Control theory and ecology
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- Control theory and ecology
- Resource management examples
The classic control problem is framed as follows. For a continuous time formulation, the system is given by:
where the ẋi are rates of change for the state variables, the parameters are assumed fixed (e.g. are basic physical constants) and are hence not shown, and u is the intervention vector. The problem is to identify u of this augmented model such that some desired outcome is attained. In a rocket control problem, this might be the prevention of instabilities (wobble, spin) and the reaching of a target. In resource management, it might be the timing of pesticide application to maximize crop yield. In this formalism uit might be the insect kill rate at a time t, which acts to reduce the state variable for the pest population. We can be more explicit in writing equation 1 as:
by including the environmental parameter vector p, which governs the rates. In ecological control problems, many of our interventions actually alter p. We can partition our interventions u into v (those that affect parameters such as fertilization) and w (those that directly affect state variables, such as removal of biomass). If we designate q as the new (augmented) parameter set, then we get the algebraic relation:
More detailed treatment of v, w, and q is given below, but it is clear that this aspect of ecosystem control introduces even more complexity than that posed by the non-linear relations of the ecosystem itself. Control theory methods have also been developed for discrete-time systems.
Several control theory concepts can be carried over to the ecosystem management problem to help clarify issues of resource management. These concepts include controllability, observability and identifiability. In control theory, a component is input controllable if given inputs allow that component to be regulated. In an ecological system, we might ask if manipulation of tree density can cause an alteration of herbivore abundance and thereby affect predator abundance. If so, then the predator population is at least partially controllable. This kind of question underlies every attempt to manage ecosystems, protect endangered species and extract natural resources. Thus the concept of input controllability applies to ecosystems. However, in contrast to classic control theory, control need not only be in terms of altered mass inputs but may also alter process rates (discussed below). It is rare for controllability to be specifically analysed for an ecological system. Yet a priori analysis of controllability could provide a method for feasibility analysis for proposed management actions, which at present are often undertaken based on general concepts (e.g. patch connectivity and source–sink dynamics).
The concept of observability (Cobelli, Lepschy & Romanin-Jacur 1979) is that if the outputs from a system uniquely define certain internal (not monitored) states, then those states are observable. Because ecological systems are not single valued (van Nes & Scheffer 2003), components that are not monitored cannot necessarily be inferred from other components. For example, a given predator population may be exploiting one of several herbivores, depending on their abundances. Predator numbers could be low because of disease, present low food levels or past low food levels. Thus, we cannot necessarily infer (observe) the herbivore population level from the predator population level, or conversely. We may be able to do this if we have a long time series of data that includes multiple components and we have a proper model of the system, but it is not guaranteed. Observability is rarely assessed explicitly in ecology.
Another concept is identifiability (Beck 1979; Cobelli, Lepschy & Romanin-Jacur 1979). In linear control theory, the structure and parameters of a system can be identified (parameters accurately estimated) from a sufficient set of system outputs (Leigh 2004; Walter et al. 2004). In ecology this is rarely true. Either the system has non-linearities (e.g. the functional response to prey density) or certain key processes are difficult to observe (e.g. root respiration). In the latter case, the observable components do not define a unique system.
To clarify the concepts of observability, controllability and system identification, it is useful to consider a specific case. Figure 1 shows a simple plant model. The plant is conceived as growing alone and being similar to a grass with leaves and roots. The masses of leaves and roots both affect the rate of photosynthesis. Both leaves and roots are subject to death and herbivory. Leaves leach photosynthate and roots have exudates by which carbon is lost. Above-ground carbon is translocated below-ground. Observability for a plant like this is not intuitively obvious. We can sample the leaves and even the roots, but doing so requires destructive sampling. Various experimental methods have been devised to get around this problem, such as growing large numbers of plants and destructively sampling some of them at given times, or counting the number of leaves and their length and comparing this with regression models for biomass. Most of the transfers of mass in this system are not easily observable in a direct way. Leaves that die can be counted, but roots that die tend to become lost in the soil. The leaching of compounds from leaves and losses from roots via exudates are so hard to observe that for decades these processes were not suspected at all. These mass transfers have been identified following years of detailed experiments using tracers, careful observation and other methods. Some processes can be inferred from mass balance calculations (they are indirectly observable) but experimental error precludes this method for small magnitude transfers such as leaching from leaves. Some processes cannot be observed at all, such as the effect of root mass on photosynthesis. These processes must be inferred entirely from the use of models and a comparison of model output with experimental data.
