Can rate constant depend upon the concentration of reactants (or other species involved in the reaction)?
It seems possible that you mix the terms. There is the reaction rate constant -- in your equation $k_r$, and there is the rate of reaction -- in your equation $r$ and sometimes expressed by $v$.
Depending on the order of a reaction, the overall rate of reaction may, but need not depend on the concentration of the reactants available.  There are reaction types known by experience to proceed typically one kinetic order (e.g., heterogeneous hydrogenation); however, reading the stoichiometric reaction equation often does not provide sufficient information and the rate law has to be determined experimentally.
Examples of kinetic reaction order are zero, one, two, three, but fractional orders are known, too.  It is possible to perform reactions which are of second order ($v = k_r[\ce{A}][\ce{B}]$) in a pseudo-first order regime if the concentration of reagent A is so much much higher, than the one of [B], that you detect after consumption of B still much of A.  This however assumes that the reaction still proceeds by the same pathway, thus requires experimental evidence to be supported.
This contrasts to $k_r$, which is independent from the concentration of the reagents, however depends on the temperature $T$ (keyword Boltzmann, and even more, Arrhenius equation).