Suppression gene drives are designed to bias their inheritance and increase in frequency in a population, disrupting an essential gene in the process. When the frequency is high enough, the population will be unable to reproduce above the replacement level and could be eliminated. CRISPR suppression drives based on the homing mechanism have already seen success in the laboratory, particularly in malaria mosquitoes. However, several models predict that the use of these drives in realistic populations with spatial structure may not achieve complete success. This is due to the ability of wild-type individuals to escape the drive and reach empty areas with reduced competition, allowing them to achieve high reproductive success and leading to extinction-recolonization cycles across the landscape. Here, we extend our continuous space gene drive framework to include two competing species or predator-prey species pairs. We find in both discrete-generation and mosquito-specific models that the presence of a competing species or predator can greatly facilitate drive-based suppression, even for drives with modest efficiency. However, the presence of a competing species also substantially increases the frequency of outcomes in which the drive is lost before suppression is achieved. These results are robust in models with seasonal population fluctuations. We also found that suppression can be somewhat more difficult if targeting a predator with strong predator-prey interactions. Our results illustrate the difficulty of predicting outcomes of interventions that could substantially affect the populations of interacting species in complex ecosystems. However, our results are also potentially promising for the prospects of less powerful gene drives in achieving successful elimination of target pest populations.