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This paper is concerned with the existence, nonexistence and minimal wave speed of traveling waves of a nonlocal dispersal delayed SIR model with constant external supplies and Holling-II incidence rate. We find that the existence and nonexistence of traveling waves of the system are not only determined by the minimal wave speed c ∗ , but also by the so-called basic reproduction number R 0 of the corresponding reaction system. That is, we establish the existence of traveling waves for R 0 > 1 and each wave speed c ⩾ c ∗ , and the nonexistence for R 0 > 1 and any 0 < c < c ∗ or R 0 < 1 . We also discuss how the latency of infection and the spatial movement of the infective individuals affect the minimal wave speed. Biologically speaking, the longer the latency of infection in a vector is, the slower the disease spreads. [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with a delayed three-species Lotka–Volterra food chain model with diffusion effects and homogeneous Neumann boundary conditions. By taking the sum of delays as the bifurcation parameter, spatially homogeneous and nonhomogeneous Hopf bifurcations at the positive constant steady state are proved to occur for a sequence of critical values of the delay parameter. In addition, sufficient conditions for global asymptotic stability of the positive constant steady state are derived by using an iteration technique. Furthermore, in order to determine the direction of spatially homogeneous Hopf bifurcation and the stability of bifurcated periodic solutions, the formulas are given by using the normal form theory and the center manifold reduction for PFDEs. Finally, to verify our theoretical predictions, some numerical simulations are also included. [Copyright &y& Elsevier]

Abstract: We establish the uniqueness of pulsating wave fronts for reaction–diffusion equations in time periodic media with monostable nonlinearities. For the Kolmogorov–Petrovsky–Piskunov (KPP) type nonlinearity, this result provides a complete classification of all types of KPP pulsating fronts. [Copyright &y& Elsevier]

Abstract: This paper is concerned with two-species spatial homogeneous and inhomogeneous predator–prey models with Beddington–DeAngelis functional response. For the spatial homogeneous model, the asymptotic behavior of the interior equilibrium and the existence of Hopf bifurcation of nonconstant periodic solutions surrounding the interior equilibrium are considered. Furthermore, the direction of Hopf bifurcation and the stability of bifurcated periodic solutions are investigated. For the model with no-flux boundary conditions, Turing instability of the interior equilibrium solution is studied. In particular, Turing instability region regarding the parameters is established. Finally, to verify our theoretical results, some numerical simulations are also included. [Copyright &y& Elsevier]

Abstract: The main aim of this paper is to study the existence and global exponential stability of periodic solution for high-order bidirectional associative memory (BAM) neural networks with and without impulses. Easily verifiable sufficient conditions are established. The method is based on coincidence degree theory as well as a priori estimates and Lyapunov functional. It is shown that the convergence characteristics of periodic solution for the impulsive system are preserved by the corresponding nonimpulsive system with some restriction imposed on the impulse effect. Numerical simulation results are given to support the theoretical predictions. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the uniqueness and globally exponential stability of traveling wave solutions in a lattice delayed differential equation for a single species with two age classes and a fixed maturation period living in a two-dimensional spatial unbounded environment. Under bistable assumption and realistic assumption on the birth function, we construct various pairs of super- and subsolutions and employ the squeezing technique to prove that the equation has a uniqueness traveling-wave solution. Moreover, the wave front is globally asymptotic stable with phase shift. Comparing with the previous results, we consider the symmetry of the traveling wave solutions. Our results can be expended to the case with multi-dimension lattice. [Copyright &y& Elsevier]

Abstract: Let be a time scale such that . By the Schauder fixed-point theorem and the upper and lower solution method, we present some existence criteria of the positive solution of m-point singular p-Laplacian dynamic equation with boundary conditions , where with , is continuous for and nonincreasing if . The nonlinear term may be singular in its dependent variable and is allowed to change sign. Our results are new even for the corresponding differential and difference equations . As an application, an example is given to illustrate our result. [Copyright &y& Elsevier]

Abstract: In this paper, a delayed predator–prey epidemiological system with disease spreading in predator population is considered. By regarding the delay as the bifurcation parameter and analyzing the characteristic equation of the linearized system of the original system at the positive equilibrium, the local asymptotic stability of the positive equilibrium and the existence of local Hopf bifurcation of periodic solutions are investigated. Moreover, we also study the direction of Hopf bifurcations and the stability of bifurcated periodic solutions, an explicit algorithm is given by applying the normal form theory and the center manifold reduction for functional differential equations. Finally, numerical simulations supporting the theoretical analysis are also included. [Copyright &y& Elsevier]

