  In this article, we find three isometric models of the Funk disc: Finsler
upper half of the hyperboloid of two sheets model, the Finsler band model and
the Finsler upper hemi sphere model; and we also find two new models of the
Finsler-Poincar\'e disc. We explicitly describe the geodesics in each model.
Moreover, we compute the Busemann function and consequently describe the
horocycles in the Funk and the Hilbert disc. Finally, we prove the asymptotic
harmonicity of the Funk disc. We also show that, the concept of asymptotic
harmonicity of the Finsler manifolds {\it tacitly} depends on the measure, in
{\it contrast} to the Riemannian case.
