Spreading Speeds And Traveling Wave Solutions For Three-species Discrete-time Competitive Systems

In this paper,we investigate the asymptotic speeds of spreading and traveling wave solutions of the following three-species discrete-time competitive system(?)where un(x),un(x),wn(x)denote the population densities of species u,v,and w at time n and location x ?R,respectively;the dispersal kernels k1(x),k2(x),k3(x):R ? R+ represent probability density functions;x?R,n ? N,r1,r2,r3,a1,a2,a3,a4are positive.We focus on the propagation dynamics of the above system under the monostable condition,which models the invading process of species v to the habitats of species u and w.By introducing a new cooperative system and using the abstract results of the time-discrete recursive systems,we prove the corresponding initial value problem of the new system has a single spreading speed,and obtain the sufficient conditions for linear determinacy.These results show that the invasion speed of species v and the extinction speeds of species u and w are the same one in the limit sense.Then the traveling wave solutions of this system are studied.By constructing appropriate upper and lower solutions,the existence of the monostable traveling wave solution of the above system is proved.We certify that the asymptotic speed is minimal wave speed by using comparison principle.These results show that species v will eventually successfully invade the habitats of species u and species w.In addition,by the relationship between minimum wave speed and spreading speed,another sufficient condition for the linear determinacy of the asymptotic speed of spreading is obtained.