Abstract: In this paper, we study the global stability of the difference equation , where the parameters for , , and . We prove that the unique positive equilibrium is globally asymptotically stable if and only if it is locally asymptotically. Also we provide sufficient condition for it to be globally asymptotically stable and our results solve the open problem proposed by Kulenović and Ladas (Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman & Hall/CRC, Boca Raton, 2002). [Copyright &y& Elsevier]

Abstract: Sufficient conditions are obtained for the existence of periodic positive solution of the neutral impulsive delay predator–prey system. This is the first time that positive solution for impulsive delay predator–prey system is obtained by using the theory of coincidence degree. Our results in this paper indicate that under the appropriate linear periodic impulsive perturbations, the neutral impulsive delay predator–prey system preserves the original periodicity of the neutral nonimpulsive delay predator–prey system. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the boundedness, periodic character, invariant intervals and the global asymptotic stability of all nonnegative solutions of the difference equationwhere a, b, A, B are positive real numbers, k ⩾1 is a positive integer, and the initial conditions x −k ,…, x −1, x 0 are nonnegative real numbers. It is shown that the zero equilibrium of this equation is globally asymptotically stable if b ⩽ A − a and the unique positive equilibrium is globally asymptotically stable if A − a < b < A + a. The results obtained solve a open problem proposed by Kulenovic and Ladas [M.R.S. Kulenovi, G. Ladas, Dynamics of Second Order Rational Difference Equations, Chapman & Hall/CRC, Boca Raton, FL, 2002, p. 129]. [Copyright &y& Elsevier]

Abstract: Let be a time scale such that , a i ⩾0 for i =1,…, m −2. Let ξ i satisfy 0< ξ 1 < ξ 2 <⋯< ξ m−2 < ρ(T) and . We consider the following p-Laplacian m-point boundary value problem on time scaleswhere a ∈ C ld ((0, T),[0,∞)) and f ∈ C ld ((0, T)×[0,∞),[0,∞)). Some new results are obtained for the existence of at least twin or triple positive solutions of the above problem by applying Avery-Henderson and Leggett-Williams fixed point theorems respectively. In particular, our criteria extend and improve some known results. [Copyright &y& Elsevier]

Abstract: This paper is concerned with a delayed predator–prey system with same feedback delays of predator and prey species to their growth, respectively. We prove that a sequence of Hopf bifurcations occur at the positive equilibrium as the delay increases monotonously from zero. By using the theory of normal norm and center manifold reduction, an explicit algorithm for determining the direction of Hopf bifurcations and the stability of bifurcating periodic solutions is derived. In addition, the global existence results of periodic solutions bifurcating from Hopf bifurcations are established by using a global Hopf bifurcation result due to Wu [J. Wu, Symmetric functional differential equations and neural networks with memory, Trans. Amer. Math. Soc. 350 (1998) 4799–4838]. Finally, a numerical example supporting our theoretical prediction is also given. Our findings are contrasted with recent studies on a delayed predator–prey system with the feedback time delay of prey species to its growth by Song and Wei [Y. Song, J. Wei, Local Hopf bifurcation and global periodic solutions in a delayed predator-prey system, J. Math. Anal. Appl. 301 (2005) 1–21]. As the feedback time delay τ increases monotonously from zero, the positive equilibrium of the latter switches k times from stability to instability to stability. In contrast, the positive equilibrium of our system appears to lose the above property. [Copyright &y& Elsevier]

Abstract: In this paper the global asymptotic stability of the nonlinear difference equationis investigated, where α, β, A, B, C >0 are real numbers, and the initial conditions x −1 is nonnegative real numbers and x 0 is a positive real number. We show that the unique positive equilibrium of the equation is globally asymptotically stable. In particular, our results solve two conjectures proposed by Kulenovic and Ladas in their monograph [M.R.S. Kulenovic, G. Ladas, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman & Hall/CRC, Boca Raton, 2002] and by Kulenovic et al. in their paper [M.R.S. Kulenovic, G. Ladas, L.F. Martins, I.W. Rodrigues, The dynamics of facts and conjectures, Comput. Math. Appl. 45 (2003) 1087–1099]. [Copyright &y& Elsevier]

