  Non-autonomous differential equations exhibit a highly intricate dynamics,
and various concepts have been introduced to describe their qualitative
behavior. In general, it is rare to obtain time dependent invariant compact
attracting sets when time goes to plus infinity. Moreover, there are only a few
papers in the literature that explore the geometric structure of such sets. In
this paper we investigate the long time behaviour of cooperative
$n$-dimensional non-autonomous Lotka-Volterra systems is population dynamics.
We provide sufficient conditions for the existence of a globally stable
(forward in time) entire solution in which one species becomes extinct, or
where all species except one become extinct. Furthermore, we obtain the precise
geometrical structure of the non-autonomous forward attractor in one, two, and
three dimensions by establishing heteroclinic connections between the globally
stable solution and the semi-stable solutions in cases of species permanence
and extinction. We believe that understanding time-dependent forward attractors
paves the way for a comprehensive analysis of both transient and long-term
behavior in non-autonomous phenomena.
