The effect of initial values on extinction or persistence in degenerate diffusion competition systems

When the asymptotic spreading for classical monostable Lotka-Volterra competition diffusion systems is concerned, extinction or persistence of the two competitive species is completely determined by the dynamics of the corresponding kinetic systems, while the size of initial values does not affect the final states. The purpose of this paper is to demonstrate the rich dynamics induced by the initial values in a class of degenerate competition diffusion systems with weak Allee effect. We present various extinction or persistence results by selecting different initial values although the corresponding kinetic system is fixed, which also implies the existence of balance between degenerate nonlinear reaction and diffusion. For example, even if the positive steady state of the corresponding kinetic system is globally asymptotically stable, we observe four different spreading-vanishing phenomena by selecting different initial values. In addition, the interspecific competition of one species may be harmful to the persistence of the other species by taking proper initial values. Our results show that the superior competitor in the sense of the corresponding kinetic system is not always unbeatable, it can be wiped out by the inferior competitor in the sense of the corresponding kinetic system depending on the size of initial habitats as well as the intensity of Allee effect.