We consider the following system of difference equations: where , , , , are positive constants and are initial conditions. This system has interesting dynamics and it can have up to seven equilibrium points as well as a singular point at , which always possesses a basin of attraction. We characterize the basins of attractions of all equilibrium points as well as the singular point at and thus describe the global dynamics of this system. Since the singular point at always possesses a basin of attraction this system exhibits Allee’s effect.

We consider the following system of difference equations: xn+1=axn21+xn2+cyn, yn+1=byn21+yn2+dxn, n=0,1,… , where a, b, c, d are positive constants and x0,y0≥0 are initial conditions. This system has interesting dynamics and can have up to nine equilibrium points. The most complex and perhaps most interesting case is the case of nine equilibrium points, four of which are local attractors, four of which are saddle points, and one of which is a repeller. Using recent results of Kulenović and Merino we are able to characterize the basins of attractions of all local attractors and thus to describe the global dynamics of this system. This case can be considered as a two-dimensional version of the Allee effect for competitive systems. MSC:39A10, 39A30, 37G35.

We investigate global dynamics of the following systems of difference equations: {xn+1=b1xn2A1+yn2,yn+1=a2+c2yn2xn2,n=0,1,2,…, where the parameters b1, a2, A1, c2 are positive numbers and the initial condition y0 is an arbitrary nonnegative number and x0 is a positive number. We show that this system has rich dynamics which depends on the part of a parametric space. We find precisely the basins of attraction of all attractors including the points at ∞. MSC:39A10, 39A30, 37E99, 37D10.

, ...,0 } are nonnegative real numbers. We investigate the asymptotic behavior of the solutions of the considered equation. We give easy-to-check conditions for the global stability and global asymptotic stability of the zero or positive equilibrium of this equation.

We investigate the local stability and the global asymptotic stability of the difference equation , with nonnegative parameters and initial conditions such that , for all . We obtain the local stability of the equilibrium for all values of parameters and give some global asymptotic stability results for some values of the parameters. We also obtain global dynamics in the special case, where , in which case we show that such equation exhibits a global period doubling bifurcation.

We investigate the local and global character of the equilibrium and the local stability of the period-two solution of the difference equation xn+1=βxnxn−1+γxn−12+δxnBxnxn−1+Cxn−12+Dxn where the parameters β, γ, δ, B, C, D are nonnegative numbers which satisfy B+C+D>0 and the initial conditions x−1 and x0 are arbitrary nonnegative numbers such that Bxnxn−1+Cxn−12+Dxn>0 for all n≥0. MSC:39A10, 39A11, 39A30.

By using the KAM theory we investigate the stability of equilibrium solutions of the Gumowski-Mira equation: x n+1 = (2ax n)/(1 + x n 2) − x n−1, n = 0,1,…, where x −1, x 0 ∈ (−∞, ∞), and we obtain the Birkhoff normal forms for this equation for different equilibrium solutions.

We investigate the basins of attraction of equilibrium points and minimal period-two solutions of the difference equation of the form x n+1 = x n−1 2/(ax n 2 + bx n x n−1 + cx n−1 2), n = 0,1, 2,…, where the parameters a,  b, and  c are positive numbers and the initial conditions x −1 and x 0 are arbitrary nonnegative numbers. The unique feature of this equation is the coexistence of an equilibrium solution and the minimal period-two solution both of which are locally asymptotically stable.

Consider the difference equation where and the initial conditions are real numbers. We investigate the existence and nonexistence of the minimal period-two solution of this equation when it can be rewritten as the nonautonomous linear equation , where and the functions . We give some necessary and sufficient conditions for the equation to have a minimal period-two solution when .