Equilibriums and cycles of some nonautonomous difference equations

Sufficient conditions of existence of positive and asymptotically stable equilibrium for nonautonomous discrete exponential predator-prey model are obtained. If
$$ r\in\Biggl(0,\frac1a+\frac1{a\sqrt{1-4a\gamma}}\Biggr),\qquad r\ne\frac1{2a}+\frac1{2a\sqrt{1-4a\gamma}}, $$
then the equation of the nonautonomous “Consensus” model
$$ x_{n+1}=x_n\exp\Bigl(r_n\Bigl(-a+\frac1{x_n}-\frac\gamma{x^2_n}\Bigr)\Bigr),\qquad r_n>0,\quad a>0,\quad\gamma>0,\quad a\gamma<\frac14, $$
has positive and asymptotically stable equilibrium.