Generalized Propagating Speeds And Positive Entire Solutions Of Lattice KPP Equations With Time And Space Dependence

Lattice dynamical systems usually refer to infinite systems of ordinary differential equations on discrete space or infinite systems of differential equations.On the one hand,lattice differential equation is used to describe the model with discrete behavior in real life,such as the phenomenon of patches in population,crystal growth in material physics.On the other hand,they also arise as the spatial discretization of partial differential equations.Therefore,it is more meaningful and valuable in theory and practice to study such equations.In this paper,we consider entire solutions and spreading speed interval for lattice KPP equations.Firstly,we mainly discuss the research background and give a brief introduction to the problem of this article.Secondly,we prove the existence,uniqueness,and stability of entire positive solutions.Finally,we study various fundamental properties of the spreading speed intervals,including the boundedness,minimality,and natural spatial spread properties.Also,when the model is time periodic,it is shown that the spreading speed interval is a singleton(called the spreading speed).Moreover,we construct appropriate sub-solutions and super-solutions.After that,we establish lower and upper bounds of the spreading speed intervals by combing comparison principle with sub-super solution method.