We investigate the asymptotic behavior of a proposed ordinary differential equation (ODE) model for Genetic Toggle switches from Gardner et. al. and I. Rajapakse and S. Smale: dxdt=a1+ym−x and dydt=b1+xn−y where a,b,m,n>0 and x(t),y(t)≥0. We also investigate the asymptotic behavior of the Euler discretization of this system: xn+1=a1xn+b11+ynm=f(xn,yn) and yn+1=a2yn+b21+xnn=g(xn,yn), where 1−h=a1, 1−k=a2, ah=b1 and bk=b2, a1,a2∈(0,1) and h,k>0 are steps of discretizations. Here, x and y represent protein concentrations at a particular time in both genes and a,b,m,n>0, respectively, above. We will apply the theory of competitive maps to find the basins of attractions of different equilibrium points and period-two solutions of systems of difference equations.

In this paper we give a characterization of monotone discrete systems of equations in terms of associated signature matrix and give some properties of certain invariant surfaces of codimension 1, which often give the boundary of attraction of some fixed points. We present several examples that illustrate our results in the case of k-dimensional systems where $ k \geq 3 $ k≥3.

The third-order difference equation yn+1=a1yn21+yn2+a2yn−121+yn−12+a3yn−221+yn−22, as a potential discrete time model of population dynamics with three generation involved, is studied. The parts of the basins of attraction of three equilibrium points that this equation admits are described. Some results about period-two and period-three solutions have been established.

This paper investigates the dynamics of non-autonomous cooperative systems of difference equations with asymptotically constant coefficients. We are mainly interested in global attractivity results for such systems and the application of such results to evolutionary population cooperation models. We use two methods to extend the global attractivity results for autonomous cooperative systems to related non-autonomous cooperative systems which appear in recent problems in evolutionary dynamics.

This paper investigates the rate of convergence of a certain mixed monotone rational second-order difference equation with quadratic terms. More precisely we give the precise rate of convergence for all attractors of the difference equation $x_{n+1}=\frac{Ax_{n}^{2}+Ex_{n-1}}{x_{n}^{2}+f}$, where all parameters are positive and initial conditions are non-negative.The mentioned methods are illustrated in several characteristic examples. 2020 Mathematics Subject Classification. 39A10, 39A20, 65L20.

We investigate the global character of the difference equation of the form $$ x_{n+1} = f(x_n, x_{n-1},\ldots, x_{n-k+1}), \quadn=0,1, \ldots $$ with several equilibrium points, where $f$ is increasing in all its variables. We show that a considerable number of well known difference equations can be embeded into this equation through the iteration process. We also show that a negative feedback condition can be used to determine a part of the basin of attraction of different equilibrium points, and that the boundaries of the basins of attractions of different locally asymptotically stable equilibrium points are in fact the global stable manifolds of neighboring saddle or non-hyperbolic equilibrium points.   2000 Mathematics Subject Classification. 39A10, 39A11

We investigate the period-two trichotomies of solutions of the equation $$x_{n+1} = f(x_{n}, x_{n-1},x_{n-2}), \quad n=0, 1, \ldots $$ where the function $f$ satisfies certain monotonicity conditions. We give fairly general conditions for period-two trichotomies to occur and illustrate the results with numerous examples.   1991 Mathematics Subject Classification. 39A10, 39A11

This paper investigates an autonomous discrete-time glycolytic oscillator model with a unique positive equilibrium point which exhibits chaos in the sense of Li–Yorke in a certain region of the parameters. We use Marotto’s theorem to prove the existence of chaos by finding a snap-back repeller. The illustration of the results is presented by using numerical simulations.

Sufficient conditions are given for planar cooperative maps to have the qualitative global dynamics determined solely on local stability information obtained from fixed and minimal period-two points. The results are given for a class of strongly cooperative planar maps of class $ C^1 $ C1 on an order interval. The maps are assumed to have a finite number of strongly ordered fixed points, and also the strongly ordered minimal period-two points. Some applications are included.