Referring to Fig. 5, which of the following statements is true about the different equilibria?

Fig. 5

Question

Referring to Fig. 5, which of the following statements is true about the different equilibria?

Fig. 5

BlackTom AI · Accepted Answer

For a one-dimensional iterative map X_{t+1} = F(X_t) the relevant stability concept is local return after a small perturbation: an equilibrium X* is stable if nearby states move back toward X*, unstable if they move away, and semi‑stable if one side returns while the other side departs. A useful diagnostic is the slope of F at the fixed point: if |F'(X*)| < 1 small perturbations decay (stable); if |F'(X*)| > 1 they grow (unstable); if F'(X*) = 0 or the curve is tangent at X* the linear test is inconclusive and one checks one‑sided behavior (semi‑stable). Step‑by‑step reasoning using the geometry in Fig. 5: 1. Point A lies on the left, rising limb of the hump. Geometrically F increases there and crosses the reference level such that when X is slightly below A the image F(X) is larger than X (so X_{t+1} > X_t, the population grows back toward A), and when X is slightly above A the image F(X) is smaller than X (so X_{t+1} < X_t, the population falls back toward A). Both perturbations return to A, therefore A is locally attracting — stable. This corresponds to a slope magnitude less than 1 at A. 2. Point B is at the top (the peak) where the graph is locally flat (or where the curve crosses in the direction that causes perturbations to move away on both sides). A tiny perturbation either side of B moves the state away from B (the sign of X_{t+1}-X_t is such that both left and right perturbations do not return). That one‑sided / two‑sided divergence identifies B as unstable. 3. Point C lies on the right, descending limb of the hump. The curve is tangent in such a way that a perturbation on one side is pushed back toward C while a perturbation on the other side moves away. That one‑sided attraction/repulsion is the definition of semi‑stable. Why the other choices are incorrect: - Any choice that labels A as unstable and C as stable contradicts the directional sign analysis around the intersections (left limb vs right limb give opposite restoring behavior). - Any choice that makes B stable contradicts the fact that B, at the peak, does not attract perturbations on both sides. - Any option claiming all unstable or all stable ignores the one‑sided behavior at the peak and the opposite stability properties of the two limbs. Therefore the correct classification is: Point C is semi‑stable, Point B is unstable, and Point A is stable, which matches the provided answer.

Referring to Fig. 5, which of the following statements is true about the different equilibria? Fig. 5单项选择题

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