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From solution (7.19) and transformation (7.18) we can see that as time goes to infinity, the solution x(i) tends to the normalized eigenvector corresponding to the largest of the eigenvalues ao,... ,an.

The exact solution corresponding to the initial condition z(0) = (1,0,..., 0)T can be found. The appropriate coefficients in equation (7.19)

For the ith component, we obtain,

The short-time behavior of this quantity is given by

Zi(t) = ( bk J t\ 1 <i<n, ajt < 1 Vj. \k=l

The expression for zn+1 can also be obtained but is slightly more cumbersome.

Mutation cascades. Let us assume that ao < an, and in addition we have at — a^i ~ a^ ~ 1. Then the system exhibits the following behavior (Figure 7.3).

Starting from the "all Xo" state, the fraction of Xq goes down steadily, and the population acquires some amount of x\ (they may or may not be the majority). Upon reaching a maximum, the fraction of xj decreases and the fraction of x2 experiences a "hump", to be in turn replaced by xs, etc. The characteristic time at which each type experiences its maximum abundance can be estimated if we replace the expressions for Zj(t) in (7.21) by the leading term, i.e. the term which has the largest exponential, so that

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