Spatial Dynamics Of Periodic Lattice Dynamical Systems With Delay And Global Interaction

In ecology,the reaction-diffusion equation(system)is always used to describe the population growth.Actually,in order to simulate the density change of population more precisely,we need to consider various factors comprehensively.For example,the living environment of species may not always be continuous or homogeneous.The periodic environment is the most common heterogeneous environment.At the same time,the reproduction of species is affected by gestation and maturation,time delay is also not negligible.Therefore,it is necessary to explore the spatial dynamics of the model by taking into account these factors.In this paper,we study the spatial dynamics of periodic lattice dynamical systems(spatially discrete reaction diffusion equation)with delay and global interactions based on the population growth model of a single species,which including the qualitative properties of traveling waves and entire solutions.In the first part of this paper,we study the qualitative properties of the noncritical pulsating traveling waves of a periodic lattice differential model with delay and non-local effect.Firstly,by establishing the discrete Harnack inequality and using a slide method,we prove the monotonicity of the noncritical pulsating traveling waves.Secondly,we obtain the exponential upper and lower bounds of the pulsating traveling waves at minus infinity by establishing two new comparison principles.Finally,the uniqueness of all noncritical pulsating traveling waves and their exact asymptotic behavior are also proved by using the strong comparison principle.In the second part of this paper,we investigate the front-like entire solutions of a more general lattice periodic dynamical system with delay and global interaction.By establishing various comparison theorems,the front-like entire solutions are constructed by mixing the traveling fronts with different directions of propagation and a spatially periodic solution connecting unstable equilibrium and stable equilibrium.Some properties of these entire solutions are also considered.These entire solutions are different from the the traveling fronts,which can exhibit new characteristic behaviors in the front dynamics.