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In this paper we investigate the presence of oscillations and simple dynamics in $x'(t)=-\nu x(t)+f(x(t))g(x(t-\tau))$. Specifically, we prove that the global attraction towards a nontrivial equili...

In this paper we extend the classical “decomposing+embedding” method for systems of delay differential equations. Our extension has two advantages: (1) enhancing the range of applicability of the classical method and (2) providing delay dependent criteria of global attraction. The leading ideas are, on the one hand, the extension of the notion of dominance introduced first for difference equations and, on the other hand, the use of some monotone ideas developed by H.L. Smith and collaborators. We apply our results to some classical models of population dynamics, mainly models with patch structure.

The oscillation of a first order differential equation with several non-monotone delays is studied. A standard technique is improved and used to prove some new oscillation criteria of lim sup type. The applicability and strength of the obtained results are assured using an analytic approach as well as a practical example.

Consider the first-order linear differential equation with several retarded arguments x ′ ( t ) + ∑ k = 1 n p k ( t ) x ( τ k ( t ) ) = 0 , t ≥ t 0 , $$\begin{array}{} \displaystyle x^{\prime }(t)+\sum\limits_{k=1}^{n}p_{k}(t)x(\tau _{k}(t))=0,\;\;\;t\geq t_{0}, \end{array} $$ where the functions pk , τk ∈ C([t 0, ∞), ℝ+), τk (t) < t for t ≥ t 0 and lim t→∞ τk (t) = ∞, for every k = 1, 2, …, n. Oscillation conditions which essentially improve known results in the literature are established. An example illustrating the results is given.

In this paper we deduce criteria of global attraction in systems of delay differential equations. Our methodology is new and consists in “dominating” the nonlinear terms of the system by a scalar function and then studying some dynamical properties of that function. One of the crucial benefits of our approach is that we obtain delay-dependent results of global attraction that cover the best delay-independent conditions. We apply our results in a gene regulatory model and the classical Nicholson's blowfly equation with patch structure.

We study the global attractivity of the unique positive equilibrium of a population model with distributed delays and nonlinear death rate. Both delay dependent and delay independent criteria are obtained which generalize, unify and improve known criteria. These results will be applied to some models with bounded and unbounded death functions.

We study the dynamics of the differential equation $$u'(t)=-\gamma u(t)+ b f(u(t-\tau))- c f(u(t-\sigma))$$ with two delayed terms, representing a positive and a negative feedback. We prove delay-dependent and absolute global stability results for the trivial and for the positive equilibrium. Our theorems provide new mathematical results as well as novel insights for several biological systems, including three-stage populations, neural models with inhibitory and excitatory loops, and the blood platelet model of Belair and Mackey. We show that, somewhat surprisingly, the introduction of a removal term with fixed delay in population models simplifies the dynamics of the equation.

New global attractivity criteria are obtained for the second-order difference equation via a Lyapunov-like method. Some of these results are sharp and support recent related conjectures. Also, a necessary and sufficient condition for the oscillation of this equation is obtained using comparison with a second-order linear difference equation with positive coefficients.

This paper contains new estimates for the distance between adjacent zeros of solutions of the first order delay differential equation x ′ ( t ) + p ( t ) x ( t − τ ) = 0 where p is a nonnegative continuous function. Our results are new and improve some known criteria. Illustrative examples are given to show the strength of some obtained results over the known ones.

In this paper, we study the rational difference equation where p, , and are sequences of positive integers such that , for all and . It is proved that this equation has either one or two equilibria. The unique equilibrium point is proved to be the global attractor of all solutions. The dynamics, when the system has two equilibria, is also investigated.

We provide further insight on the dynamics of Clark's delayed recruitment equation, depending on the relevant parameters involved in the model. We pay special attention to the stability and bifurcations from the positive equilibrium, and to the existence and attraction properties of nontrivial cycles. A detailed analysis is worked out for a three-dimensional example.

The global attractivity character of nonlinear higher order difference equations of the form is investigated when g is dominated by an interval scalar map. Some basic properties of the interval map are obtained and used to prove new global attractivity criteria for the above equation with no monotonicity restrictions on g. Our results are applied to many models from mathematical biology and economy. The derived global attractivity criteria of these models are either new or improve substantially known ones.

