How to combine two methods to restore populations cost effectively

Ecosystems worldwide have become degraded due to global change, and therefore, restoration of these ecosystems is critical for the prolonged provision of ecosystem services. Specifically, major restoration efforts are directed toward the restoration of key species that provide important services and functions. There are often several alternative methods to restore a species population, such as the reintroduction of its individuals, improvement of its habitat quality, and removal of competing invasive species. However, these methods can be expensive, and hence, it is important to determine how to cost effectively combine them over time. In this paper, we use optimal control theory and we find a general rule of thumb for combining two restoration methods. The general rule, which applies to a wide variety of ecosystems, is that cost-effective restoration entails one of the following two strategies: (1) using a single method until the system approaches a restoration threshold or (2) combining both methods to approach an investment benchmark, which is a certain configuration of the system that does not depend on the system's initial state. After either the restoration threshold or the investment benchmark has been approached, investment should stop and the system should be left to recover naturally. Therefore, finding the restoration threshold and the investment benchmark is key for guiding effective restoration, and we demonstrate a simple method for finding them.