Run Brilli, Run!

Brilli the ant is trapped in a $2014 \times 2014$ grid, along with $n$ anteaters!

All the anteaters, and Brilli herself, can only stay on the vertices of the unit squares. In each move, Brilli and the anteaters can,

• either stay in the same vertex,
• move to a vertex which is at a distance of $1$ unit from the current vertex.

In each move, Brilli moves first, and the anteaters move after Brilli.

Brilli faces a tragic end if an anteater comes to the same vertex as her. Brilli does her best to survive.

Given that Brilli and all the anteaters are Brilliant and try to reach their goals, what is the minimum value of $n$ for which the anteaters definitely win?