An Example

Lake eutrophication can provide a good model for explaining hysteresis using the ecosystem approach. Such cases have been modeled using structural dynamic models and using models based on catastrophe theory. Both types of models can demonstrate hysteretic effects. In the case of lake eutrophication, the possible community equilibria are defined in Figure 3 and they vary between turbid (high algal biomass) and clear water stable states (usually with aquatic plants or macrophytes dominating). Consider what happens when we start the system in the clear water state (left bottom corner of Figure 3 ). Total environmental phosphorus (TP) available for

n rB

Figure 2 Hysteresis arises when parameter changes occur and alter the surface upon which the ball sits. When the dynamics are governed by parameter set P1, one stable equilibrium point exists (labeled A). As the parameter set is changed toward P2 the state of the community tracks the route indicated by the solid arrows, until it finally arrives at the equilibrium point indicated in panel (d) (labeled B). However, if the parameters are then moved back toward P1, the community returns via a different route (hysteresis), indicated by the dotted arrows. In panels b) and c) two equilibria exist, but which is adopted depends on the history of the past perturbations. Reproduced with permission from Beisner BE, Haydon D, and Cuddington KL (2003) Alternative stable states in ecology. Frontiers in Ecology and the Environment 1: 376-382, © Ecological Society of America.

primary production of algae increases along the x-axis, and increases in TP are matched by an erosion of the basin of attraction defining the clear water state. Eventually, a two-state region develops in the middle of the graph, but, since the trajectory originated at the clear water state, observers will still see the lake as clear. The resilience of this clear water state will continue to erode, however, as TP increases across this middle region. (Resilience in this case is defined as the size of further perturbation the system can withstand and still remain in its current state.) In this region where two states are possible, stochastic events can initiate a 'flip' of the system to the turbid water state, as defined by the upper line of the S-curve. Such state shifts can occur in lakes, for

Turbid-water equilibrium Unstable equilibrium Clear-water equilibrium

Figure 3 Alternate states in lake algal biomasses are possible along a TP gradient. Dark solid lines represent stable states with the dashed line joining them indicating an unstable state that defines the boundary (separatrix) between the two basins of attraction. The bar to the right of the graph indicates algae conditions associated with each state. When raising nutrient levels (moving to the right), the state change is observed at a higher TP than when lowering TP, such as in lake remediation: a hysteretic response.

Turbid-water equilibrium Unstable equilibrium Clear-water equilibrium

Total phosphorus (TP)

Figure 3 Alternate states in lake algal biomasses are possible along a TP gradient. Dark solid lines represent stable states with the dashed line joining them indicating an unstable state that defines the boundary (separatrix) between the two basins of attraction. The bar to the right of the graph indicates algae conditions associated with each state. When raising nutrient levels (moving to the right), the state change is observed at a higher TP than when lowering TP, such as in lake remediation: a hysteretic response.

example, when especially heavy winter snowpack results in an unusually high pulse of nutrient-rich runoff to the lake in spring. Once TP increases beyond the second bifurcation point, the lake, if it has not already done so, flips to the turbid, or high algae state, because this is the only stable equilibrium possible. Models have been constructed to simulate the particular lakes where turbid and clear water states have been observed to vary from year to year. With these models it is possible to show that the switch to a turbid state from a macrophyte dominated one should occur in shallow lakes at TP levels greater than 0.15-0.25 mgl-1, at which point large zooplankton grazers can no longer control the rapidly reproducing phyto-plankton and recycling of TP from the sediment under anoxic conditions adds to productivity levels.

From a lake management perspective, a turbid state is a highly undesirable regime as frequent nuisance, and occasionally noxious, algal blooms reduce water quality. To return the lake to a clear water state, TP input needs to be reduced (Figure 3). On the return path, the lake will return via the turbid water state line in the region of TP parameter space where two states are possible because anoxic sediments created under turbid conditions continue to release TP into the water column. Before the clear water state can once again be realized, TP must be reduced considerably below that at which the switch to high algal biomass first occurred: a hysteretic response. Lake models also demonstrate this hysteretic response with a shift to a clear state, and the dominance of macro-phytes on the return path generally occurring at TP levels of 0.10 mgl~ . Turbid states show a certain buffering capacity in the face of nutrient reductions because of internal loading of TP to the lake from the sediments and because time lags in the responses of zooplankton and fish to a reduction in productivity levels.

The hysteretic response that is demonstrated clearly in lake models arises because at intermediate levels of TP

(between 0.1 and 0.25 mgl-1), there are two possible stable equilibrium solutions. Which one is actually observed depends on the state of the system in the previous timeframe. It has been shown with structured dynamic models that multiple solutions to the model arise because at intermediate TP levels, there is more than one way in which system properties (e.g., species) are best arranged to maximize survival.

Managers and ecologists are interested in the potential for hysteresis because it implies that communities and ecosystems might be easily pushed into some configurations from which it may prove more difficult to reverse. In the case of nutrient-enriched lakes, not only are alternative states likely, but also the hysteretic nature of these changes make them very expensive to clean up.

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