Special Session 32: Propagation Dynamics in Nonlocal Dispersal Systems

Dynamics of Nonlocal Dispersal SIS Models in Heterogeneous Environments

Wan-Tong Li Lanzhou University Peoples Rep of China Co-Author(s):

Abstract: In this talk we consider a nonlocal dispersal SIS epidemic model, where the spatial movement of individuals is described by a nonlocal diffusion operator, the transmission rate and recovery rate are spatially heterogeneous. We first define the basic reproduction number R_0 and discuss the existence, uniqueness and stability of steady states of the nonlocal dispersal SIS epidemic model in terms of R_0. Then we study the asymptotic profiles of the endemic steady states for large and small diffusion rates to illustrate the persistence or extinction of the infectious disease. We also observe the concentration phenomenon which occurs when the diffusion rate of the infected individuals tends to zero. The obtained results indicate that the nonlocal movement of the susceptible or infectious individuals will enhance the persistence of the infectious disease. In particular, our analytical results suggest that the spatial heterogeneity tends to boost the spread of the infectious disease.

The propagation dynamics for three species competitive-cooperative reaction-diffusion systems

Yan Li Xidian University Peoples Rep of China Co-Author(s):    Xiaoqiang Zhao

Abstract: In this talk, we will introduce the sign of bistable wave speed and spreading properties of solutions for a class of three species competitive-cooperative reaction-diffusion systems. We first establish sufficient conditions for a species to be the winner, and then show that the species with large invasion speed can always establish a wave moving into open space with its own speed, while the others will be driven out or invade successfully, depending on the parameters, at a lower speed. This is the co-work with professor Xiaoqiang Zhao in Memorial University of Newfoundland.

Spatial dynamics for a time-periodic epidemic model in discrete media

Shi-Liang Wu Xidian University Peoples Rep of China Co-Author(s):

Abstract: This talk considers the spreading speed and periodic traveling waves for a time-periodic epidemic model in discrete media. We first characterize the spreading speed of the system which can be used to estimate how fast the disease spreads. Then, based on constructing two different pairs of explicit subsolutions and supersolutions, we establish the existence of supercritical and critical periodic traveling waves. We further derive the non-existence of periodic traveling waves when the wave speed is smaller than the spreading speed.

Propagation phenomena of time heterogeneous reaction-diffusion equations in a cylinder

Bao Xiongxiong Chang`an University Peoples Rep of China Co-Author(s):

Abstract: In this talk, we are devoted to the study of generalized traveling fronts of time heterogeneous reaction-diffusion equations in a cylinder. Here the reaction term is of Fisher-KPP type and depends in a general way on $t\in\Bbb{R}$. We first investigate the existence and non-existence of generalized traveling wave solutions for such equations. Then we prove some spreading properties for the solution of the corresponding Cauchy problem with compactly supported initial values. We also show the stability of generalized traveling wave solutions by proper modifications of the squeezing technique.

The stability of monostable traveling waves for a class of asymmetric diffusion system with nonlocal effects and delay

Yun-Rui Yang Lanzhou Jiaotong University Peoples Rep of China Co-Author(s):

Abstract: In this talk, the stability of monostable traveling waves for a class of asymmetric diffusion system with nonlocal effects and delay is considered, where the system can be quasi-monotone or non-quasimonotone and the kernel functions in diffusion terms and nonlocal reactions can both be asymmetric. Firstly, the global stability of monostable wavefronts for the asymmetric nonlocal diffusion system is established by the Fourier transform and the anti-weighted energy method, and a new technique is developed to control the real part of the Fourier transform of the asymmetric kernels by a bounded function, which is different from the case of symmetric kernels. Secondly, if the system can be non-quasimonotone, the global stability of monostable waves with the decay rate of the form an exponential function multiplying by an algebraic function is also obtained. Moreover, some concrete examples and numerical stimulations are included to confirm the theoretical conclusions.