The occurrence of tracking or tipping situations for a transition equation $x'=f(t,x,\G(t,x))$
with asymptotic limits $x'=f(t,x,\G_\pm(t,x))$ is analyzed. The approaching condition is just
$\lim_{t\to\pm\infty}(\G(t,x)-\G_\pm(t,x))=0$ uniformly on compact real sets, and so
there is no restriction to the dependence on time of the asymptotic equations. The hypotheses
assume concavity in $x$ either of the maps $x\mapsto f(t,x,\G_\pm(t,x))$ or of their derivatives with respect
to the state variable (d-concavity), but not of $x\mapsto f(t,x,\G(t,x))$ nor of its derivative.}
The analysis provides
a powerful tool to analyze the occurrence of critical transitions for one-parametric
families $x'=f(t,x,\G^c(t,x))$. The new approach significatively widens the field
of application of the results, since the evolution law of the transition
equation can be essentially different from those of the limit equations.
Among these applications, some scalar population dynamics models subject
to non trivial predation and migration patterns are analyzed, both theoretically and numerically.
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Some key points in the proofs are: to understand the transition equation
as part of an orbit in its hull which approaches the \upalfa-limit and
\upomeg-limit sets; to observe that these sets concentrate all the ergodic measures;
and to prove that in order to describe the dynamical possibilities of the equation
it is sufficient that the concavity or d-concavity conditions hold for a complete measure subset of the
equations of the hull.
