  This paper analyzes the structure of the set of positive solutions of a class
of one-dimensional superlinear indefinite bvp's. It is a paradigm of how
mathematical analysis aids the numerical study of a problem, whereas
simultaneously its numerical study confirms and illuminates the analysis. On
the analytical side, we establish the fast decay of the positive solutions as
$\lambda\downarrow -\infty$ in the region where $a(x)<0$ (see (1.1)), as well
as the decay of the solutions of the parabolic counterpart of the model (see
(1.2)) as $\lambda\downarrow-\infty$ on any subinterval of $[0,1]$ where
$u_0=0$, provided $u_0$ is a subsolution of (1.1). This result provides us with
a proof of a conjecture of [28] under an additional condition of a dynamical
nature. On the numerical side, this paper ascertains the global structure of
the set of positive solutions on some paradigmatic prototypes whose intricate
behavior is far from predictable from existing analytical results.
