The spread of an invasive species represents the final stage of invasion once a population has been successfully transported and established. My aim was to investigate the pathways and patterns of spread in urban areas, as well as the management of two iconic invasive plant species by combining genetic and mathematical tools. In the southeastern United States, Pueraria montana var. lobata (kudzu) and Wisteria (likely a hybrid of Wisteria sinensis and Wisteria floribunda) are two invasive perennial leguminous vines. I found that persistence of kudzu populations varied across the invasive range, with high persistence in the core of the range. Furthermore, the genetic composition of populations over the growing season or dormant period was constant in some populations but varied drastically in others. Tracking gene flow within core populations of kudzu and Wisteria requires searching for potentially rare events. Therefore, genetic data from many samples was necessary, but next-generation sequencing techniques that are affordable for large sample sizes often result in excessive missing data. I developed an economical reduced-representation library preparation method, RADcap, to collect a repeatable fraction of the genome from thousands of samples. Using this method and landscape genetic models, I found that roads and rivers likely facilitate the spread of kudzu and Wisteria in urban areas. Finally, I constructed a mathematical model to optimize invasive species management. I compared management that prioritizes control of high-density patches versus previously managed low-density patches that result from incomplete control of perennial plants, and found that for vines which mature slowly, but once mature, have a high colonization rate, like kudzu and Wisteria, treating high-density patches would be optimal. For invaders that mature quickly, it is optimal to re-treat low-density patches. If elimination is deemed unattainable and the goal of management is to convert all heavily invaded patches to low-density patches, prioritizing the treatment of high-density patches is more effective. My work combines multiple scales of analysis to illustrate the dynamics of kudzu populations within a year and over multiple years, how kudzu and Wisteria spread across the urban landscape, and how to best manage their spread.