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We investigate the global character of the difference equation of the form

xn+1 = f (xn, xn–1), n = 0,1, . . .

with several period-two solutions, where f is increasing in all its variables. We show that the boundaries of the basins of attractions of different locally asymptotically stable equilibrium solutions or period-two solutions are in fact the global stable manifolds of neighboring saddle or non-hyperbolic equilibrium solutions or period-two solutions. As an application of our results we give the global dynamics of three feasible models in population dynamics which includes the nonlinearity of Beverton-Holt and sigmoid Beverton-Holt types.

MSC: 39A10; 39A20; 37B25; 37D10

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Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 License.