Invasive alien species continue to threaten global biodiversity. CRISPR-based gene drives, which can theoretically spread through populations despite imparting a fitness cost, could be used to suppress or eradicate pest populations. We develop an individual-based, spatially explicit, stochastic model to simulate the ability of CRISPR-based homing and X-chromosome shredding drives to eradicate populations of invasive mice (Mus muculus) from islands. Using the model, we explore the interactive effect of the efficiency of the drive constructs and the spatial ecology of the target population on the outcome of a gene-drive release. We also consider the impact of polyandrous mating and sperm competition, which could compromise the efficacy of some gene-drive strategies. Our results show that both drive strategies could be used to eradicate large populations of mice. Whereas parameters related to drive efficiency and demography strongly influence drive performance, we find that sperm competition following polyandrous mating is unlikely to impact the outcome of an eradication effort substantially. Assumptions regarding the spatial ecology of mice influenced the probability of and time required for eradication, with short-range dispersal capabilities and limited mate-search areas producing `chase' dynamics across the island characterised by cycles of local extinction and recolonization by mice. We also show that highly efficient drives are not always optimal, when dispersal capabilities are low, rapid local population supression around the introduction sites can cause loss of the gene drive before it can spread to the entire island. We conclude that, although the design of efficient gene drives is undoubtedly critical, accurate data on the spatial ecology of target species is critical for predicting the result of a gene-drive release.