= Solution
For an autonomous scalar differential equation, <linear stability> of a simple <equilibrium point> $N_e$ is determined by the <derivative> $f_N(N_e,a)=P'(N_e)$: a negative <derivative> attracts nearby solutions and a positive <derivative> gives <repulsion in a dynamical system> of nearby solutions.
Here $f(0,a)=1+a\ge1/2>0$ throughout the prescribed <mathematical model parameter> range, while $f(N,a)\to-\infty$ as $N\to\infty$. If there is only one <equilibrium point>, the <function graph> must cross the axis on one of its decreasing branches, outside the interval between its two extrema. The crossing is simple and $P'(N_e)<0$. Thus \b[the unique <equilibrium point> is linearly <asymptotically stable>]. At a <saddle-node bifurcation> there are two distinct <equilibrium points>, one of them double, so that case does not fall under the one-equilibrium hypothesis.
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