Free Boundary Problems Of Two-Species Competition Model With Time Delays

In recent years,reaction-diffusive free boundary problems described by classical Stefan conditions have been widely discussed by many scholars.Based on the diffusion Lotka-Volterra competitive free boundary problems with time delays,the dynamics of two competing species are studied by using the basic theory of parabolic and elliptic equations.Firstly,the research backgrounds of diffusion Lotka-Volterra competition models and free boundary problems are described.Meanwhile,the current developments of free boundary problems for a single population and two competitive populations are introduced.It also describes the main works.Secondly,taking age structure as a starting point,a time-delayed diffusion Lotka-Volterra competition model with the same free boundary for both species in a one-dimensional habitat is investigated.By a compatible condition with the initial habitat,the well-posed global solution is established.Under the case of weak competition,the long-time behaviors and the criteria for spreadingvanishing are expounded.Furthermore,we estimate the upper and lower bounds of asymptotic spreading speed by using delay-induced semi-wave problems if the spreading happens.It reveals that the asymptotic spreading speed cannot be faster than the minimal speed of traveling wavefront solutions corresponding to the Cauchy problem.Lastly,considering the interaction between the invasive population and the native population in biological ecology,a diffusion Lotka-Volterra competition model with time delay is constructed,where there is the only invasive population with a free boundary.The well-posed global solution is established by using the compatible condition.The long-time behaviors and spreading-vanishing dichotomy criteria are given respectively under the case of weak-strong competition,strong-weak competition,and weak competition.Then under the case of strong-weak competition,we obtain the exact asymptotic spreading speed by a new delay-induced semi-wave problem if the spreading of invasive species happens.And we find that the maturation period of invasive species slows down the asymptotic spreading speed.