Travelling wavefronts in nonlocal diffusion equations with nonlocal delay effects

This paper deals with the existence, monotonicity, uniqueness and asymptotic behaviour of travelling wavefronts for a class of temporally delayed, spatially nonlocal diffusion equations

Dynamics of a Leslie-Gower predator-prey system with cross-diffusion

A Leslie–Gower predator–prey system with cross-diffusion subject to Neumann boundary conditions is considered. The global existence and boundedness of solutions are shown. Some sufficient conditions ensuring the existence of nonconstant solutions are obtained by means of the Leray–Schauder degree theory. The local and global stability of the positive constant steady-state solution are investigated via eigenvalue analysis and Lyapunov procedure. Based on center manifold reduction and normal form theory, Hopf bifurcation direction and the stability of bifurcating timeperiodic solutions are investigated and a normal form of Bogdanov–Takens bifurcation is determined as well

Invariant measures and random attractors of stochastic delay differential equations in Hilbert space

This paper is devoted to a general stochastic delay differential equation with infinite-dimensional diffusions in a Hilbert space. We not only investigate the existence of invariant measures with either Wiener process or Lévy jump process, but also obtain the existence of a pullback attractor under Wiener process. In particular, we prove the existence of a non-trivial stationary solution which is exponentially stable and is generated by the composition of a random variable and the Wiener shift. At last, examples of reaction-diffusion equations with delay and noise are provided to illustrate our results

Existence and stability of mild solutions to parabolic stochastic partial differential equations driven by Lévy space-time noise

This paper is concerned with well-posedness and stability of parabolic stochastic partial differential equations. Firstly, we obtain some sufficient conditions ensuring the existence and uniqueness of mild solutions, and some $\mathcal{H}$-stability criteria for a class of parabolic stochastic partial differential equations driven by Lévy space-time noise under the local/non-Lipschitz condition. Secondly, we establish some existence-uniqueness theorems and present sufficient conditions ensuring the $\mathcal{H}'$-stability of mild solutions for a class of parabolic stochastic partial functional differential equations driven by Lévy space-time noise under the local/non-Lipschitz condition. These theoretical results generalize and improve some existing results. Finally, two examples are given to illustrate the effectiveness of our main results

Invariant measures and random attractors of stochastic delay differential equations in Hilbert space

This paper is devoted to a general stochastic delay differential equation with infinite-dimensional diffusions in a Hilbert space. We not only investigate the existence of invariant measures with either Wiener process or Lévy jump process, but also obtain the existence of a pullback attractor under Wiener process. In particular, we prove the existence of a non-trivial stationary solution which is exponentially stable and is generated by the composition of a random variable and the Wiener shift. At last, examples of reaction-diffusion equations with delay and noise are provided to illustrate our results

Dynamics of a Leslie-Gower predator-prey system with cross-diffusion

A Leslie–Gower predator–prey system with cross-diffusion subject to Neumann boundary conditions is considered. The global existence and boundedness of solutions are shown. Some sufficient conditions ensuring the existence of nonconstant solutions are obtained by means of the Leray–Schauder degree theory. The local and global stability of the positive constant steady-state solution are investigated via eigenvalue analysis and Lyapunov procedure. Based on center manifold reduction and normal form theory, Hopf bifurcation direction and the stability of bifurcating time-periodic solutions are investigated and a normal form of Bogdanov-Takens bifurcation is determined as well

Synchronous dynamics of a delayed two-coupled oscillator

This paper presents a detailed analysis on the dynamics of a delayed two-coupled oscillator. Linear stability of the model is investigated by analyzing the associated characteristic transcendental equation. By means of the equivariant Hopf bifurcation theorem, we not only investigate the effect of time delay on the spatio-temporal patterns of periodic solutions emanating from the trivial equilibrium, but also derive the formula to determine the direction and stability of Hopf bifurcation. Moreover, we illustrate our results by numerical simulations