In this paper, we investigate the periodic character, invariant intervals, oscillation and global stability of all positive solutions of the equationwhere the parameters p and q and the initial conditions x-k,…,x0 are nonnegative real numbers. As might be expected, the two cases p⩽q and p>q give rise to different dynamic behaviors. In particular, our results solve the open problem proposed by Kulenvic and Ladas in their monograph [Dynamics of Second Order Rational Difference Equations: with Open Problems and Conjectures, Chapman & Hall/CRC, Boca Raton, 2001]. [Copyright &y& Elsevier]

By using the continuation theorem base on Gaines and Mawhin''s coincidence degree, sufficient and realistic conditions are obtained for the global existence of positive periodic solutions for a discrete time nonautonomous food chain model, and sufficient conditions for the global stability of positive periodic solutions are also obtained. Some known results are extended. [Copyright &y& Elsevier]

Necessary and sufficient conditions are obtained for the existence of positive solutions of a second-order half-linear dynamic equation on time scales. Relations between this equation and an advanced type dynamic equation are also discussed. [Copyright &y& Elsevier]

In this paper, the global asymptotic stability, the global attractivity, the boundedness character, the periodic nature and chaotic behavior of solutions of the difference equationare investigated, where α∈R is a real number, and the initial conditions x-1, x0 are arbitrary real numbers. As might be expected, the six cases α<-1, α=-1, -1<α⩽1, 1<α⩽2, 2<α<3 and α>3 give rise to substantially different dynamic behavior. [Copyright &y& Elsevier]

In this paper, we study the qualitative behavior of solutions of the discrete population modelwhere p∈(0,1), q,r∈(0,∞), p, m∈(0,∞) and k is a nonnegative integer. We obtain sufficient and necessary conditions for the oscillation of all eventually positive solutions about the positive equilibrium. Furthermore, we also show that such model is uniformly persistent, and that all its eventually positive solutions are bounded. Finally, we prove that the unique positive equilibrium x* is globally asymptotically stable if and only if x* is locally asymptotically stable and provide sufficient condition for x* to be globally asymptotically stable. [Copyright &y& Elsevier]

In this paper we consider the following boundary value problem of discrete systemwith the Dirichlet boundary conditionSome new results of the existence, nonexistence and multiplicity are obtained by using Krasnosel''skii''s fixed point theorem in a cone. In particular, it is proved that the above boundary value problem has N positive solutions under suitable conditions, where N is an arbitrary positive integer. [Copyright &y& Elsevier]

Some oscillation criteria for the second order neutral nonlinear differential equationare established. New oscillation criteria are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of [t0,∞), rather than on the whole half-line. Our results are more natural according to the Sturm Separation Theorem. [Copyright &y& Elsevier]

This paper is concerned with the discrete Wazewska and Lasota model where μ∈(0,1), ν,p∈(0,∞) and k∈{0,1,2…}. We show that every positive solution of is persistent, and then we provide sufficient conditions for oscillation of all solutions about x*. Finally, we give sufficient conditions for x* to be stable or attractive. [Copyright &y& Elsevier]

By using the continuation theorem base on Gaines and Mawhin''s coincidence degree, sufficient and realistic conditions are obtained for the global existence of positive periodic solution for a delayed predator–prey system without dominating instantaneous negative feedback. Indeed, our result are applicable to statedependent delays. [Copyright &y& Elsevier]

A Lotka–Volterra periodic delay model with m-predators and n-preys is studied in this paper. Sufficient conditions for the global existence of a positive periodic solutions of this delay model are obtained by means of topological degree. Finally, a suitable example is given to illustrate that the conditions of the main theorem are feasible. [Copyright &y& Elsevier]

In this paper we consider the following boundary value problem of discrete systemwith the Dirichlet boundary value conditionSome sufficient conditions are obtained for the existence of one or two positive solutions to the above system by using nonlinear alternative of Leray–Schauder type and Krasnosel''skii''s fixed point theorem in a cone. [Copyright &y& Elsevier]

With the help of continuation theorem base on Gaines and Mawhin''s coincidence degree, sufficient and realistic conditions are obtained for the global existence of positive periodic solutions for both the first and the second Leslie–Gower predator–prey models with distributed or state dependent delays. [Copyright &y& Elsevier]

A classification scheme for the eventually positive solutions of a class of two-dimensional Volterra nonlinear difference equations is given in terms of their asymptotic magnitudes, and necessary as well as sufficient conditions for the existence of such solutions are provided. [Copyright &y& Elsevier]

