Geodesic
Equilibriums and cycles of some nonautonomous difference equations
Matematičeskoe modelirovanie, Tome 21 (2009) no. 3, pp. 120-126

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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. 
@article{MM_2009_21_3_a9,
     author = {A. V. Lasunsky},
     title = {Equilibriums and cycles of some nonautonomous difference equations},
     journal = {Matemati\v{c}eskoe modelirovanie},
     pages = {120--126},
     publisher = {mathdoc},
     volume = {21},
     number = {3},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MM_2009_21_3_a9/}
}
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A. V. Lasunsky. Equilibriums and cycles of some nonautonomous difference equations. Matematičeskoe modelirovanie, Tome 21 (2009) no. 3, pp. 120-126. http://geodesic.mathdoc.fr/item/MM_2009_21_3_a9/