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In this short review, we describe some recent developments on the spatial propagation for diffusion problems in shifting environments, including single species models, competition/cooperative models and chemotaxis models submitted to classical reaction-diffusion equations (with or without free boundaries), integro-difference equations, lattice differential equations and nonlocal dispersal equations. The considered topics may typically come from modeling the threats associated with global climate change and the worsening of the environment resulting from industrialization which lead to the shifting or translating of the habitat ranges, and also arise indirectly in studying the pathophoresis as well as some multi-stage invasion processes. Some open problems and potential research directions are also presented. [ABSTRACT FROM AUTHOR]

The current paper is devoted to the study of the existence and stability of generalized transition waves of the following time-dependent reaction-diffusion cooperative systemHere depends on in a general way. Recently, the spreading speeds and linear determinacy of the above time-dependent system have been studied by Bao et al. [J. Differential Equations 265 (2018) 3048-3091]. In this paper, using the principal Lyapunov exponent and principal Floquent bundle theory of linear cooperative systems, we prove the existence of generalized transition waves in any given direction with speed greater than the spreading speed by constructing appropriate subsolutions and supersolutions. When the initial value is uniformly bounded with respect to a weighted maximum norm, we further show that all solutions converge to the generalized transition wave solutions exponentially in time. [ABSTRACT FROM AUTHOR]

Time-varying matrix inversion (TVMI) and tracking control of vehicular inverted pendulum (VIP) system are usually considered as two important benchmarking examples in mathematical science and control fields. In this paper, Zhang neural dynamics (ZND) method is used for solving these two problems. Besides, in order to investigate the effect of time delay (especially the time delay caused by the derivative approximation) on the ZND method when solving the above-mentioned two problems, backward difference rules in the form of time-delay differential equation are elegantly applied to approximating the ZND method, which is simply termed time-delay ZND method. Moreover, two theorems about time-delay ZND method are presented together with theoretical proofs. Finally, simulative tests of TVMI and tracking control of VIP system are conducted to show the effectiveness and accuracy of non-time-delay and time-delay ZND methods. [ABSTRACT FROM AUTHOR]

Abstract This paper is concerned with a time periodic competition–diffusion system { u t = u x x + u (r 1 (t) − a 1 (t) u − b 1 (t) v) , t > 0 , x ∈ R , v t = d v x x + v (r 2 (t) − a 2 (t) u − b 2 (t) v) , t > 0 , x ∈ R , where u (t , x) and v (t , x) denote the densities of two competing species, d > 0 is some constant, r i (t) , a i (t) and b i (t) are T -periodic continuous functions. Under suitable conditions, it has been confirmed by Bao and Wang (2013) [2] that this system admits periodic traveling fronts connecting two stable semi-trivial T -periodic solutions (p (t) , 0) and (0 , q (t)) associated to the corresponding kinetic system. In the present work, we first investigate the asymptotic behavior of periodic bistable traveling fronts with non-zero speeds at infinity by a dynamical approach combined with the two-sided Laplace transform method. With these asymptotic properties, we then obtain some key estimates. As a result, by applying the super- and subsolutions techniques as well as the comparison principle, we establish the existence and various qualitative properties of the so-called entire solutions defined for all time and the whole space, which provides some new spreading ways other than periodic traveling waves for two strongly competing species interacting in a heterogeneous environment. [ABSTRACT FROM AUTHOR]

This paper deals with the spatial propagation for reaction–diffusion cooperative systems. It is well-known that the solution of a reaction–diffusion equation with monostable nonlinearity spreads at a finite speed when the initial condition decays to zero exponentially or faster, and propagates fast when the initial condition decays to zero more slowly than any exponentially decaying function. However, in reaction–diffusion cooperative systems, a new possibility happens in which one species propagates fast although its initial condition decays exponentially or faster. The fundamental reason is that the growth sources of one species come from the other species. Simply speaking, we find a new interesting phenomenon that the spatial propagation of one species is accelerated by the other species. This is a unique phenomenon in reaction–diffusion systems. We present a framework of fast propagation for reaction–diffusion cooperative systems. [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with the existence and stability of periodic pyramidal traveling fronts for reaction–diffusion equations with bistable time-periodic nonlinearity in with . It is well known that two-dimensional periodic traveling curved fronts exist and are stable. In this paper, by constructing various of supersolutions and subsolutions, we first show that there exist three-dimensional periodic pyramidal traveling fronts, and then we prove that such periodic pyramidal traveling fronts are asymptotically stable. Finally, we further prove that our existence result holds for with . [Copyright &y& Elsevier]

