Fig. 1: The impact of observability on controllability for a natural resource harvesting system.

The colored regions show the controllability sets for a natural resource extraction system for our 3 observability conditions as labeled, the color representing the time to reach the target equilibrium. The black curves with arrows are system trajectories starting from different initial conditions A–E. Trajectories starting outside the controllability set can never reach the target equilibrium indicated by the red star while those starting inside do. Equilibria are stable when biomass or effort is observable but unstable when the only the harvest is observable in the case of the MSY target Panel (c). In Panels (a–c), the goal state, shown by the red star, is the equilibrium where the biomass isocline (red curve) intersects the effort isocline (green line/curve). When biomass Panel (a) and effort Panel (b) are observable, the equilibrium is stable whereas the equilibrium is degenerate when harvest Panel (c) is observable. Starting at point D, the system very slowly converges to the target. However, if the system exceeds this target (e.g. an effort a little bit higher than u_MSY due to, for example, a measurement error), the system will slowly collapse because the harvest isocline is tangent with the biomass isocline. This means that from initial conditions around the target (point E for instance), even though the system will approach the equilibrium and `slow down' to give the illusion that the system is stabilizing, the system will eventually collapse in the long term. Comparing Panel (d) with Panels (a and b) shows that more conservative targets are required to compensate for instabilities generated by policies based on observing output (harvest) that combine information about system states to achieve the the same level of controllability when observing system states (biomass and effort) directly. Comparing Panels (c and d) to Panels (a and b) shows how good decisions made with the wrong observation can fundamentally change the dynamics of controlled systems.