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Regardless of the value of c, when we linearize around (1,0) one eigenvalue will be positive and one will be negative. We thus conclude that (1, 0) is a saddle point; the isocline analysis tells us that one of the directions moving away from (1,0) moves towards (0, 0), as shown in Figure 2.22c.

Now, the situation around (0, 0) is a little bit more complicated. First, note that if c < 2, the eigenvalues will be complex, with a negative real part. This means that (0, 0) will be a stable focus. However, if the origin is a focus, trajectories will spiral around it -which includes visiting values of U that are negative. Now, W can be negative, since it is the derivative of U, but U itself cannot be negative. We thus conclude on a biological basis that c > 2, in which case the origin is a stable node. As time increases, trajectories will approach the origin in the direction of the smaller (in absolute value) of the two eigenvalues.

Exercise 2.18 (E)

The condition c > 2 pertains to the scaled time and space variables. How does it translate to the original variables, involving the strength of selection (or maximum per capita growth rate for the ecological case) and the diffusion coefficient?

For a traveling wave to exist, a trajectory must come out of the saddle point at (1, 0) and go directly into the stable node at (0, 0) along the smaller of the two eigenvalues. The results of Exercise 2.11 tell us that the vector which joins the point (1, 0) and (1 — e , — e[(—c + Vc2 + 4)/2]) is the eigenvector corresponding to the positive eigenvalue at (1, 0). We move to the point (1 — e , —e[(—c + Vc2 + 4)/2]) and integrate Eq. (2.64) forward in time, given c > 2. In this way, we construct the trajectory that comes out of the saddle point directly into the origin, along its eigenvector (Figure 2.22d).

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