Abstract: This paper is concerned with the existence of positive solutions of -Laplacian dynamic equation subject to boundary conditions or , where with . By using the five functionals fixed-point theorem, we prove that the boundary value problem has at least three positive solutions. As an application, an example is given to illustrate the result. [Copyright &y& Elsevier]

Abstract: We investigate a class of singular -point -Laplacian boundary value problem on time scales with the sign changing nonlinearity. By using the well-known Schauder fixed point theorem and upper and lower solutions method, some new existence criteria for positive solutions of the boundary value problem are presented. These results are new even for the corresponding differential and difference equations , as well as general time scales setting. As an application, an example is given to illustrate the results. [Copyright &y& Elsevier]

Abstract: Entire solutions for monostable reaction–diffusion equations with nonlocal delay in one-dimensional spatial domain are considered. A comparison argument is employed to prove the existence of entire solutions which behave as two traveling wave solutions coming from both directions. Some new entire solutions are also constructed by mixing traveling wave solutions with heteroclinic orbits of the spatially averaged ordinary differential equations, and the existence of such a heteroclinic orbit is established using the monotone dynamical systems theory. Key techniques include the characterization of the asymptotic behaviors of solutions as in term of appropriate subsolutions and supersolutions. Two models of reaction–diffusion equations with nonlocal delay arising from mathematical biology are given to illustrate main results. [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: Let be a time scale such that . We consider the three-point boundary value problem for -Laplacian dynamic equations on time scales of the form for with boundary conditions , , where is symmetric in and with . By using a pseudo-symmetric technique and the five-functionals fixed-point theorem, we prove that the boundary value problem has at least three positive pseudo-symmetric solutions under some assumptions. As an application, an example is given to illustrate the result. [Copyright &y& Elsevier]

Abstract: In this paper we consider the one-dimensional p-Laplacian boundary value problem on time scales where is p-Laplacian operator, i.e., , . Some new results are obtained for the existence of at least single, twin or triple positive solutions of the above problem by using Krasnosel''skii''s fixed point theorem, new fixed point theorem due to Avery and Henderson and Leggett–Williams fixed point theorem. This is probably the first time the existence of positive solutions of one-dimensional p-Laplacian boundary value problems on time scales has been studied. [Copyright &y& Elsevier]

Abstract: In this paper we are concerned with the following nonlinear first-order periodic boundary value problem on time scales Some new existence criteria of at least one solution are established by using novel inequalities and the well-known Schaefer fixed point theorem. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the existence and multiplicity of the solutions for the fourth-order boundary value problem , , , where is continuous, and are parameters. Using the variational structure of the above boundary value problem and critical point theory, it is shown that the different locations of the pair and lead to different existence results for the above boundary value problem. More precisely, if the pair is on the left side of the first eigenvalue line, then the above boundary value problem has only the trivial solution for and has infinitely many solutions for ; if is on the right side of the first eigenvalue line and , then the above boundary value problem has two nontrivial solutions or has at least distinct pairs of solutions, which depends on the fact that the pair is located in the second or fourth (first) quadrant. [Copyright &y& Elsevier]

Abstract: In this paper, an existence criterion is established for a class of second order perturbed dynamic inclusions on time scales with boundary conditions. The main approach is a fixed-point theorem for multi-valued maps due to Dhage. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the controllability of a certain class of evolution differential inclusions in Banach spaces. A sufficient condition is established by using a recent fixed-point theorem for multi-valued maps due to Dhage combined with an evolution system. An example is also given to illustrate our result. [Copyright &y& Elsevier]

Abstract: In this paper we study the existence and multiplicity of the solutions for the fourth-order boundary value problem (BVP) , , , where is continuous, and are parameters. By means of the idea of the decomposition of operators shown by Chen [W.Y. Chen, A decomposition problem for operators, Xuebao of Dongbei Renmin University 1 (1957) 95–98], see also [M. Krasnosel''skii, Topological Methods in the Theory of Nonlinear Integral Equations, Gostehizdat, Moscow, 1956], and the critical point theory, we obtain that if the pair is on the curve satisfying , then the above BVP has at least one, two, three, and infinitely many solutions for λ being in different interval, respectively. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the existence and global stability of the positive periodic solutions of delayed discrete food chains with omnivory. With the help of continuation theorem in coincidence degree theory and Lyapunov functions, some sufficient conditions are obtained. The results show that, for such a system with omnivory, more conditions are required for the persistence of the positive periodic solutions than that of a linear food chain. On the other hand, omnivory makes no differences on the stability of the solutions. [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 the existence and nonexistence of positive solutions of the second-order nonlinear dynamic equation , , satisfying either the conjugate boundary conditions or the right focal boundary conditions , where a and f are positive. We show that there exists a such that the above boundary value problem has at least two, one and no positive solutions for , and , respectively. Furthermore, by using the semiorder method on cones of the Banach space, we establish an existence and uniqueness criterion for positive solution of the problem. In particular, such a positive solution of the problem depends continuously on the parameter λ, i.e., is nondecreasing in λ, and . [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: Sufficient conditions are obtained for the existence and global attractivity of positive periodic solutions of the delay differential system with feedback controlThe method involves the application of Krasnoselskii''s fixed point theorem and estimates of uniform upper and lower bounds of solutions. When these results are applied to some special delay population models with multiple delays, some new results are obtained and some known results are generalized. [Copyright &y& Elsevier]

