Robust population management: A methodology based on integral control
Population management or regulation aims to bring ecological systems to desired population densities, for example restocking a declining plant population by planting seedlings grown in greenhouses. Managed regulation to constant set points is ubiquitous for natural and man–made systems. It includes regulation of blood sugar by insulin; bacterial chemotaxis in cells; and navigation of supertankers across stormy seas. A key feature in all applications is that set–point regulation must be robust to parametric uncertainty and measurement error. In fact, this robustness of set-point regulation necessitates the use of integral control. Integral controllers can be implemented in the presence of system uncertainty and using only limited system information. These two features make integral control appealing for management of ecological systems. We develop a strategy for replenishing a managed, but declining, population. We assume stage-structured populations. For management purposes, we need access to observations of the stage(s) of interest and, crucially, such observations are the only information available for determining the management strategy. The replenishment strategy must be independent of the initial population structure and be robust to model uncertainty. This problem is ubiquitous in feedback control engineering and we exploit ideas from this field here in an ecological context.
Our strategy for managed replenishment uses a feedback mechanism. This mechanism is widely applicable but is most easily described for an endangered plant in a context of restocking by planting seedlings. The number of seeds planted in each time-step is adjusted additively by an amount proportional to an offset between the current observed stage and the desired set point. We demonstrate that this simple “integral control” strategy works and is robust to numerous concurrent sources of uncertainty. These uncertainties may include parametric model uncertainty or stochastic noise. We can also incorporate natural constraints on the available resources for replenishment. In the case of planting seedlings this might mean a constraint arising from a fixed greenhouse capacity. By adding functionality to the strategy, by including a component in the strategy that is directly proportional to the offset, we can speed up the convergence of the managed population to the desired level. We can also make the parameters (feedback gains) in the strategy “adaptive”, meaning that the strategy does not need to know anything about the model. We show that the same management strategy can be applied to other ecological models that include spatial components, for example integral projection models.