Alternative Stable States

The current condition of an ecological system is its state. Because , it is possible for them to occupy more than one state for a given set of environmental conditions. In other words, while a system may have the Which alternative state is occupied depends on the history of the system and the perturbations to which it has been exposed. Alternative states are stable when they are maintained following further small perturbations. In the following article we will learn more about alternative stable states in ecology and develop a simple model that reveals how important related concepts are defined.

There are many observed examples that infer the presence of ASS (see Beisner et al. 2003b, Folke et al. 2004 and Knowlton 2004 for reviews), including the collapse of fisheries owing to food web interactions and harvesting (Steele 1998, Jackson et al. 2001) and the shift from healthy coral reef communities to macroalgae (Hughes 1994) in marine environments. In the Serengeti, there are shifts between woodlands and herbaceous plants regulated by elephant activity and their pests (Dublin et al. 1990). Another commonly studied example is a shift between clear and turbid water states in ponds and lakes (e.g. Scheffer 1993, Beisner et al. 2003a). There are fewer examples from experimentally-induced ASS (but see Van de Koppel et al. 2001, Petraitis and Dudgeon 2004, Schröder et al. 2005) and one reason may be because the scale at which disturbances must occur are often too large for experimentation (Petraitis and Latham 1999). ASS are not restricted to ecological systems however; rather they are a generic feature of dynamical systems. Thus we can also find examples from the collapse of societies (Diamond 2005), in the tendencies of stock markets, as well as crime and disease epidemics (Gladwell 2000) among others.

The fact that ASS can exist is important for a number of reasons. Ecologists are interested in observing and understanding their study systems so as to be able to predict what they might do in the future. The problem with ASS is that if we do not know that they are possible, they present surprises. Thus, our ecological frameworks need to include them, where they may occur, to predict when and why a sudden change might happen. Just knowing the habitat conditions is not good enough if it is possible that two states might persist there. We also need to know if a state change will be easily reversed or not, and how this might best be done if the outcome is an undesirable one. Ecologists must therefore understand the dynamics of their systems to avoid surprising catastrophic changes that are difficult to reverse. To understand dynamics, we need a good theoretical framework, best accomplished with a mathematical model. We will now explore how one might build such theoretical understanding by using a simple example. Through this modelling example we will be able to clearly see how important related concepts to the study of ASS, like resilience and hysteresis, are defined and fit together. Also the catastrophic potential for mistakes that can arise from managing an ecological system as if ASS weren't present become evident through a modelling framework.

(i) The model:

One of the earliest frameworks for ASS was developed by Noy-Meir (1975) for grazing animals on pasture land. In the model, vegetation grows based on resource availability, but the density of herbivores is controlled by human managers who decide how many grazers to allow on a patch of land. The dynamics of the vegetation (V) through time (t) can be written generally as:

(Eq. 1)

where G(V) is a function defining vegetation growth and c(V) is a function describing the consumption of vegetation by herbivores (H).

A graphical approach provides the simplest glimpse into the dynamics of the system (Fig. 1). The vegetation population grows (G(V)) logistically, with high growth rates when plant biomass is low and resources are abundant, but then declining at higher biomasses as competition for limited resources increases (Figure 1a). Vegetation is consumed (c(V)) by the herbivore according to a Holling Type III functional response (Fig. 1b). The total amount of vegetation removed depends on the stocking density of the herbivores (H), as set by the human managers of the pastureland.

(ii) Defining the "states":

Now we can ask, what is the state (i.e. equilibrium biomass or V^*) of the vegetation in the pasture? And how does this state vary with the number of herbivores stocked. Equilibria occur when the vegetation biomass is no longer changing, thus when dV/dt = 0. Following through algebraically for Eq.1, that means G(V) = c(V)H; and graphically, that the equilibria occur at the intersection of our two sets of lines (Figures 1a and 1b), when plotted together (Figure 1c).

(iii) Stability of "states":

We can now examine the dynamics around each of the equilibria (states) in Figure 1c by asking what happens if we change V very slightly from V^* for different levels of herbivore stocking. V will simply return to V^* in cases when there are many or few H (top two panels). Graphically, this is because moving the dot to the right along the G line results in more herbivory than plant growth (c line > G line). Likewise, a slight decrease in V (moving dot to the left along the G line) leads to greater plant growth than herbivory (G line > c line). Thus the system always returns to the original equilibrium point. These states or equilibria are therefore called stable.

However in the case of moderate herbivory (Figure 1c; bottom panel), a different situation occurs. Note that there are three equilibrium points now (two filled and one empty dot; Figure 1c). While the two extreme ones (U and L) are stable (as in the cases discussed already), the middle one (M) is unstable. For the middle equilibrium point, a slight increase in V leads to dominance of plant growth (G line > c line) and a continued shift to the right as plants will grow, until they reach the upper equilibrium point (U). A slight decrease in V leads to greater herbivory (c line > G line) and thus further reduction in V^* to the lower stable equilibrium (L). For this intermediate herbivore stocking density therefore, we have an example of ASS: there is a low V^* (L) and a high V^* (U) separated by an unstable equilibrium (M).

(iv) Generating S-curves:

Another way of examining the dynamics of equilibria is by using phase space to plot all possible equilibria in a graph of V vs. H. Understanding such plots is important to interpreting ASS, as one of their common defining features is a phase space plot with a characteristic S-curve (Figure 2) (Schröder et al. 2005). Figure 2 shows how an S-curve is generated for our model: take the value of V^* (read off the X-axis) for each equilibrium dot in Fig. 2a and plot this on the Y-axis of Fig. 2b at the value of H to which it corresponds. For example, the left-most (blue) point in Figure 2b thus represents the case where V is high and H is low (blue point in Figure 2a). Each equilibrium is thus transcribed to the new figure and then the points are joined together with a (reversed) S-curve to represent all possible equilibria. The lines marked f₁ and f₂ in Figure 2a represent important herbivore densities (called bifurcation points). Between these two H lines is the region within which alternative states are possible (i.e. three equilibria). Arrows around each equilibrium show the changes in the biomass of V that would result with slight perturbations from V^* as discussed in the previous section. Unstable equilibrium points occur along the dotted line in this region.

(v) Resilience:

S-curves allow us identify the resilience of the stable states. Resilience is the amount of change or disruption required to transform a system from being maintained by one set of mutually reinforcing processes and structures, to a different set of processes and structures (Peterson et al. 1998). It is an important concept, because it defines how close our system may be to a drastic shift. Returning to our example, for equilibria occurring along the top solid blue line in Figure 2b, all are infinitely resilient as long as H remains low: there will always be high vegetation biomass if there are few herbivores. For equilibria along the lowest end of the S-curve in red, this low V equilibrium is also infinitely resilient as long as herbivores are abundant. It is only for medium H's (orange lines) that resilience changes. Travelling from left to right, equilibria along the top part of the S-curve have decreasing resilience until f₂ is reached, at which point resilience of the high V^* state is 0. For equilibria occurring along the bottom arm of the orange part of the S-curve, moving again from left to right, resilience increases from 0 at f₁. The difference between each solid line and the dotted line defines resilience in each case. Thus, some factor (e.g. a lawn mower, an insect outbreak) that removed more vegetation than that difference for the top solid line would cause a shift to the lower equilibrium line. The opposite case would apply with respect to the resilience of the lower equilibrium line in the face of a "perturbation", such as land managers deciding to plant new vegetation (causing a tip toward the upper equilibrium line). When resilience of either state is low, small stochastic perturbations alone in V^* may result in a shift in state even without changing H.