Yuming Chen

Global stability of neural networks with distributed delays

Using the Lyapunov functional approach, we study the global stability of a class of neural networks with distributed delays. Without assuming the boundedness, monotonicity and differentiability of the activation functions and symmetry of the interconnection matrix, we give some new sufficient conditions on the existence, uniqueness and global asymptotic stability of the equilibrium point. Our results improve, extend and complement some existing ones.

Multiple periodic solutions of delayed predator–prey systems with type IV functional responses

In this paper, we considered a periodic predator–prey system with a type IV functional response, which incorporates the periodicity of the environment. Sufficient conditions for the existence of multiple positive periodic solutions are established by applying the continuation theorem. This is the first time that multiple periodic solutions are obtained by using the theory of coincidence degree. Moreover, unlike other types of functional responses, a type IV functional response declines at high prey densities. Thus the existing arguments for obtaining bounds of solutions to the operator equation Lx=λNx are inapplicable to our case and some new arguments are employed for the first time.

Linear stability and Hopf bifurcation in a three-unit neural network with two delays

Abstract Considered is a system of delay differential equations modeling a time-delayed connecting network of three neurons without self-feedback. We investigate the linear stability of the trivial solution and Hopf bifurcation of this system. The general formula for the direction, the estimation formula of period and stability of Hopf bifurcating periodic solution are also given.

Threshold dynamics of a delayed reaction diffusion equation subject to the Dirichlet condition.

We establish the threshold dynamics of a delayed reaction diffusion equation subject to the homogeneous Dirichlet boundary condition when the delayed reaction term is non-monotone. We illustrate the main results by two examples, including the delayed Nicholsons blowflies diffusion equation.

Analysis of sexually transmitted disease spreading in heterosexual and homosexual populations.

Sexually transmitted diseases can pose major health problems so scientists and health agencies are very concerned about the spread of these diseases. Sexually transmitted diseases spread through a network of contacts created by the formation of sexual partnerships. In the paper, the spreading of sexually transmitted diseases on bipartite scale-free graphs, representing heterosexual and homosexual contact networks, is considered. We propose an SIS model on sexual contact networks. We analytically derive the expression for the epidemic threshold and its dependence with the ratio of female and male in finite populations. It is shown that if the basic reproduction number R0 is less than 1 then the disease-free equilibrium is globally asymptotically stable; if R0>1 then the disease-free equilibrium is unstable and there is a unique endemic equilibrium, which asymptotically attracts all nontrivial solutions. These theoretical results are supported by numerical simulations. We also carry out some sensitivity analysis of the basic reproduction number R0 in terms of various model parameters.

Traveling wavefronts of a prey-predator diffusion system with stage-structure and harvesting

From a biological point of view, we consider a prey-predator-type free diffusion fishery model with stage-structure and harvesting. First, we study the stability of the nonnegative constant equilibria. In particular, the effect of harvesting on the stability of equilibria is discussed and supported with numerical simulation. Then, employing the upper and lower solution method, we show that when the wave speed is large enough there exists a traveling wavefront connecting the zero solution to the positive equilibrium of the system. Numerical simulation is also carried out to illustrate the main result.

Population dynamics of plateau pika under lethal control and contraception control

The overabundance of plateau pika brings a great damage to the alpine meadow. Rodenticide and sterilant have been used to control this mammalian pest. In this article, we proposed models to incorporate these controls and the seasonal cycle of breeding and non-breeding. It is shown that when the basic reproduction number is less than 1 then the trivial equilibrium is globally asymptotically stable; if the basic reproduction number is greater than 1 then the trivial equilibrium is unstable and there is a positive equilibrium which attracts all positive solutions. Then we study the effects of controls on the existence of the positive equilibrium and the population size. These theoretical results are supported by numerical simulations. We also propose the possible strategies to be implemented in practice.Mathematical Subject Classification: 39A30; 39A60; 92D25.

Threshold dynamics of an SIRS model with nonlinear incidence rate and transfer from infectious to susceptible

Abstract In this paper, we propose an SIRS epidemic model with a nonlinear incidence and transfer from infectious to susceptible. Applying LaSalle’s invariance principle and Lyapunov direct method, we establish a threshold dynamics completely determined by the basic reproduction number.

Periodic Solutions and the Global Attractor in a System of Delay Differential Equations

We consider a system of delay differential equations with a delayed excitatory feedback loop and instantaneous damping. We develop the Cao–Krisztin–Walther technique and establish the existence and uniqueness of periodic solutions with prescribed oscillation frequencies (characterized by the values of a discrete Lyapunov functional). We then use the Poincare-Bendixson theorem due to Mallet-Paret and Sell to show that the global attractor of such a system is the union of the unstable sets of stationary points and periodic orbits.

Imitation dynamics of vaccine decision-making behaviours based on the game theory.

ABSTRACT Based on game theory, we propose an age-structured model to investigate the imitation dynamics of vaccine uptake. We first obtain the existence and local stability of equilibria. We show that Hopf bifurcation can occur. We also establish the global stability of the boundary equilibria and persistence of the disease. The theoretical results are supported by numerical simulations.