强弱Lotka-Volterra竞争系统的尖锐对数修正
Sharp logarithmic corrections for a strong-weak Lotka-Volterra competition system
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中文总结 AI 辅助
本研究针对一维强弱Lotka-Volterra竞争扩散系统，通过转化为等价合作抛物系统建立动边界比较原理，推导了较强物种最右侧水平集的尖锐对数渐近展开，分类出五种长时间动力学 regime，捕捉了波剖面的细微相位偏移。
中文摘要 AI 辅助
我们研究一维强弱Lotka-Volterra竞争扩散系统，其方程为\( u_t=u_{xx}+u(1-u-av)\)、\( v_t=dv_{xx}+rv(1-v-bu)\)，初始数据具有紧支集，参数条件为\(0<a<1<b\)。现有文献仅给出全参数空间内波前的主导阶传播速度渐近行为，未包含精细的对数修正。我们将该竞争系统转化为等价的合作抛物系统，并建立动边界比较原理；基于热核估计，推导了较强物种\(u\)最右侧水平集的尖锐对数渐近展开。根据三个特征速度的大小，分类出五种长时间动力学 regime，包括被拉、非局部被拉、被推及临界过渡波前，带有明确的对数和双对数相位修正。我们的结果捕捉到了先前主导阶理论所忽略的波剖面细微长时间相位偏移。
英文摘要
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.
