V.L. Kocic

Global behavior of solutions of the nonlinear difference equation

We study the global asymptotic behavior of solutions of the nonautonomous difference equation where {p n } is a positive bounded sequence and the initial conditions are positive. We obtain sufficient conditions for the boundedness and persistance of solutions and for the existence of unbounded solutions. In addition, we obtain global attractivity results. The results are applied to the case when {p n } is periodic with prime period k.

Global behaviour of solutions of a nonautonomous delay logistic difference equation, II

Our aim in this paper is to investigate the asymptotic behavior of solutions of the following difference equation where {a n } is a positive bounded sequence and k is a positive integer. Sufficient conditions for boundedness and attractivity are obtained. The results are applied to the special case when {a n } is a periodic sequence with prime period p.

Global behavior of solutions of

Our aim is to investigate the boundedness character and the oscillatory nature of solutions of the difference equation where .

Global behavior of solutions of a periodically forced Sigmoid Beverton–Holt model

Our aim in this paper is to investigate the boundedness, the extreme stability, and the periodicity of positive solutions of the periodically forced Sigmoid Beverton–Holt model: where {a n } is a positive periodic sequence with period p and δ>0. In the special case when δ=1, the above equation reduces to the well-known periodic Pielou logistic equation which is known to be equivalent to the periodically forced Beverton–Holt model.

Global stability of nonlinear delay difference equations

Usung a Liapunov-Razumikhin type function, we find conditions under which the zero solution of an autonomous delay difference equation is globally asymptoticallly stable. Our results are applied to discrete logistic equation with mulitidelays and to a model in population dynamics.

Monotone unstable solutions of difference equations and conditions for boundedness

In this paper we obtain sufficient conditions for the existence of solutions of the difference equation which converge monotonically to unstable equilibria. We also obtain sufficient conditions for all solutions to be bounded and sufficient conditions for the existence of unbounded solutions.

A note on the nonautonomous delay Beverton–Holt model

It is well known that the periodic cycle of a periodically forced nonlinear difference equation is attenuant (resonant) if where {K n } is the carrying capacity of the environment and (arithmetic mean of the p-periodic cycle {t n }). In this article, we extend the concept of attenuance and resonance of periodic cycles using the geometric mean for the average of a periodic cycle. We study the properties of the periodically forced nonautonomous delay Beverton–Holt model where {K n } and {r n } are positive p-periodic sequences; (K n >0, r n >1) as well as k and l are nonnegative integers. We will show that for all positive solutions {x n } of the previous equation In particular, in the case where is a p-periodic solution of the above equation (assuming that such solution exists) and r n =r>1, the periodic cycle is g-attenuant, that is Surprisingly, the obtained results show that the delays k and l do not play any role.

Generalized attenuant cycles in some discrete periodically forced delay population models

It is well known that the periodic cycle of a periodically forced non-linear difference equation is attenuant (resonant) if , where {K n } is the carrying capacity of the environment and (arithmetic mean of the p-periodic cycle {t n }). In this paper, we introduce the concept of g-attenuance and g-resonance of periodic cycles by using the geometric mean for the average of a periodic cycle instead of the arithmetic mean. For the general class of periodically forced population models with delay where (i) ; (ii) ; (iii) (g(t) − 1) (t − 1) < 0 for t ∈ (0, ∞), t ≠ 1 and (iv) tg′(t)/g(t) is 1-1 function on (0, ∞); {K n } is p-periodic positive sequence, p ≥ 2 and k ≥ 0 are integers. We prove that the periodic cycle is g-attenuant, that is the geometric mean of the periodic cycle is less than the geometric mean of the carrying capacity: The obtained result shows that the delay k does not play any role. The results are applied to classical population models with delay.

Dynamics of a discontinuous discrete Beverton–Holt model

Our aim is to investigate the global asymptotic behaviour, oscillation, periodicity and bifurcation in discontinuous Beverton–Holt type difference equationwhere x0>0, functions k and r are discontinuous piecewise constant functionssatisfyingand h is Heaviside function.

Resonance in Beverton–Holt population models with periodic and random coefficients

An increase in the mean population density in a fluctuating environment is known as resonance. Resonance has been observed in laboratory experiments and has been studied in discrete-time population models. We investigate this phenomenon in the Beverton–Holt model with either periodic or random variables for two biologically relevant coefficients: the intrinsic growth rate and the carrying capacity. Three types of resonance are defined: arithmetic, geometric and harmonic. Conditions are derived for each type of resonance in the case of period-2 coefficients and some results for period p>2. For period 2, regions in parameter space where each type of resonance occurs are shown to be subsets of each other. For the case of random coefficients with constant intrinsic growth rate, it is shown that the three types of resonance do not occur. Numerical examples illustrate resonance and attenuance (decrease in the mean population density) in the Beverton–Holt model when the coefficients are discrete random variables.