Dr

The steady states of Eq. (2.64) are (U, W) = (0, 0) and (1, 0). This is very handy, since we know that U = 1 corresponds to r !—1 and U = 0 corresponds to r !i. The isoclines are also easy to compute: the line W = 0 (i.e. the U-axis) is the isocline for U and the parabola W = [—U(1 — U)]/c is the isocline for W. These are shown in Figure 2.22a and b respectively. Our next step will be to characterize the steady states.

Show that the eigenvalues of Eq. (2.64) when linearized around (1, 0) are 2 = (—c ± x/c2 + 4)/2 and when linearized around (0, 0) are

Figure 2.22. Analysis of the traveling wave solution of the Fisher equation. (a) The isocline for U is the line W = 0.

(b) Isocline for W is the parabola W = [-U(1 - U)]/c.

(c) The eigenvalue analysis tells us that (1,0) is a saddle point and that if c > 2 that (0,0) is a stable node. (d) The traveling wave comes out of the saddle point and moves towards the stable node.

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