On competition models under allee effect: Asymptotic behavior and traveling waves

Abstract

In this article, we study a reaction-diffiusion model on infinite spa- tial domain for two competing biological species (u and v). Under one-side Allee effect on u-species, the model demonstrates complexity on its coexis- tence and u-dominance steady states. The conditions for persistence, perma- nence and competitive exclusion of the species are obtained through analysis on asymptotic behavior of the solutions and stability of the steady states, in- cluding the attraction regions and convergent rates depending on the biological parameters. When the Allee effect constant K is large relative to other bio- logical parameters, the asymptotic stability of the v-dominance state (0; 1) indicates the competitive exclusion of the u-species. Applying upper-lower so- lution method, we further prove that for a family of wave speeds with speciffic minimum wave speed determined by several biological parameters (including the magnitude of the u-dominance states), there exist traveling wave solutions owing from the u-dominance states to the v-dominance state. The asymptotic rates of the traveling waves at ξ → 1 are also explicitly calculated. Finally, numerical simulations are presented to illustrate the theoretical results and population dynamics of coexistence or dominance-shifting.