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We address the global stability issue for some discrete population models with delayed-density dependence. Applying a new approach based on the concept of the generalized Yorke conditions, we establish several criteria for the convergence of all solutions to the unique positive steady state. Our results support the conjecture stated by Levin and May in 1976 affirming that the local asymptotic stability of the equilibrium of some delay difference equations (including Ricker's and Pielou's equations) implies its global stability. We also discuss the robustness of the obtained results with respect to perturbations of the model.

We address the problem of existence of periodic solutions for the differential delay equationεx˙(t)+x(t)=f(x(t-1)),00,where |m|0,B>0. We show that when the map x↦f(x) has an attracting 2-cycle then the delay differential equation has a periodic solution, which is close to the square wave corresponding to the limit (as ε→0+) difference equation x(t)=f(x(t-1)).

We present some explicit sufficient conditions for the global stability of the zero solution in nonautonomous higher order difference equations. The linear case is discussed in detail. We illustrate our main results with some examples. In particular, the stability properties of the equilibrium in a nonlinear model in macroeconomics is addressed.

Asymptotic estimates are established for higher-order scalar difference equations and inequalities the right-hand sides of which generate a monotone system with respect to the discrete exponential ordering. It is shown that in some cases the exponential estimates can be replaced with a more precise limit relation. As corollaries, a generalization of discrete Halanay-type inequalities and explicit sufficient conditions for the global exponential stability of the zero solution are given.

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.

For a scalar delayed differential equation $\dot x(t)=f(t,x_t)$, we give sufficient conditions for the global attractivity of its zero solution. Some technical assumptions are imposed to insure boundedness of solutions and attractivity of non-oscillatory solutions. For controlling the behaviour of oscillatory solutions, we require a very general condition of Yorke type, together with a 3/2-condition. The results are particularly interesting when applied to scalar differential equations with delays which have served as models in populations dynamics, and can be written in the general form $\dot x(t)=(1+x(t))F(t,x_t)$. Applications to several models are presented, improving known results in the literature., Fundo Europeu de Desenvolvimento Regional (FEDER) - project BFM2001-3884-C02-02., Spain. MCT., Fondo Nacional de Desarrollo Científico y Tecnológico (FONDECYT) - project 1030992., International project HP2003-0080., Fundação para a Ciência e a Tecnologia (FCT).

In this paper we derive new discrete Halanay-type inequalities and show some applications in the investigation of the asymptotic behavior of nonlinear di4erence equations. In particular, di4erence equations involving the “maximum” functional are considered. ? 2003 Elsevier Ltd. All rights reserved.

For a scalar Lotka–Volterra-type delay equation ẋ(t) = b(t)x(t)[1 − L(xt)], where L: C([−r, 0];R) → R is a bounded linear operator and b a positive continuous function, sufficient conditions are established for the boundedness of positive solutions and for the global stability of the positive equilibrium, when it exists. Special attention is given to the global behaviour of solutions for the case of L a positive linear operator. The approach used for this situation is applied to address the global asymptotic stability of delayed logistic models in the more general form ẋ(t) = b(t)x(t)[a(t) − L(t, xt)], with L(t, ·) being linear and positive.

EXISTENCE OF PERIODIC SOLUTIONS FOR FUNCTIONAL EQUATIONS WITH PERIODIC DELAY

We present an extension of the well-known 3/2-stability criterion by Yorke for two term functional differential equations. We prove the exact nature of the obtained stability region which coincides with the Yorke result in the special case when the decay term is absent. Moreover, we reveal some interesting links existing between the Yorke conditions, Halanay inequalities and differential equations with maxima, all of them essentially involving the maximum functionals.

We prove the existence of extremal solutions for a first order functional differential equation subject to nonlinear boundary conditions of functional type. Moreover, the functions that define our problem are allowed to be discontinuous. The proof of our main result is based on a generalized iterative technique.

We study a periodic boundary value problem for a first-order differential equation from a new point of view, which permits us to obtain an existence result involving upper and lower solutions with “jump” discontinuities. We present some applications of our result with illustrative examples.

We review some dynamical effects induced by constant effort harvesting in single-species discrete-time population models. We choose three different forms for the density-dependent recruitment function, which include the overcompensatory Ricker map for semelparous species; a modified Ricker model allowing for adult survivorship; and a model with both strong Allee effect and overcompensation which results from incorporating mate limitation in the Ricker model. We show that these simple models exhibit some interesting (and sometimes unexpected) phenomena such as the hydra effect; bubbling; sudden collapses; and essential extinction. We underline the importance of two often underestimated issues that turn out to be crucial for management: census timing and intervention time.