Figure 1. A simple plant system showing mass transfers (arrows) and controls (bowties) on photosynthesis and translocation. Most of the mass transfers cannot be directly observed on a living plant, but must be inferred from experiments, as must the functional forms for flow controls.
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System identification for such a system is again not entirely intuitive from an engineering standpoint. We do not have the exactly correct physical model for the growth of the plant. Even with error-free measurements of the states of the system, such as leaves and roots, we might still have an indeterminate system mathematically because we do not have direct estimates of the parameters governing processes, such as death, herbivory and photosynthesis. Thus, standard practice is to combine experimental estimates of some of the process rates with models containing free parameters that are then fit to experimental data. Because a physical model of the system is not available, and because many of the parameters are estimated indirectly, there is considerable sloppiness in this type of system identification procedure.
When considering system controllability, it is tempting to view the diagram in Fig. 1 as if all of the arrows could be manipulated. In fact, very few of the processes in this system can be directly manipulated. It is possible to add herbivores above-ground or kill them, but herbivory below-ground is almost impossible to control. Photosynthesis can be manipulated by altering the carbon dioxide content of the air, temperature, light levels and other factors, but translocation below-ground cannot be affected by experimental means. Thus, great care must be taken when moving from a diagram or model of an ecological system to questions of controllability and manipulation.
This is the fundamental problem of resource management. The system itself, although ‘visible’ in the literal sense, is not necessarily ‘observable’ in the systems theory sense. Without exact physical models of the system and with poor observability, system predictability is poor, as is our level of understanding.
A second major difficulty in controlling ecosystems is that feasible management options are generally extensive rather than intensive. The cheapest way to apply herbicide is aerially, but this method does not allow precision. Spot application of herbicide is possible, but costs much more. Larger timber harvest units have lower per volume harvest costs.
The manner in which control can be exerted on ecosystems puts serious constraints on what environmental or research management goals can be achieved. Highlighting the available options helps clarify why classic control theory does not always work in resource management. Manipulation of a biotic system can be achieved via three possible routes: state control, parametric (environmental) control and physical habitat control. State control is the addition or removal of biotic components. For example, mowing, burning, herbicide application, timber harvest, hunting, planting and stocking fish are in this category. Parametric control refers to alteration of environmental factors that influence organism vital rates and responses. For example, pollution, urban heat islands, fertilization and irrigation all alter organism growth and survival rather than their mass directly. Physical habitat control is the actual restructuring of the physical habitat. Examples include damming a river, addition of nest boxes, submerging cars to create coral reefs and installing drain tiles. These control actions can be characterized by spatial extent, intensity (degree of hunting or number of animals introduced), timing and spatial and/or temporal pattern. Only in the case of domesticated plants and animals do we expect any direct control over internal constants of the system, such as plant morphology and photosynthetic response.
In each of these dimensions, human interventions are generally extensive and not necessarily precise. It is not too hard to set hunting limits for deer that are enforceable, but it turns out to be very hard to get hunters to go precisely where the deer need to be shot or to change hunting limits rapidly. Thus hunting pressure is spatially uneven. It is very hard to design fishing nets that catch only the species desired (Vaca-Rodríguez & Enríquez-Andrade 2006). Cattle are controlled only generally by fencing them into large enclosures, but their daily movements cannot be controlled. Fire can be used, but not with spatial precision. Fire will burn too hot in some spots and will jump other spots. A forest can be fertilized, but only broadcast fertilization is economical.
