Propagation Dynamics Of Multi-Species Lotka-Volterra Diffusion-Competition System

In recent years,the research and application of reaction-diffusion equations in population dynamics,chemistry,physics,materials science and other disciplines have attracted widespread attention,among which the study on propagation dynamics mainly focuses on spreading speed,traveling wave solutions and entire solutions.In this paper,we first studied spreading speed and forced traveling wave solutions in a two-species competition-diffusion model with a shifting habitat.Secondly,we studied traveling wave solutions and entire solutions in a three-species competition system with non-local dispersal.Firstly,we consider a two-species Lotka-Volterra competition-diffusion model with a shifting habitat.It is assumed that the growth rate of each species is non-decreasing along the x-axis,positive near ? and negative near-?,and shifting rightward at a speed c.We investigate the spatial population dynamics of the com-petition model where in the absence of its rival,each species persists and spreads at a speed determined by the maximum linearized growth rate and the diffusion coef-ficient.We study two cases:(?)one species is competitively stronger near ? and has a slower spreading speed,and(?)both species may coexist near ?.We obtain conditions under which the outcome of competition depends critically on a number c(?)given by the model parameters.We show that under appropriate conditions,if c(?)>c then the species with the faster spreading speed spreads into the open space at its own speed and the species with the slower spreading speed spreads into its rival at speed c(?),and if c(?)<c then the species with the slower spreading speed eventually dies out in space.Our results particularly demonstrate the possi-bility that a competitively weaker species with a faster spreading speed can drive a competitively stronger species with a slower spreading speed to extinction.Secondly,we considered the existence of forced traveling waves for the above model.Under some conditions,we found some existence and nonexistence results:for case(?),there exists a critical speed c(?)such that for any c>c(?),we have established a forced traveling wave connecting extinction steady state with semi-trivial steady state and for c<c(?),there is no such waves;for case(?),on one hand,we reached the same conclusions as case(?),on the other hands,for any speed c>0,we also show the existence of a forced traveling wave connecting extinction steady state with positive steady state,which is different from the previous related conclusions.Our results show that for the same speed c,there are different types of forced traveling wave solutions.Finally,we studied monostable traveling wave solutions,bistable traveling wave solutions and related entire solutions for a three-species competition model with nonlocal dispersal.We first establish the existence and asymptotic behavior of the monostable traveling wave solutions,then we establish a new type of entire solutions which behave as two monostable traveling wave solutions coming from both sides of x-axis and obtains some qualitative properties.Then,we established bistable traveling waves and entire solutions for the above competition model with nonlocal anisotropic dispersal.Specifically,we establish the existence and monotonicity of traveling waves on truncation problems by the limiting argument.Then using Ike-hara's lemma,we demonstrate the asymptotic behavior of traveling waves,by which the uniqueness of wave profile and wave speed is investigated.Finally,a class of front-like entire solutions are constructed by comparison principle and super/sub-solutions,and some qualitative properties such as the monotonicity and smoothness of these entire solutions are also obtained.