We discuss the global dynamics of some logistic models governed by delay-differential equations. We focus on models of exploited populations, and study the changes in the dynamics as the harvesting effort is increased. We get new results and highlight the link among different logistic equations usually employed in population models.

a b s t r a c t We study the global behavior of the price dynamics in a commodity market governed by a balance between demand and supply. While the dependence of demand on price is considered instantaneous, the supply term contains a delay, leading to a delay–differential equation. A discrete model is naturally defined as a limit case of this equation. We provide a thorough study of the discrete case, and use these results to get new sufficient conditions for the global convergence of the solutions to the positive equilibrium in the continuous case. For when the equilibrium is unstable, we provide some bounds for the amplitude of the oscillations that are quite sharp when the delay is large.

We analyze the effects of a strategy of constant effort harvesting in the global dynamics of a one-dimensional discrete population model that includes density-independent survivorship of adults and overcompensating density dependence. We discuss the phenomenon of bubbling (which indicates that harvesting can magnify fluctuations in population abundance) and the hydra effect, which means that the stock size gets larger as harvesting rate increases. Moreover, we show that the system displays chaotic behaviour under the combination of high per capita recruitment and small survivorship rates.

In this paper, we extend the result in [Y. Muroya, Persistence, contractivity and global stability in logistic equations with piecewise constant delays, J. Math. Anal. Appl. 270 (2002) 602–635] to nonautonomous logistic equations with piecewise constant arguments and establish two sufficient conditions for contractivity‐like property, from which the global asymptotic stability of the solutions of this equation is derived, even if the effect of delays dominates over the instantaneous feedback. We offer two examples to illustrate our results.

We consider a class of boundary value problems for first‐order fuzzy differential equations and prove the existence of solution under the approach of Hukuhara differentiability.

The method of lower and upper solutions provides, as well as results of existence, other important properties such as location of solution, extremal solutions,…, which have been under used and, moreover, its potential has not been optimized, either in theory either in applications. This work will present some cases to emphasize both items: two fourth order problems with functional boundary conditions (including an application to a continuous model for the deformation of the human spine under the action of some forces) and a third order periodic problem where unbounded nonlinearities are allowed, provided that an one‐sided Nagumo‐type condition is verified.

We consider populations growth models with Allee effect, proportional to beta densities with shape parameters p and 2, where the dynamical complexity is related with the Malthusian parameter r. For p>2, these models exhibit a population dynamics with natural Allee effect. However, in the case of 1

In this survey we first summarize the results on the existence of analytic normalization for general analytically integrable systems to their normal forms and for analytically integrable symplectic Hamiltonian systems to their Birkhoff normal forms. Next we summarize the results on the existence of first integrals for analytic differential systems in a neighborhood of a singularity. Also part of them is related to the existence of embedding flows of analytically integrable diffeomorphisms.

The paper investigates singular nonlinear problems arising in hydrodynamics. In particular, it deals with the problem on the half‐line of the form (p(t)u′(t))′ = p(t)f(u(t)), u′(0) = 0, u(∞) = L. The existence of a strictly increasing solution (a homoclinic solution) of this problem is proved by the dynamical systems approach and the lower and upper functions method.

In this paper we present existence and location results for two‐point boundary value problems for third and fourth order fully nonlinear differential equations.Nonlinearities are assumed to satisfy a sign‐type Nagumo condition which allows an asymmetric unbounded behavior. The arguments make use of lower and upper solutions method and degree theory.

In this paper we collect some concepts about viability theory that we can use in order to obtain some results for the existence of solutions (and, in many cases, extremal solutions) of discontinuous differential equations. Concretely, we show in this way some results for initial value problems, given in [1], functional value problems, given in [2], and we also give a new result for impulsive problems.

It is well known that as the intrinsic growth rate r increases, the Ricker map exhibits an irreversible period doubling route to chaos. If a constant positive perturbation is introduced, then there is a break of chaos giving birth to a two‐cycle. We study a similar effect in a 2‐dimensional spatial setting where each cell of the lattice is influenced by its nearest neighbours only. Chaos is not observed for r large enough, and the model can incorporate cells that experience two‐cycle oscillations with stable dynamics at some locations. A Ricker map with a negative perturbation (depletion) is discussed, as well as some generalizations of the problem.

We show how some recent work of Webb and Infante, which gave a unified method of tackling many nonlocal boundary value problems, can be applied to some higher order boundary value problems with more general nonlocal boundary conditions than previously studied. This improves some recent work on problems with conjugate type boundary conditions.

