  In this paper a possibility of the accelerated expansion of the large scale
universe with a varying polytropic fluid of a certain type is presented. About
a special role of non-gravitational interactions between dark energy and dark
matter, in particular, about a possibility of improvement and solution of
problems arising in modern cosmology, has been discussed for a long time. This
motivates us to consider new models, where non-gravitational interactions
between varying polytropic fluid and cold dark matter are allowed. Mainly
non-linear interactions of a specific type is considered, found in recent
literature. In order to understand the behavior of suggested cosmological
models, besides cosmographic analysis, $Om$ analysis is applied. Moreover, with
different datasets, including a strong gravitational lensing dataset, the
observational constraints on the model parameters are obtained using $\chi^{2}$
analysis.
