= Solution
A steady population $N_*$ is a root $h(N_*)=0$. To linearize one needs $h$ differentiable near that root, an extra local regularity assumption beyond the source's continuity. Setting $N=N_*+\eta$ gives $\dot\eta=h'(N_*)\eta+o(\eta)$, hence $\eta(t)=\eta(0)e^{h'(N_*)t}$ at linear order.
Thus \b[$h'(N_*)<0$ gives linear asymptotic stability and $h'(N_*)>0$ gives instability]. If $h'(N_*)=0$, <linearization> is inconclusive. In a one-dimensional <phase line>, the more general nonlinear test is whether the <vector field> points towards the root from both sides. At the physical boundary $N=0$, only the right-hand side matters. Continuity alone does not guarantee uniqueness of trajectories, so a standard well-defined flow also presumes a local uniqueness condition, such as local Lipschitz continuity.
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