= Solution
\b[Yes: there is a quasistatic <hysteresis> loop.] <Equilibrium points> lie on the <mathematical model parameter> curve $a=-P(N)$. Its two <saddle-node bifurcations> occur at the <stationary points> computed in part (a), at <mathematical model parameter> values
$$
a_{\rm down}=-P(N_M)=-\frac5{54}-\frac{7\sqrt7}{108}\simeq-0.264076,
$$
$$
a_{\rm up}=-P(N_m)=-\frac5{54}+\frac{7\sqrt7}{108}\simeq0.0788913.
$$
Between these values there are two <attracting> branches separated by a <repelling> branch. Starting below $a_{\rm down}$ and increasing $a$ slowly, a solution follows the lower <attracting> branch. This branch persists through the <bistability> interval and ends only at $a_{\rm up}$, where it meets the <repelling> branch in a <saddle-node bifurcation>. The solution then moves to the upper <attracting> branch.
If the direction of <mathematical model parameter> change is reversed, the upper branch persists until $a_{\rm down}$, where a different <saddle-node bifurcation> forces the downward transition. The two transition values differ, and both lie inside the allowed range $[-1/2,1/2]$. Thus the <biological population> selected at an intermediate <mathematical model parameter> value depends on its prior branch. This is the <equilibrium point> <hysteresis> shown in the branch diagram; finite-rate passage can additionally delay the jumps near the <saddle-node bifurcations>.
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