By using an inequality due to Hardy, Littlewood and Polya and averaging functions, new interval oscillation criteria are established for the half-linear functional differential equationthat are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of [t0,∞), rather than on the whole half-line. Our results extend and improve some previous oscillation criteria and handle the cases which are not covered by known results and implies that the delay τ(t)=t±τ does not effect the oscillation, where τ>0 is a constant. In particular, several examples that dwell upon the sharp conditions of our results are also included. [Copyright &y& Elsevier]

With the help of continuation theorem in coincidence degree theory and Lyapunov functions, we derive sufficient and realistic conditions that guarantee the existence of a positive periodic solution for a delayed discrete predator–prey system, and also obtain sufficient conditions for the global stability of positive periodic solutions. [Copyright &y& Elsevier]

This paper is concerned with the following systemΔ2ui(k)+fi(k,u1(k),u2(k),…,un(k))=0, k∈[0,T], i=1,2,…,nwith the Dirichlet boundary conditionui(0)=ui(T+2)=0, i=1,2,…,n.Some new results are obtained for the existence and multiplicity of positive solutions to the above system by using nonlinear alternative of Leray–Schauder type, Krasnosel''skii''s fixed point theorem in a cone and Leggett–Williams fixed point theorem. In particular, it proves that the above system has N positive solutions under suitable conditions, where N is an arbitrary integer. [Copyright &y& Elsevier]

In this paper, necessary and sufficient conditions for the existence and classification of positive solutions of the higher-order neutral difference equationΔrnΔkxn−∑i=1mpnixn−τi+f(n,xn−σ1,…,xn−σl)=0are obtained, where rn>0, pni&ges;0, i=1,2,…,m, and f(n,s1,…,sl) is continuous with respect to n and sj, j=1,…,l. [Copyright &y& Elsevier]

In this paper, we investigate the global stability of all negative solutions of the difference equationxn+1=α+xn−k/xn, n=0,1,…,where α<1 is a real number, k&ges;1 is an integer, and the initial conditions x−k,…,x0 are arbitrary real numbers. We show that the unique negative equilibrium of the equation is a global attractor with a basin that depends on certain conditions of the coefficient. [Copyright &y& Elsevier]

In this paper, we investigate an optimal harvesting problem for linear periodic age-dependent population dynamics. Namely, we consider the Lotka–Mckendrick model with periodic vital rates and a periodic forcing term that sustains oscillations. By Mazur''s Theorem, we demonstrate existence of solutions of the optimal control problem (OH) and by the conception of normal cone, we also obtain the first order necessary conditions of optimality for problem (OH). Finally, under suitable assumptions, we give uniqueness of solutions of the optimal control problem (OH). Our results extend some known criteria. [Copyright &y& Elsevier]

Our aim in this paper is to investigate the global attractivity of all positive solutions of the higher order nonlinear difference equationxn+1=a−bxn/A−∑ki=0bixn−i, n=0,1,…,where A,b,bk∈(0,∞), a,b0,…,bk−1∈[0,∞), k∈{1,2,…} and the initial conditions x−k,…,x0 are arbitrary real numbers. We show that one positive equilibrium of the equation is a global attractor with a basin that depends on certain conditions posed on the coefficients. [Copyright &y& Elsevier]

This paper is concerned with the following systemΔ3u1(k)+f1(k,u1(k),u2(k))=0,k∈[0,T],Δ3u2(k)+f2(k,u1(k),u2(k))=0,k∈[0,T],with the Dirichlet boundary conditionu1(0)=u1(1)=u1(T+3)=0=u2(0)=u2(1)=u2(T+3).By using Leggett–Williams fixed point theorem, sufficient conditions are obtained for the existence of three positive solutions to the above system. [Copyright &y& Elsevier]

With the help of continuation theorem based on Gaines and Mawhin’s coincidence degree, sufficient and realistic conditions are obtained for the global existence of positive periodic solutions for the delay differential system with feedback control. When these results are applied to some special delay population models, some new results are obtained, and some known results are generalized. [Copyright &y& Elsevier]

By using two inequalities due to Ho¨lder and Hardy, Littlewood and Polya as well as averaging functions, several interval oscillation criteria are established for the second-order quasi-linear differential equation with forced term of the form (r(t)|y′(t)|α−1y′(t))′+p(t)|y′(t)|α−1y′(t)+q(t)|y(t)|β−1y(t)=e(t) that are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of [t0,∞), rather than on the whole half-line, where β>α>0. Our results make use of the oscillatory properties of the forcing term and extend a recent result which is obtained by means of a Picone identity. [Copyright &y& Elsevier]

