations and it is her results we give below.
Consider an expansion of the form a = ac + a = ac + (sa\ + e a2 + •••)• Substituting this into (5.69) we get the system
+ higher-order terms,
where the superscript c denotes evaluation at the critical set. The change in the critical parameter a only occurs in a and so its effect can be isolated in the analysis.
Substituting the expansions for all of the small variables ( ), and collecting and equating terms of like order in e, we obtain systems of equations for each order in e. For notational simplicity, the superscript c is omitted in the result, and for the remainder of the analysis all parameter values are from a critical set. To show what these equations look like we just give the O (e) and O (e2) systems although to carry out the nonlinear analysis it is necessary to also have the O (e3) system which is algebraically extremely complicated. We do not need them to sketch the procedure. The O(e) equations are
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