标题： 通过格子差分方程的波传播动力学 显示英文标题

标题： Wave Propagation Dynamics via Lattice Difference Equations

摘要： 我们开发并分析了一个格子差分方程（LDE）框架，用于模拟种群入侵的空间动态。该框架超越了经典的积分差分和反应扩散模型，通过更真实地纳入空间离散性和生境破碎化，使其特别适用于城市和斑块状景观。我们表征了平衡点的局部稳定性，并证明了行波解的存在。一个重点是探讨扩散核——从高斯到柯西——如何与阿尔效应相互作用，从而影响波形形成、传播速度和入侵成功率。我们的数值模拟表明，长尾核可以通过播种效应克服阿尔阈值，显著加速波前。这些发现对媒介控制策略有直接的意义，为异质环境中的最优释放阈值和空间定位提供了依据。此外，我们推导了爆发规模的随机表征，作为泊松-二项分布，提供了对局部感染负担的概率见解，并为基于双峰性的释放策略的新最小化标准铺平了道路。我们的结果为基于{\it 沃尔巴克氏体}的媒介控制项目的空间扩散预测和释放策略优化提供了数学基础。这项工作连接了理论与应用，为理解离散生境中的空间流行病学提供了分析见解和计算工具。 显示英文摘要

摘要： We develop and analyze a lattice difference equation (LDE) framework to model the spatial dynamics of invasion in populations. This framework extends beyond classical integro-difference and reaction-diffusion models by incorporating spatial discreteness and habitat fragmentation more faithfully, making it well-suited for urban and patchy landscapes. We characterize the local stability of equilibria, and demonstrate the existence of traveling wave solutions. A key focus is on how dispersal kernels--ranging from Gaussian to Cauchy--interact with the Allee effect to influence wave formation, propagation speed, and invasion success. Our numerical simulations reveal that long-tailed kernels can overcome the Allee threshold through seeding effects, significantly accelerating wave fronts. These findings have direct implications for vector control strategies, informing optimal release thresholds and spatial targeting in heterogeneous environments. Furthermore, we derive a stochastic characterization of outbreak size as a Poisson-binomial distribution, offering probabilistic insight into local infection burden and paving the way to a novel minimization criteria of release strategies based on bi-modality. Our results provide a mathematically grounded basis for predicting spatial spread and optimizing release strategies for {\it Wolbachia}-based vector control programs. This work bridges theory and application, providing both analytical insights and computational tools for understanding spatial epidemiology in discrete habitats.