In this paper, we study the qualitative behavior of solutions of the discrete delay Logistic equation with several delaysxn+1=xn exp∑i=1mri1−xn−ki/K, n=0,1,2,…,where ri∈(0,∞) for i=1,…,m, ki (i=1,2,…,m) are positive integers and K∈(0,∞). We obtain sufficient conditions for the global attractivity of all positive solutions about the positive equilibrium. Furthermore, we show that such model is uniformly persistent and that all its eventually positive solutions are bounded. The oscillation about the positive equilibrium is also discussed. [Copyright &y& Elsevier]

We present new oscillation criteria for the second-order sublinear neutral delay difference equationΔanΔxn+pnxn−τ+qnxn−σγ=0,where 0<γ<1 is a quotient of odd positive integers and ∑n=0∞1/an=∞. [Copyright &y& Elsevier]

In this paper, the existence of multiple positive solutions to the second-order Neumann BVPs−u′′+Mu=f(t,u), u′(0)=u′(1)=0andu′′+Mu=f(t,u), u′(0)=u′(1)=0are proved by a fixed point theorem in cone due to Krasnoselskii. [Copyright &y& Elsevier]

This paper studies the existence, global stability and uniform persistence of positive periodic solutions of a periodic predator–prey system with Holling type III functional response. By using the continuation theorem of coincidence degree theory and Liapunov functional, some sufficient conditions are obtained. [Copyright &y& Elsevier]

Our aim in this paper is to investigate the periodic character, invariant intervals and the global attractivity of all positive solutions of the nonlinear difference equationxn+1=α+βxn/γ−xn−1, n=0,1,…,where α&ges;0, β,γ>0. We show that the positive equilibrium of the equation is a global attractor with a basin that depends on certain conditions posed on the coefficients. [Copyright &y& Elsevier]

This paper is concerned with the following systemΔ2u1(k)+f1(k,u1(k),u2(k))=0, k∈[0,T],Δ2u2(k)+f2(k,u1(k),u2(k))=0, k∈[0,T],u1(0)=u1(T+2)=0=u2(0)=u2(T+2).By using Leggett–Williams fixed point theorem, sufficient conditions are obtained for the existence of three positive solutions to the above system. [Copyright &y& Elsevier]

In this paper, we consider two-dimensional nonlinear difference systems of the formΔxn=anf(yn),Δyn−1=bng(xn).We classify their solutions according to asymptotic behavior and give some necessary and sufficient conditions for the existence of solutions of such classes. [Copyright &y& Elsevier]

Our aim in this paper is to investigate the global attractivity of the recursive sequencexn+1=α−βxn/γ−xn−1, n=0,1,…,where α&ges;0, β,γ>0. We show that one positive equilibrium point of the equation is a global attractor with a basin that depends on certain conditions posed on the coefficients. [Copyright &y& Elsevier]

Abstract: In this paper, we study the dynamics of a predator–prey system with Holling type III functional response of the formWhen h =0, Chen and Zhang (J.P. Chen, H.D. Zhang, The qualitative analysis of two species predator–prey model with Holling’s type III functional response, Appl. Math. Mech. 7 (1) (1986) 73–80) and Kazarinoff and Driessche (N.D. Kazarinoff, P. Van Den Driessche, A model predator–prey system with functional response, Math. Biosci. 39 (1–2) (1978) 125–134) gave qualitative analysis of the above system and obtained some conditions for the existence and stability of positive equilibria and limit cycles. The dynamical phenomena they concluded are normal, and there is no degenerate singularities. But when the constant harvesting is present, we will show that the system may have more complex and more rich dynamics. We carry out bifurcation analysis by choosing the death rate and the harvesting rate of the predator as the bifurcation parameters and show that the system can undergo the Bogdanov–Takens bifurcation. Also the figures of all degenerate structures are given. [Copyright &y& Elsevier]

In this paper, we investigate optimal policies for three age-dependent populations in a predator–prey system, which is controlled by fertility. By using Dubovitskii–Milyutin''s general theory, the maximum principles are obtained for problems with free terminal states, infinite horizon and target sets, respectively. [Copyright &y& Elsevier]