A Krasnoselskii type compression‐expansion fixed point theorem is adapted for the treatment of systems of semi‐linear equations. The compression‐expansion conditions are given componentwise which allows the nonlinear term of a system to have different behaviors both in components and in variables. Applications to boundary value problems for systems of second order differential equations are included.

A long‐standing question on possibility of construction of the Prandtl’ problem classic solution without some kind of kinematic or static hypotheses as well as on an existence of the other, “non‐classic,” solutions which the Prandtl’ hypothesis is not correct for (and which cannot be verified in real experiment), is discussed and resolved. On the basis of asymptotic analysis with natural low geometric parameter, the exact solution (in sense of finite number of asymptotic terms) coinciding with the generalized Prandtl’ solution in case of arbitrary plate roughness coefficient, is obtained. A nonapplicability of this asymptotic near the middle cross‐section of layer is precisely shown.

We study the existence of positive solutions for the differential equation u″(x) = a(x)u(x)−g(u(x)), with u′(0) = u(+∞) = 0, where a is a positive function and g satisfies some growth hypotheses. The main motivation is to check that some well known results concerning the existence of homoclinics for the autonomous case (where a is constant) extend to the nonautonomous equation.

In this note, we present a brief survey on the global attractivity of positive equilibria for multi‐species Lotka‐Volterra models with delays. For a class of Lotka‐Volterra systems with distributed delays and dominant instantaneous negative feedbacks, a sharp criterion for the global attractivity independently of the choice of delays is given. When the positive equilibrium is not a global attractor, we also address the possible existence of singularities.

A class of (1+1)‐dimensional partial differential equations of reaction‐diffusion‐convection type in nonuniform media (i.e. with nonconstant rates) is studied from the Lie symmetry point of view, by considering a variant of the Bluman‐Cole method. Explicit solutions for such differential equations, with several types of data, are obtained.

In this work, we study a nonlinear problem suggested by the Black–Scholes model for option pricing with stochastic volatility, namely, {12σ2S2∂2f∂S2+12σ2V2∂2f∂σ2+ρσ2VS∂2f∂S∂σ−12ρσ2V∂f∂σ+rS∂f∂S = rγ(f)in Ω,f(S,σ) = h(S,σ),on ∂Ω| where the variables S and σ are respectively the asset value and the market volatility ([1], [2]). In [1], an analogous nonlinear problem has been investigated with a nonlinearity γ of the following type γ(f) = g(f)f. It has been used an iterative procedure under the hypothesis that g(f)f is nondecreasing, applying upper and lower solutions. The function g was assumed to be C2 and this regularity was used in computations in the proofs.We consider a Holder continuous nonlinearity γ(f) and, assuming certain conditions on the potential Γ of γ, we prove the existence of a positive solution. The method of the proof, which is based on the construction of upper and lower solutions, obtained as solutions of an auxiliary initial value problem, also yields information on the localization of f.

In this paper, using Schauder fixed point theorem, we prove existence results concerning radial solutions for Dirichlet problems associated with some systems involving mean curvature operators in Euclidean and Minkowski spaces. The p‐Laplacian case is also considered.

In this note we explain the main motivations Norbert Wiener had for the creation of his Generalized Harmonic Analysis [13] and his Tauberian Theorems [14]. Although these papers belong to the most pure mathematical tradition, they were deeply based on some Engineering and Physics Problems and Wiener was able to use them for such diverse areas as Optics, Brownian motion, Filter Theory, Prediction Theory and Cybernetics.

We calculate the amplitude and period of the limit cycle of the following classes of generalized Lienard equations by using the method of Lopez and Lopez‐Ruiz ([2]): ẍ+e(x2−1)ẋ3+x = 0; ẍ+e(x2m−1)(ẋ)2p−1+x2n−1 = 0; ẍ+e( ∑ i = 0mb2ix2i)(ẋ)2p−1+x2n−1 = 0; ẍ+e(|x|m−1)(ẋ)2p−1+sign(x)⋅|x|n = 0; ẍ+e(x2m−1)(ẋ)2p−1+x12n−1 = 0, where m, n et p∈N. We give numerical results.

In this work, we are concerned about a singular boundary value problem for a nonlinear ordinary differential equation involving the N‐dimensional p‐laplacian. This equation may be considered as a generalized Emden‐Fowler equation and arises in models of fluid mechanics, elasticity theory and other fields of physics. The main feature of the considered boundary value problem is that it has two singularities at the endpoints of the considered interval. We analyze the asymptotic behavior of the solutions near these singularities and propose computational methods that take this behavior into consideration. Numerical examples are presented and dicussed.

