AI 中文总结
研究具有空间异质性的混合反应扩散流行病模型，证明了脉动行波存在性及前沿指数衰减率，给出\(c^*_R\)和\(c^*_L\)不同的例子，突出该系统与标量KPP型方程差异。
AI 中文摘要
我们考虑一个一维空间中的两种群反应扩散系统，它源自具有野生型和突变型两种病原体的空间周期环境中的流行病学模型。该系统具有混合性质，部分合作部分竞争，但都不完全。因此比较原理不成立。在之前的工作中，我们研究了此系统柯西问题解的传播性质，表明左右前沿的传播速度\(c^*_R\)和\(c^*_L\)可用某些主特征值表征，研究了空间周期\(L\)趋于\(0\)时的均匀化极限，还讨论了前沿后解的长期行为。本文证明了对于任意\(c\geq c^*_R\)（分别地，\(c\geq c^*_L\)），存在向右（分别地，向左）速度为\(c\)的脉动行波，其中\(c^*_R\)和\(c^*_L\)表示上述左右方向的传播速度。我们还证明了任何行波的前沿具有形式线性分析预期的指数衰减率，从而将Hamel 2008的部分结果扩展到方程组。最后，通过考虑一个多尺度奇异极限问题给出了\(c^*_R\)和\(c^*_L\)不同的例子。该结果突出了我们的系统与标量KPP型方程之间的显著差异。
英文摘要
We consider a two-species reaction-diffusion system in one space dimension that is derived from an epidemiological model in a spatially periodic environment with two types of pathogens: the wild type and the mutant. The system is of a hybrid nature, partly cooperative and partly competitive, but neither of these entirely. As a result, the comparison principle does not hold. In the previous work, we studied the propagation properties of the solutions to the Cauchy problem for this system and showed, among other things, that the spreading speeds of the fronts to the right and to the left directions, denoted by $ c^*_R$ and $ c^*_L$, can be characterized by using certain principal eigenvalues, and studied the homogenization limit as the spatial period $L$ tends to $0$, and also discussed the long-time behavior of solutions behind the fronts. In the present paper we prove the existence of pulsating traveling waves in the right direction (resp. left direction) with speed $c$ for any $c\geq c^*_R$ (resp. $c\geq c^*_L$), where $c^*_R$ and $c^*_L$ denote the aforementioned spreading speeds in the right and left directions. We also prove that the leading edge of any traveling wave has the exponential decay rate that is anticipated from formal linear analysis, thus extending part of the results of Hamel 2008 to systems of equations. Finally, we present an example in which the two speeds $c^*_R$ and $c^*_L$ are different. This is done by considering a multi-scale singular limit problem. This result highlights a marked difference between our system and scalar KPP type equations.
CommentsarXiv admin note: substantial text overlap with arXiv:2408.07501
