具有空间异质性的混合反应-扩散流行病模型中的波前传播 第一部分：传播速度与渐近行为
Front propagation in hybrid reaction-diffusion epidemic models with spatial heterogeneity. Part I: Spreading speed and asymptotic behavior
AI总结：
该研究针对含野生型与突变型病原体的空间异质混合反应-扩散流行病模型，分析了波前传播速度、齐次与快速振荡系数下解的渐近收敛性及均匀化极限。
摘要：
我们研究了一维空间中的两物种反应-扩散系统，该系统源自空间周期环境下的流行病学模型，包含野生型和突变型两类病原体。该系统具有混合特性，部分为合作型、部分为竞争型，但二者均不占主导，因此比较原理不适用于整个系统。我们研究了初始感染局域化时解的波前传播特性，证明系统在左右两个方向均存在明确定义的传播速度，且该速度可通过传播波前前沿处的线性化方程计算得到。随后我们研究了系数空间齐次的情形，证明当传播发生时，柯西问题的所有解都会在$t\to\infty$时收敛到唯一的正平稳解。最后我们考虑了系数快速振荡的情形，即系数的空间周期（记为$\varepsilon$）非常小的情况，证明此时存在唯一的正平稳解，且柯西问题的所有正解都会在$t\to\infty$时收敛到该平稳解，我们还讨论了$\varepsilon\to 0$时的均匀化极限。
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 for the whole system. We study spreading properties of solution fronts when the infection is localized initially. We show that there is a well-defined spreading speed both in the right and left directions and that it can be computed from the linearized equation at the leading edge of the propagation front. Next we study the case where the coefficients are spatially homogeneous and show that, when spreading occurs, every solution to the Cauchy problem converges to the unique positive stationary solution as $t\to\infty$. Finally we consider the case of rapidly oscillating coefficients, that is, when the spatial period of the coefficients, denoted by $\varepsilon$, is very small. We show that there exists a unique positive stationary solution, and that every positive solution to the Cauchy problem converges to this stationary solution as $t\to\infty$. We then discuss the homogenization limit as $\varepsilon\to 0$.
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