New explicit sufficient conditions for the asymptotic stability of the zero solution of higher order difference equations are obtained. These criteria can be applied to autonomous and nonautonomous equations. The celebrated Clark asymptotic stability criterion is improved. Also, applications to models from mathematical biology and macroeconomics are given.

We address the global stability properties of the positive equilibrium in a general delayed discrete population model. Our results are used to investigate in detail a well-known model for baleen whale populations.

New oscillation results are obtained for the second order nonlinear difference equation Δ ( r n f ( Δ x n − 1 ) ) + g ( n , x n ) = 0 , and its functional form Δ ( r n f ( Δ x n − 1 ) ) + g ( n , x τ n ) = 0 . The role played by the argument τ n on the oscillation of the functional equation is explored. In particular, we characterize a class of sequences { τ n } which have a harmless effect on the oscillation of this type of equations. Some of our results generalize, improve or unify known fundamental oscillation results for several particular cases of the above equations.

For a family of difference equations xnþ1 ¼ axn þ f ðxn2kÞ; n ¼ 0; 1; ... ; wherea [ ð0; 1Þ; k [ {1; 2; ... }; and f : ½0; 1Þ ! ð0; 1Þ is continuous and decreasing, we find sufficient conditions for the convergence of all solutions to the unique positive equilibrium. In particular, we improve, up to our knowledge, all previous results on the global asymptotic stability of the equilibrium in the particular cases of the discrete Mackey – Glass and Lasota–Wazewska models in blood-cells production.

The dynamical characteristics of scalar difference equations of the form¶xn+1=f1(xn−τ1)+f2(xn−τ2), n=0,1,2,...,¶are investigated. A necessary and sufficient condition is obtained for all positive solutions to be oscillatory about a unique positive equilibrium point and sufficient criteria for the global attractivity of the equilibrium are established. Also, the stability and periodicity of more general equations are studied via comparison with the corresponding properties of an associated first-order non-linear equation.

A nonlinear delay integral equation modeling the response of a nenron under external stimulus and threshold effects in investigated. We study the global attractivity of the equilibrium as well as the existence of a pcriodic solutions of the system. Sufficient Conditions for oscillation (nonoscillation) of all solutions are obtained.

This paper is concerned with nonlinear difference equations of the formwhere δ is the forward difference operator defined by δun−1 =un−un−1 δ2un−1= δ(δun-1) and {an} is a real sequence which is not assumed to be nonnegative. The functionfis such thatuf(u) < 0 for allu≠ 0 andf(u) −f(v) =g(u, v)(u − v), for allu, v≠ 0, and for some nonnegative functiong. Our results are not only new but also improve and generalise some recent oscillation criteria. Examples illustrating the importance of our main results are also given.

In this paper we are concerned with some new criteria for the oscillation and nonoscillation of the second-order nonhomogeneous linear difference equations of the form Δ( c n − 1 Δ x n − 1 ) + q n x n = f n , n = 1, 2,…, where { c n }, { f n }, and { q n } are real sequences, c n > 0 for n ≥ 0, and Δ x n = x n + 1 − x n is the forward difference operator. The discrete analogs of some of the known results in the continuous case are presented.

In this paper we investigate the oscillatory character of the second order nonlinear difference equations of the forms ∆(cn−1∆(xn−1 + pnxσn)) + qnf(xτn) = 0, n = 1, 2, . . . and the corresponding nonhomogeneous equation ∆(cn−1∆(xn−1 + pnxσn)) + qnf(xτn) = rn, n = 1, 2, . . . via comparison with certain second order linear difference equations where the function f is not necessarily monotonic. The results of this paper are essentially new and can be extended to more general equations.

We investigate the oscillatory behavior of all solutions of a new class of first order nonlinear neutral difference equations. Several explicit oscillation criteria are established. Our main results are supported by illustrative examples.

The global attractivity of the unique positive equilibrium of the equation x n +1 = g ( x n − k 1 , x n − k 2 , …, x n − k m ), n = 0,1,2,…, is investigated where g is a nonincreasing continuous function in each of its arguments and the k are positive integers. We are able to use our results to extract some important global attractivity criteria of the positive equilibria of the discrete Clark model x n +1 = αx n + f ( x n − k ) and the rational recursive sequence x n+1 = (a + bx n ) (A + x n−1 ) as well.