Forced Waves Of The 2-D Lattice Differential Equation With Time Period

As one of the important research directions of biostatistics,the population dynamics is used to study the nature of the interaction between population and population as well as between population and environment.It also can be used to describe,predict,regulate and control the development process and trend of species.The population dynamics model is widely used in the quantitative development and management of resources,assessment and management of environment,prevention and control of catastrophe,etc.In recent years,due to global climate change,many species have begun to migrate slowly to the polar region.In order to reveal the impact of migration on the survival of species,scholars have begun to be interested in the dynamic behavior of species with shifting habitat.Studying the dynamic behavior of populations can help people judge whether species will continue to exist or go extinct in the future,which also makes scholars to study the forced waves of reaction-diffusion equations with shifting habitat.In this paper,the existence,uniqueness and stability of forced waves for the 2-D lattice differential equation with time period are studied.First,considering the corresponding initial value problem and establish the comparison principle.Then investigating the asymptotic spreading speed_*c and periodic traveling waves for a 2-D lattice differential equation without shifting habitat,and by constructing a pair of upper and lower solutions and using the monotone iteration technique,the existence of periodic forced waves with the speed at which the habitat is shifting can be proved.Then the uniqueness of the periodic forced wave is proved by the sliding technique.Finally,the asymptotic stability of the periodic forced wave is established by the comparison principle,and then the exponential stability of the periodic forced wave is proved.