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.

This paper investigates the dynamics of non-autonomous competitive 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 the evolutionary population of competition models of two species.

<jats:p>We present the bifurcation results for the difference equation <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"> <msub> <mrow> <mi>x</mi> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow> <msubsup> <mi>x</mi> <mi>n</mi> <mn>2</mn> </msubsup> </mrow> <mo>/</mo> <mfenced open="(" close=")" separators="|"> <mrow> <mi>a</mi> <msubsup> <mi>x</mi> <mi>n</mi> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <mi>f</mi> </mrow> </mfenced> </math> </jats:inline-formula> where <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2"> <mi>a</mi> </math> </jats:inline-formula> and <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M3"> <mi>f</mi> </math> </jats:inline-formula> are positive numbers and the initial conditions <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M4"> <msub> <mrow> <mi>x</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msub> </math> </jats:inline-formula> and <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M5"> <msub> <mrow> <mi>x</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> </math> </jats:inline-formula> are nonnegative numbers. This difference equation is one of the perturbations of the sigmoid Beverton–Holt difference equation, which is a major mathematical model in population dynamics. We will show that this difference equation exhibits transcritical and Neimark–Sacker bifurcations but not flip (period-doubling) bifurcation since this difference equation cannot have period-two solutions. Furthermore, we give the asymptotic approximation of the invariant manifolds, stable, unstable, and center manifolds of the equilibrium solutions. We give the necessary and sufficient conditions for global asymptotic stability of the zero equilibrium as well as sufficient conditions for global asymptotic stability of the positive equilibrium.</jats:p>

We use the epidemic threshold parameter, R0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {R}}}_{0}$$\end{document}, and invariant rectangles to investigate the global asymptotic behavior of solutions of the density-dependent discrete-time SI epidemic model where the variables Sn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{n}$$\end{document} and In\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_{n}$$\end{document} represent the populations of susceptibles and infectives at time n=0,1,…\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n = 0,1,\ldots $$\end{document}, respectively. The model features constant survival “probabilities” of susceptible and infective individuals and the constant recruitment per the unit time interval [n,n+1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[n, n+1]$$\end{document} into the susceptible class. We compute the basic reproductive number, R0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {R}}}_{0}$$\end{document}, and use it to prove that independent of positive initial population sizes, R0<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {R}}}_{0}<1$$\end{document} implies the unique disease-free equilibrium is globally stable and the infective population goes extinct. However, the unique endemic equilibrium is globally stable and the infective population persists whenever R0>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {R}}}_{0}>1$$\end{document} and the constant survival probability of susceptible is either less than or equal than 1/3 or the constant recruitment is large enough.

It is shown that locally asymptotically stable equilibria of planar cooperative or competitive maps have basin of attraction \begin{document}$ \mathcal{B} $\end{document} with relatively simple geometry: the boundary of each component of \begin{document}$ \mathcal{B} $\end{document} consists of the union of two unordered curves, and the components of \begin{document}$ \mathcal{B} $\end{document} are not comparable as sets. The boundary curves are Lipschitz if the map is of class \begin{document}$ C^1 $\end{document} . Further, if a periodic point is in \begin{document}$ \partial \mathcal{B} $\end{document} , then \begin{document}$ \partial\mathcal{B} $\end{document} is tangential to the line through the point with direction given by the eigenvector associated with the smaller characteristic value of the map at the point. Examples are given.