  The occurrence of tracking or tipping situations for a transition equation
$x'=f(t,x,\Gamma(t,x))$ is analyzed under the assumptions on concavity in $x$
either of the maps giving rise to the asymptotic equations
$x'=f(t,x,\Gamma_\pm(t,x))$ or of their derivatives with respect to the state
variable (d-concavity), but without assuming these conditions on the transition
equation itself. The approaching condition is just
$\lim_{t\to\pm\infty}(\Gamma(t,x)-\Gamma_\pm(t,x))=0$ uniformly on compact real
sets, and so there is no restriction to the dependence on time of the limit
equations. The analysis provides a powerful tool to analyze the occurrence of
critical transitions for one-parametric families $x'=f(t,x,\Gamma^c_\pm(t,x))$.
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.
  Some key points in the proofs are: to understand the transition equation as
part of an orbit in its hull which approaches the $\alpha$-limit and
$\omega$-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 suffices that the concavity or d-concavity conditions hold for
a complete measure subset of the equations of the hull.

Dynamical Systems