In this paper, we investigate the problem of existence and nonexistence of positive solutions for the nonlinear boundary value problem of fractional order: Dαu(t)+λa(t)f(u(t)) = 0, 0

Sufficient conditions for the solvability of the problem u″(t) = l(u)(t)+F(u)(t); u(a) = 0, u(b) = u(t0) are established, where l:C([a,b];R)→L([a,b];R) is a linear bounded operator, F:C([a,b];R)→L([a,b];R) is a continuous operator, and t0∈(a,b).

One of the most studied planar polynomial differential systems are the so‐called Lienard polynomial differential equations ẍ+fn(x)ẋ+gm(x) = 0, where gm(0) = 0 and fn and gm are polynomials of degree n and m respectively. The main objective of this survey is to present the old and the new results on the limit cycles of these Lienard polynomial differential systems related with the Hilbert’s 16th problem. We also summarize the results on their algebraic limit cycles.

As can be seen from the definition of extended operations on fuzzy numbers, subtraction and division of fuzzy numbers are not the inverse operations to addition and multiplication, respectively. Hence for solving equations or system of equations, we must use methods without using inverse operators. In this paper, we propose a novel method to find the nonzero solutions of fully fuzzy linear systems (shown as FFLS). System’s parameters are Split to two groups of non positives and non negatives by solving one multi objective linear program (MOLP) and employing embedding method to transform n×n (FFLS) to 2n×2n parametric form linear system and hence, transform operations on fuzzy numbers to operations on functions. And finally, numerical examples are used to illustrate this approach.

In this paper an Improved Predictor Corrector (IPC) method to solve the “fuzzy initial value problem” is proposed. The IPC method is obtained by combining an explicit three‐step method and an implicit two‐step method. These methods are compared with existed the methods, where the proposed methods proved to have more accuracy. The convergence and stability of the proposed methods are also presented in detail. In addition, these methods are illustrated by solving some examples.

We consider two types of Fucik like spectra, which have somewhat peculiar features. The first one stretches along the bysectrix of the plane of parameters, the second one may contain separated components. Both spectra are significant for existence of solutions to nonlinear boundary value problems with nonlinearities having asymmetric asymptotics.

The maximum principles present one of the classical parts in the qualitative theory of ordinary and partial differential equations. Although assertions about the maximum principles for functional differential equations can be interpreted in a corresponding sense as analogs of corresponding classical ones in the case of ordinary differential equations, they do not imply important corollaries, reached on the basis of finite dimensional fundamental systems. For example, results associated with the maximum principles in contrast with the cases of ordinary and even partial differential equations do not add so much in problems of existence and uniqueness. In this paper we obtain the maximum principles for functional differential equations and on this basis new results on existence and uniqueness of solutions for various boundary value problems are proposed.

Two nondimensional density functional models of homogeneous nucleation are analyzed. A shooting procedure has been developed to solve the boundary value problems that define the critical density profiles in these models. The numerical procedure takes advantage of some asymptotic properties fulfilled by such density profiles. The knowledge of such an asymptotic behavior also facilitates the accurate evaluation of the corresponding nucleation barriers.

Auditory evoked cortical potentials (AECPs) have extensively been applied in studies related to diagnosis and treatment of hearing disorders as well as cognitive and behavioral mechanisms. Regarding the mechanisms of attention and habituation, numerous studies involving electroencephalographic and magnetic resonance imaging techniques, emphasize the role of prominent cortico‐subcortical brain structures as being implicated in a bidirectional processing and flux of sensory information with specific consequences for these processes. In spite of such progress, the effect of the interplay between prominent cortico‐subcortical structures reflected in AECPs remains poorly understood. To address this issue, we propose a neuronal mean field approach for the study of neural correlates of selective attention and habituation in the case of the auditory modality. Such a framework is endowed with a neurophysiological interpretation so that we can formulate hypothesis concerning the mechanisms of selective attention and habituation. It is concluded that our approach represents a useful methodology for the study of neural correlates reflected in large‐scale potentials.

In this paper, we establish sufficient conditions under which Lotka‐Volterra cooperative systems are permanent for the n‐dimensional case. We improve the result of [G. Lu and Z. Lu, Permanence for two species Lotka‐Volterra systems with delays, Math. Biol. Engi. 5 (2008), pp. 477–484] for the 2‐dimensional case in that no restrictions of the size of time delays are needed. When the interval of time delays is constant, we further show that the restriction of the size of time delays is not required for the case n = 2, but it is required for the case n⩾3 to obtain lower bounds of solutions. An example is offered to illustrate the feasibility of our results.