M. R. S. Kulenović

A coupled system of rational difference equations

Abstract We investigate the global stability properties and asymptotic behavior of solutions of the recursive sequence where the parameters a, b, c, and d are arbitrary positive numbers, and the initial conditions x0 and y0 are arbitrary nonnegative numbers.

On the recursive sequence

In this section we present some open problems and conjectures about some interesting types of difference equations. Please submit your problems and conjectures with all relevant information to G. Ladas.

Global asymptotic behavior of a two-dimensional difference equation modelling competition

We investigate the global asymptotic behavior of solutions of the system of difference equations xn+1 = xn/ a + cyn, yn+1 = yn/ b + dxn, n =0,1,..., where the parameters a and b are in (0, 1), c and d are arbitrary positive numbers and the initial conditions x0 and y0 are arbitrary nonnegative numbers. We show that the stable manifold of this system separates the positive quadrant into basins of attraction of two types of asymptotic behavior. In the case where a = b we find an explicit equation for the stable manifold.

Environmental periodicity and time delays in a “food-limited” population model

Abstract Sufficient conditions are obtained for the existence of a globally attracting positive periodic solution of the “food-limited” population system modelled by the equation N (t) = r(t)((K(t) − N(t − mω)) (K(t) + c(t)r(t) N(t − mω))) , where m is a nonnegative integer and K, r, c are continuous positive periodic functions of period ω.

Global attractivity in population dynamics

We established sufficient conditions for the global attractivity of the positive equilibrium of the delay differential equation N(t)=−mu;N(t)+∑i=1m pi exp[−γN(t−τi)], t⩾0, m⩾1. For m = 1, equation (1) was used by Wazewska-Czyzewska and Lasota as a model for the survival of red-blood cells in an animal.

A rational difference equation

Abstract We investigate the boundedness character, the periodic nature, and the global asymptotic stability of all positive solutions of the equation in the title with positive parameters and nonnegative initial conditions.

INVARIANT MANIFOLDS FOR COMPETITIVE DISCRETE SYSTEMS IN THE PLANE

Let T be a competitive map on a rectangular region , and assume T is C1 in a neighborhood of a fixed point . The main results of this paper give conditions on T that guarantee the existence of an invariant curve emanating from when both eigenvalues of the Jacobian of T at are nonzero and at least one of them has absolute value less than one, and establish that is an increasing curve that separates into invariant regions. The results apply to many hyperbolic and nonhyperbolic cases, and can be effectively used to determine basins of attraction of fixed points of competitive maps, or equivalently, of equilibria of competitive systems of difference equations. These results, known in hyperbolic case, have been used to determine the basins of attraction of hyperbolic equilibrium points and to establish certain global bifurcation results when switching from competitive coexistence to competitive exclusion. The emphasis in applications in this paper is on planar systems of difference equations with nonhyperbolic equilibria, where we establish a precise description of the basins of attraction of finite or infinite number of equilibrium points.

On the Trichotomy Character of x n +1 =(α+β x n +γ x n −1 )/( A + x n )

We show that the equation in the title with nonnegative parameters and nonnegative initial conditions exhibits a trichotomy character concerning periodicity, convergence, and boundedness which depends on whether the parameter n is equal, less, or greater than the sum of the parameters g and A .

Oscillations and global attractivity in models of hematopoiesis

AbstractLetP(t) denote the density of mature cells in blood circulation. Mackey and Glass (1977) have proposed the following equations: