The problem of biological invasions has largely arisen as an artifact of globalization. Human movement and trade have increased during the last two centuries and this has been accompanied by the mostly inadvertent movement of species to new world regions where they have no prior evolutionary history; as a result, these species sometimes exhibit explosive population growth accompanied by undesirable ecological and economic impacts.

Given the immense impacts that these species can have, there may be good economic justifications for managing invasions. This management can be generalized as 1) preventing species from arriving in non-native habitats, 2) eradicating species once they have arrived and 3) limiting the spread of non-native species that have already established.

While considerable resources are expended to manage invasions, there are several good open questions related to how best to allocate those resources. Different species arrive in non-native habitats and there are unresolved questions about how best to allocate efforts to prevent arrivals – should these efforts target individual species or entire pathways? Answering these questions involves balancing benefits derived from averting species establishment with impacts of quarantine effort on the benefit of international trade. Similarly, there are many open questions with regard to efforts to detect and eradicate new invasions. Some of these questions focus on the allocation of resources between surveillance and eradication, while other questions focus on how best to allocate surveillance activities among geographical regions. Finally, there are several open questions related to the optimal allocation of resources for limiting the spread of invading species – e.g., should resources focus on finding and eradicating new satellite questions or on the production of propagules which form new populations?

There are several opportunities where mathematical analysis can provide useful insight into managing invasions more effectively. Given that trends of increasing globalization are driving increases in invasions, this work is particularly important for minimizing impacts in the future.