5. Reference levels based on nonlinearities

Nonlinearities are common in nature (Beisner et al. 2003, Scheffer et al. 2001). Sudden change in ecosystem attributes can result from seemingly smooth and gradual change in physical or biological components (Muradian 2001). For instance, in kelp forests, increasing sea urchin densities initially produce small or negligible changes in habitat-providing kelp. However, above a threshold sea urchin density, declines in kelp and changes in associated ecological communities can be quite rapid (Ling et al. 2009, Steneck et al. 2002). Similarly, on coral reefs, important ecosystem functions decline rapidly with initial increases in human impacts, but thereafter change quite slowly (Knowlton and Jackson 2008, Bellwood et al. 2003). These examples illustrate that nonlinearities in functional relationships distinguish environmental conditions or types of management actions leading to smooth and proportional changes in ecosystem state from those that cause abrupt and disproportionately large changes. An understanding of nonlinearities is highly relevant in the context of managing the Puget Sound ecosystem because it presents opportunities to define clear and objective reference points (Martin et al. 2009, Samhouri et al. 2010).

Nonlinear functional relationships underpin commonly-used management reference points in fisheries and in the control of contaminants in the environment (e.g., chemicals, effluents, non-native species, etc.). For instance, the spawning stock biomass and the fishing mortality rate corresponding to maximum sustainable yield are two of many biological reference points used in single-species fisheries management (Jennings et al. 2001). The concept of maximum sustainable yield is based on the expectation that the yield from the fishery peaks at intermediate levels of population biomass and fishing mortality rate imposed on the target population. These nonlinear relationships are the consequence of assumptions in surplus production models of fish population dynamics, and make it possible to identify objectively a reference point on either side of which fishing yield is reduced. In ecotoxicology, contaminants frequently have little or no deleterious effects on biota below some minimum concentration but lead to serious sublethal or lethal effects thereafter (Figure 13 a,b). Thus, a reference point can be defined based on a threshold in such exposure-response relationships (Suter 2007). In both situations, the reference points are linked mathematically to a functional relationship of interest to managers and policymakers (Link 2005). The functional relationships most relevant in a marine EBM context fall into two broad categories (Beisner et al. 2003). In both cases, the response variables of interest are ecosystem attributes that influence ecosystem health, and might include nutrient cycling, energetic rates, and resilience. These are akin to the toxin concentrations in ecotoxicological studies. In the first category, the predictor variable (analogous to the exposure effect in ecotoxicological studies) is some environmental condition(s). For example, reductions in the amount of upwelling along the west coast of the United States are associated with an exponential increase in seabird mortality events, which appear to be indicative of broader changes in ecosystem attributes, such as productivity (Parrish et al. 2007). In the second category, the predictor variable is a factor(s) under the control of managers and policymakers. For instance, a marine food web model for northern British Columbia suggests that several ecosystem attributes show nonlinear declines with increasing fishing pressure and with reductions in nearshore habitat quantity and quality (Samhouri et al. 2010). In both cases, it is possible to define mathematically a point separating rapid and dramatic changes in the ecosystem attributes from more smooth and gradual changes (Figure 13c,d).

Reference levels for ecosystem indicators can be derived from either category of nonlinearity. The guidelines for selecting a reference point based on a functional relationship between predictable environmental conditions or factors under the control of managers and policymakers and ecosystem attributes are as follows:

1. Examine the functional relationship of interest, using data, models, or both;
2. Use information theoretic techniques (Burnham and Anderson 2002) to fit alternative linear and nonlinear mathematical functions to the relationship;
3. If the best-fit function is nonlinear, select a reference point that distinguishes the steep from the shallow portion of the curve (Samhouri et al. 2010).

Reasonable target reference levels for the sigmoidal and concave functional relationships shown in Figure 13c would correspond to portions of the curves where the value of the ecosystem attribute is high and the rate of change in the ecosystem attribute with increasing human pressure is low, i.e., where the dashed arrows intersect the curves.

The identification of nonlinear relationships between pressures and ecosystem attributes could be used productively to set target reference levels in Puget Sound. One way to detect nonlinearities relevant for food web health in particular would harness the power of a recently developed Ecopath model for the Central Basin of Puget Sound (Harvey 2010). Indeed, Samhouri et al. (2010) recently followed the methods outlined in steps 1-3 above to determine food web reference levels associated with two different stressors (fishing and habitat modification) along the British Columbia coast. Empirical examples of nonlinearities already exist as well. For instance, Rice (2007) found that there was a drastic and abrupt decline in the abundance of diving ducks and herons in Puget Sound above ~70% alongshore urban land cover. Given the potential for these species to act as reliable indicators of ecosystem health (Fulton et al. 2005, Samhouri et al. 2009), a target reference level for their abundance based on the effects of urbanization may be sensible.

A concerted effort to gather information about functional relationships between ecosystem indicators and pressures would greatly advance efforts to set target and benchmark reference levels in Puget Sound. These reference points should be considered complementary to those based on baseline conditions.

Figure 13. Examples of nonlinear relationships in ecotoxicological (a-b) and ecosystem (c-d) studies. (a) A hockey stick relationship in which the reference point could be either the LOEC (lowest observed effect concentration), i.e., the lowest concentration causing an effect that is statistically different from control (upper 95% CI of x-axis threshold estimate), or a NOEC (no observed effect concentration), i.e., the highest concentration below LOEC (could be lower 95% CI of x-axis threshold estimate). (b) A sigmoidal relationship in which the reference point is an Ecp, the concentration causing the effect in proportion p of the population (e.g., LC50). (c) It is possible to identify objectively a reference point in terms of human pressure if the relationship between the predictor variable and the ecosystem attribute is sigmoidal or concave. (d) A convex relationship suggests that management actions that reduce human pressures to steeper portions of the function will produce the greatest improvements in the ecosystem attribute. Linear functions do not allow the objective identification of a threshold-based reference point. In all figures, dashed arrows indicate possible reference points. In (c) and (d), positive values on the y-axis are assumed to represent the desired state of the ecosystem attribute.