Abstract: In this paper we are concerned with the following system of nonlinear first-order periodic boundary value problems on time scale where is continuous and there exists a constant such thatSome existence criteria of positive solution are established by using a fixed point theorem for operators on cone. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the existence and nonexistence of positive solutions of the nonlinear fourth-order beam equation , , , where is continuous and ζ, η and λ are parameters. We show that there exists a such that the above boundary value problem (BVP) has at least two, one and no positive solutions for , and , respectively. Furthermore, by using the semiorder method on cones of Banach space, we establish a uniqueness criterion for positive solution of the BVP. In particular such a positive solution of the BVP depends continuously on the parameter λ, i.e., is nondecreasing in λ, and for any . [Copyright &y& Elsevier]

In this paper we shall consider the following nonlinear impulsive delay differential equation where m and n are positive integers, V(t) and λ(t) are positive periodic continuous functions with period ω>0. In the nondelay case (m=0), we show that the above equation has a unique positive periodic solution x*(t) which is globally asymptotically stable. In the delay case, we present sufficient conditions for the global attractivity of x*(t). Our results imply that under the appropriate periodic impulsive perturbations, the impulsive delay equation shown above preserves the original periodic property of the nonimpulsive delay equation. In particular, our work extends and improves some known results. [Copyright &y& Elsevier]

Abstract: In this paper, the existence of solution on a compact interval to second order impulsive functional differential inclusions is investigated. Several new results are obtained by using suitable fixed point theorem. [Copyright &y& Elsevier]

Abstract: Sufficient and realistic conditions are obtained for the existence and global attractivity of positive periodic solutions of the delay differential system with impulses The method involves the application of the Gaines and Mawhin''s coincidence degree theory, the constructing suitable Lyapunov Functionals and estimations of uniform upper bounds on solutions. When these results are applied to some special delay population models, some new results are obtained, and some known results are generalized. In particular, our results indicate that under the appropriate linear periodic impulsive perturbations, the above impulsive delay differential equation preserves the original periodicity and global attractivity of the nonimpulsive delay differential equation [Copyright &y& Elsevier]

Abstract: Let be a time scale such that . Consider the following three-point boundary value problem on time scales: where , , , , and . By using fixed point theorems in cones, some new and general results are obtained for the existence of single and multiple positive solutions of the above problem. In particular, our criteria generalize and improve some known results. [Copyright &y& Elsevier]

Classification schemes for positive solutions of a class of second order nonlinear differential systems are given in terms of their asymptotic magnitudes, and necessary as well as sufficient conditions for the existence of these solutions are also provided. [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]

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]

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]

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 consider the existence and uniqueness of positive periodic solution for the periodic equation y′(t)=−a(t)y(t)+λh(t)f(y(t−τ(t))). By the eigenvalue problems of completely continuous operators and theory of α-concave or −α-convex operators and its eigenvalue, we establish some criteria for existence and uniqueness of positive periodic solution of above functional differential equations with parameter. In particular, the unique solution yλ(t) of the above equation depends continuously on the parameter λ. Finally, as an application, we obtain sufficient condition for the existence of positive periodic solutions of the Nicholson blowflies model. [Copyright &y& Elsevier]

We present new interval oscillation criteria for certain classes of second-order nonlinear perturbed differential equations which are based on the information only on a sequence of subintervals of [to, ∞), rather than on the whole half-line. Our results are sharper than those known in the literature and deal with the cases which are not covered by the known criteria. In particular, our results can be applied to extreme cases such as η∞q(s)ds = -∞. Finally, two examples which dwell upon the sharpness of the conditions of our results are also illustrated. [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]