Wave Propagation Dynamics via Lattice Difference Equations

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.