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This paper concerns a model of hierarchical neural network with D n × D n -symmetry and time delays. The linear stability of the steady state is obtained via an artful factorization technique of the characteristic equation corresponding to the linearized system. And we also perform a delay-induced Hopf bifurcation analysis involving spatio-temporal patterns of bifurcated periodic solutions relying on the equivariant Hopf bifurcation theory for retarded functional differential equations and the representation method of Lie groups. Some numerical simulations for a simple example are carried out to demonstrate the application of our theoretical conclusion that multiple periodic orbits could occur simultaneously from Hopf bifurcation in the model.

We consider a generalized Fisher-KPP equation with the growth function being time and space dependent in the form of “shifting with constant speed”. The main concerns are extinction and persistence, as well as spatial-temporal dynamics. By employing a new method relating to semigroup and some subtle estimates, we not only extend the main results in Li et al. [SIAM J. Appl. Math. 74 (2014), pp. 1397-1417] to a scenario when the growth function may have no sign change, but also improve the main results there by dropping some restrictions on the initial functions.

This paper is concerned with the forced traveling wave solution of a nonlocal dispersal Lotka-Volterra cooperation model under the habitat shifting with a constant speed. Particularly, we relax the condition of intrinsic production so that there may be no sign change. By constructing a pair of upper and lower solutions and using the method of monotone iteration, we prove the existence of traveling wave solution for any positive constant shifting speed.

In this paper, weighted inequalities for a certain general commutator associated to a singular integral operator satisfying a variant of Ho¨ rmander’s condition on Lebesgue spaces are obtained. To do this, some weighted sharp maximal function inequalities for the commutator are proved.

In this paper, we are concerned with existence/non-existence of traveling waves of a diffusive SIR epidemic model with general incidence rate of the form of f ( S ) g ( I ) and infinitely distributed latency but without demography. We show that the existence of traveling waves only depends on the basic reproduction number of the corresponding spatial-homogeneous system of delay differential equations, which is determined by the recovery rate, the local properties of f and g and a minimal wave speed c ∗ that is affected by the distributed delay. The proof of existence of traveling waves is by employing Schauder’s fixed point theorem, and the proof of nonexistence is completed with the aid of the bilateral Laplace transform.

In this study, we investigate the dynamic behavior on a system of neutral functional differential equations in critical case. Combining monotonicity techniques and functional methods, we develop the delicate monotonicity with the addressed system and show the convergence for every pre-compact solution with some initial condition, which improves and complements some existing ones.

A hierarchical network composed of two interacting rings each of which consists of n identical cells with an unidirectional coupling is the topic of this paper. We present a detailed discussion about the linear stability of the equilibrium by analyzing the associated characteristic equation. The local Hopf bifurcation and spatio-temporal patterns of bifurcating periodic oscillations are also given by employing the symmetric Hopf bifurcation theory for delay differential equations. In particular, by using the normal form theory and the center manifold theorem, we derive the formula determining the direction of the Hopf bifurcation and the stability of the bifurcated periodic orbits. An example with numerical simulations is presented to illustrate our theoretical results.

A coupled system of nine identical cells with delays and𝔻3×𝔻3-symmetry is considered. The individual cells are modelled by a scalar delay differential equation which includes linear decay and nonlinear delayed feedback. By analyzing the corresponding characteristic equations, the linear stability of the equilibrium is given. We also investigate the simultaneous occurrence of multiple periodic solutions and spatiotemporal patterns of the bifurcating periodic oscillations by using the equivariant bifurcation theory of delay differential equations combined with representation theory of Lie groups. Numerical simulations are carried out to illustrate our theoretical results.

We consider a predator–prey system of Lotka–Volterra type with time delays and stage structure for prey. By analyzing the corresponding characteristic equations, the local stability of the equilibria is investigated and Hopf bifurcations occurring at the positive equilibrium under some conditions are demonstrated. The mathematical tools which enable us to obtain the sufficient conditions, guaranteeing the global asymptotical stability of the equilibria, are the well-known Kamke comparison theorem and an iteration technique. Numerical simulations are carried out to illustrate our theoretical results.

We consider a four-neuron ring with self-feedback and delays. By analyzing the associated characteristic equation, linear stability is investigated and Hopf bifurcations are demonstrated, as well as the stability and direction of the Hopf bifurcation are determined by employing the normal form method and the center manifold reduction. Numerical simulations are presented to illustrate the results.

In this paper, the inequality of boundedness for the multilinear fractional singular integral operators associated to the weighted Lipschitz functions is estimated. The operators include the Calderón-Zygmund singular integral operator and the fractional integral operator. MSC:42B20, 42B25.

This paper aims at studying the problem of the dynamics of switched Cohen-Grossberg neural networks with mixed delays by using Lyapunov functional method, average dwell time (ADT) method, and linear matrix inequalities (LMIs) technique. Some conditions on the uniformly ultimate boundedness, the existence of an attractors, the globally exponential stability of the switched Cohen-Grossberg neural networks are developed. Our results extend and complement some earlier publications.