The paper deals with the analysis of the structure of the set of positive solutions of \[ -\Delta u=\lambda u-a(x)|u^p|u\quad \text{in }\Omega,\qquad u|_{\partial\Omega}=0, \] where \(\Omega\) is a bounded \(C^2\) domain of \({\mathbb R}^n\), \( N\geq 1,\) \(\lambda \in {\mathbb R}\) is regarded as a continuation parameter and \(p\in (0,\infty).\) The problem possesses a unique positive solution which is linearly asymptotically stable if the trivial state is linearly unstable and the model admits some positive solution.