In standard control theory, engineered systems can be built as designed based on analytic models. This is not necessarily true in ecology. Because models of ecological systems are non-linear, they cannot in general be solved analytically (Seppelt & Richter 1995). This means that simulation is necessary, which precludes the analytic generation of control equations. Finding an optimal control strategy then requires a link between the simulation and a non-linear optimizer of some sort, with no guarantee of finding an optimal policy. In fact, only ad hoc policies can be developed with this approach. As a simulation run over some time interval can take seconds to hours, and an optimization algorithm must call the simulation hundreds to thousands of times, it is clear that this approach can require a huge amount of computer time. It may even be impractical to use this approach with more realistic or spatial simulation models, in which case scenarios are often evaluated instead. In many cases when simulation and optimization are linked, a true control problem is not solved because the system is modelled near equilibrium rather than dynamically.
- Top of page
- Control theory and ecology
- Resource management examples
The above diagnosis of the application of control theory in ecology leads to some recommendations. Control theory clearly needs to be brought back into the spotlight for the management of ecosystems, whether for natural resources, amenity values or endangered species restoration. The design of monitoring programmes, field surveys (Ringold et al. 1999) and habitat conservation plans (Wilhere 2002) could benefit from consideration of concepts of observability (Field et al. 2004; Field, Tyre & Possingham 2005), although this is rarely done at present. Which components of the system when observed will provide information about other components? For example, at certain population levels of deer it is no longer necessary to sample the forest understorey to know that there is not much left. Such analyses can reduce sampling costs. Similarly, system identification concepts can help guide field data collection and experiments designed to parameterize a model (Walter et al. 2004). The few system identification-type studies that could be located (Lindley 2003; MacKenzie et al. 2003; Clark & Bjørnstad 2004; Wu, Fukuhara & Takeda 2005) are quite promising but very limited in number. Controllability analysis can be used to assess a priori whether given management actions can hope to influence a particular species or output, although no examples of this approach could be located.
Control theory should also be considered more seriously for the design of management guidelines and strategies. To overcome the problems presented in this paper, it is critical that the effect of discrepancies between the system model and control assumptions and the real world (Englund & Moen 2003; Jensen & Ginzburg 2005) be considered explicitly. There is often a gap between paper management plans and their implementation (Bellamy et al. 2001; Wilhere 2002; Stankey et al. 2003). For example, in the restoration of oceanic islands, removal of introduced pests (from domestic goats to snakes, rats, rabbits and birds) has been proposed as a restoration strategy. However, if every last animal is not removed, the pest population will rebound in just a few years. Removal of every animal is probably not possible, so such a policy is not feasible in the real world, even though mathematically it is a simple and optimal solution. Measurement error was shown in the fishery example to result potentially in oscillatory behaviour and management failure. It therefore needs to be considered explicitly. Some types of noise could be reduced if climate cycles (e.g. Pacific Decadal Oscillation) can be identified and factored into control policies in a feed-forward design (Yndestad 1997). Lags in response are a widespread problem in resource management. Conservation concepts when enshrined in law may take a long time to change even when everyone agrees they are out of date. For exploitation systems such as grazing systems and fisheries on public or common lands and waters, there is often resistance to reduction in harvest rates until evidence of damage and depletion is obvious and severe. More reliable forecasting based on better models and better control theory-based policies can help overcome this problem by increasing the public's confidence that predictions of future impacts are based on sound science. To cope with the reality that spatial pattern and heterogeneity can strongly influence many ecological processes, control theory needs to be extended to spatially explicit models (Hof & Bevers 1998; Akçakaya 2001). With whatever model is available, it is critical that the effect of uncertainty on the result be evaluated. For example, noise enters not only as stochastic weather over time but as uncertainty in the recruitment response equation form, as parameter estimation uncertainty and as error in implementation of the policy. The robustness of control strategies to these types of noise/error needs to be evaluated. Control strategy robustness can also be evaluated in terms of system bifurcation. If alternate stable states (Jasinski & Payette 2005) exist, the control policy can be tested for how well it prevents capture of the system by the low-yield regime. Finally, natural history does matter. Standard population dynamics models may not capture the key factors limiting a species, particularly endangered species. Management must begin with an understanding of the unique characteristics of the species. Overall, control theory offers tools and concepts indispensable to the practice of resource management.