Sharp logarithmic corrections for a strong-weak Lotka-Volterra competition system

Abstract

We study the one-dimensional strong-weak Lotka-Volterra competition-diffusion system \[ u_t=u_{xx}+u(1-u-av),\qquad v_t=dv_{xx}+rv(1-v-bu), \] with compactly supported initial data under the parameter condition $0<a<1<b$. Existing literature only gives leading-order spreading speed asymptotics without refined logarithmic corrections for wave fronts over the full parameter space. We convert the competitive system into an equivalent cooperative parabolic system and establish moving-domain comparison principles. Based on heat-kernel estimates, we derive sharp logarithmic asymptotic expansions for the rightmost level set of the stronger species $u$. According to the magnitudes of three characteristic speeds, five long-time dynamical regimes are classified, including pulled, nonlocally pulled, pushed and critical transition fronts, with explicit logarithmic and double-logarithmic phase corrections. Our results capture delicate long-time phase offsets of wave profiles neglected in previous leading-order theories.