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We investigate the existence and qualitative properties of conical traveling fronts for the bistable nonlocal equations. Based on the existence result of pyramidal traveling fronts (Li et al., 2021), we construct a sequence of pyramidal traveling fronts and take its limit. However, it is not easy to show that the limit function is a real conical traveling front, and the proofs of its qualitative properties are quite different from those of classical equations, although some of the ideas used are similar. [ABSTRACT FROM AUTHOR]

Abstract: We establish the existence of pulsating type entire solutions of reaction–advection–diffusion equations with monostable nonlinearities in a periodic framework. Here the nonlinearities include the classic KPP case. The pulsating type entire solutions are defined in the whole space and for all time . By studying a pulsating traveling front connecting a constant unstable stationary state to a stable stationary state which is allowed to be a positive function, we proved that there exist pulsating type entire solutions behaving as two pulsating traveling fronts coming from both directions, and approaching each other. The key techniques are to characterize the asymptotic behavior of the solutions as in terms of appropriate subsolutions and supersolutions. [Copyright &y& Elsevier]

Abstract: This paper is concerned with an inhomogeneous nonlocal dispersal equation. We study the limit of the re-scaled problem of this nonlocal operator and prove that the solutions of the re-scaled equation converge to a solution of the Fokker–Planck equation uniformly. We then analyze the nonlocal dispersal equation of an inhomogeneous diffusion kernel and find that the heterogeneity in the classical diffusion term coincides with the inhomogeneous kernel when the scaling parameter goes to zero. [Copyright &y& Elsevier]

Abstract: This paper is concerned with traveling wave solutions of the equation where the dispersion kernel is nonnegative and the nonlinearity is monostable type. We show that there exists such that for any , there is a nonincreasing traveling wave solution with and , where is the smallest positive solution to . Furthermore, the existence of traveling wave solutions with is also established. For , we further prove that the traveling wave solution is unique up to translation and is globally asymptotically stable in certain sense. [Copyright &y& Elsevier]

Abstract: The main purpose of this work is to investigate the effects of cross-diffusion in a strongly coupled predator–prey system. By a linear stability analysis we find the conditions which allow a homogeneous steady state (stable for the kinetics) to become unstable through a Turing mechanism. In particular, it is shown that Turing instability of the reaction–diffusion system can disappear due to the presence of the cross-diffusion, which implies that the cross-diffusion induced stability can be regarded as the cross-stability of the corresponding reaction–diffusion system. Furthermore, we consider the existence and non-existence results concerning non-constant positive steady states (patterns) of the system. We demonstrate that cross-diffusion can create non-constant positive steady-state solutions. These results exhibit interesting and very different roles of the cross-diffusion in the formation and the disappearance of the Turing instability. [Copyright &y& Elsevier]

Abstract: We consider transition fronts (generalized traveling fronts) of mono-stable reaction–diffusion equations with spatially inhomogeneous nonlinearity. By constructing a cutoff function and using an approximate method, we establish the existence of transition fronts of the equation. Furthermore, we give the uniform non-degeneracy estimates of the solutions, such as a lower bound on the time derivative on some level sets, as well as an upper bound on the spatial derivative. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the existence and asymptotic behavior of solutions of a nonlocal dispersal equation. By means of super-subsolution method and monotone iteration, we first study the existence and asymptotic behavior of solutions for a general nonlocal dispersal equation. Then, we apply these results to our equation and show that the nonnegative solution is unique, and the behavior of this solution depends on parameter in equation. For , the solution decays to zero as ; while for , the solution converges to the unique positive stationary solution as . In addition, we show that the solution blows up under some conditions. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the existence, asymptotic stability and uniqueness of traveling wavefronts in a nonlocal diffusion equation with delay. By constructing proper upper and lower solutions, the existence and asymptotic behavior of traveling wavefronts are established. Then the asymptotic stability with phase shift as well as the uniqueness up to translation of traveling wavefronts are proved by applying the idea of squeezing technique. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the monotonicity and uniqueness of traveling waves for a reaction–diffusion model with quiescent stage. We first obtain the exponential decay rate of wave profiles, and then we show that any profile is strictly monotone by using the strong comparison principle. Furthermore, we prove the uniqueness (up to translation) of all traveling waves including even the waves with minimal speed. [Copyright &y& Elsevier]

Abstract: The limit set trichotomy and dichotomy for skew-product semiflows are established under suitable monotone and concave assumptions. As an application, a delayed Hopfield-type neural network model is considered. [Copyright &y& Elsevier]

Abstract: In this paper we consider the following nonlinear first-order periodic boundary value problem with parameter on a time scale where is a parameter. For suitable , some existence, multiplicity and nonexistence criteria of positive solutions are established by using well-known results from the fixed-point index. [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: This paper is concerned with the following system on time scale : where and . By using the theory of the fixed point index, we investigate the effect of on the existence and nonexistence of positive solution for the above system in sublinear cases. The results obtained are even new for the special cases of difference system and differential system, as well as in the general time scale setting. [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: Massera type criteria are established for the existence of periodic solutions of linear and nonlinear dynamic equations on time scales. Some interesting properties of the exponential function on time scales are presented. Furthermore, a sufficient condition guaranteeing the boundedness of the solutions of the equation is presented. [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: 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 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: 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]