Search

Linear stability and Hopf bifurcation in a two-neuron network with three delays (one is due to the self-feedback) are investigated. Based on the normal form approach and the center manifold theory, we derive the formula to determine the direction and stability of Hopf bifurcation. [ABSTRACT FROM AUTHOR]

In this paper we investigate a two-species competitive chemotaxis Keller–Segel system. We first obtain the local existence and global existence of classical solutions and then study the asymptotic behavior of the solutions. [ABSTRACT FROM AUTHOR]

This paper is concerned with the dynamics of a two-species reaction–diffusion–advection competition model subject to the no-flux boundary condition in a bounded domain. By the signs of the associated principal eigenvalues, we derive the existence and local stability of the trivial and semi-trivial steady-state solutions. Moreover, the nonexistence and existence of the coexistence steady-state solutions stemming from the two boundary steady states are obtained as well. In particular, we describe the feature of the coincidence of bifurcating coexistence steady-state solution branches. At the same time, the effect of advection on the stability of the bifurcating solution is also investigated, and our results suggest that the advection term may change the stability. Finally, we point out that the methods we applied here are mainly based on spectral analysis, perturbation theory, comparison principle, monotone theory, Lyapunov–Schmidt reduction, and bifurcation theory. [ABSTRACT FROM AUTHOR]

This paper is devoted to the Neumann problem of a stationary Lotka–Volterra model with diffusion and advection. In the model it is assumed that one population growth rate is described by weak Allee effect. We first obtain some sufficient conditions ensuring the existence of nonconstant solutions by using the Leray–Schauder degree theory. And then we study a limiting system (with nonlocal constraint) which stems from the original model as diffusion and advection of one of the species tend to infinity. Finally, we classify the global bifurcation structure of nonconstant solutions of the simplified 1D case. [ABSTRACT FROM AUTHOR]

In this paper, we consider a stochastic semilinear degenerate parabolic equation with delay in a bounded domain in R N and the nonlinearity satisfying an arbitrary polynomial growth condition. The random dynamical system generated by the equation is shown to have a random attractor in C ( − τ , 0 , L p (O) ∩ D 0 1 (O , σ)) , which is a compact and invariant tempered set and attracts every tempered random subset of C ( − τ , 0 , L 2 (O)) in the topology of C ( − τ , 0 , L p (O)). In a particular case, the random attractor consists of singleton sets (i.e., a random fixed point), which generates an exponentially stable non-trivial stationary solution. This theoretical result improves some recent ones for stochastic semilinear degenerate parabolic equations. • A stochastic semilinear degenerate parabolic equation with delay in a bounded domain is studied. • The existence and regularity of pullback attractors is studied extensively. • The continuity of the solution semigroup in C ( − τ , 0 , L p (O)) with p ≠ 2 is investigated. [ABSTRACT FROM AUTHOR]

The local stability of the steady state for a general class of a reaction–diffusion model with nonlocal delay effect and nonlinear boundary condition is investigated under natural and minimal hypotheses. [ABSTRACT FROM AUTHOR]

In this paper we establish Hopf bifurcation phenomena for a diffusive Hutchinson model with delay effect and homogeneous boundary condition on a general open domain with a smooth boundary. In the process we derive criteria for the existence and direction of branches of bifurcating periodic solutions via Lyapunov–Schmidt reduction. [ABSTRACT FROM AUTHOR]

This paper studies a three-dimensional multiple chaotic system with three quadratic nonlinear terms. The system is shown to exhibit multiple periodic attractors and multiple chaotic attractors including a four-scroll chaotic attractor. It is shown analytically and numerically that any neighborhood of the four-scroll chaotic attractor contains repelling sets with positive Lebesgue measures. Moreover, in terms of incommensurate fractional order systems, a simple fractional differentiator-based controller is designed to suppress chaos. Some basic dynamical behaviors of the system are investigated through analytical techniques and numerical simulation. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of traveling wavefronts for a class of two-species chemotaxis models with Lotka–Volterra competitive kinetics. The existence of traveling wavefronts of large wave speed is established by the perturbation method. Some numerical simulations are presented to illustrate our main results. [ABSTRACT FROM AUTHOR]