Stabilization or regulation: what it all means!

Wolda (1989), in the title of his commentary on the concepts of equilibrium and density-dependence, asks "What does it all mean?" This note attempts to answer his question. In his paper, Wolda concludes that, because the concept of density-dependent population regulation rests on the a priori assumption that "populations fluctuate in size around an equilibrium value," and because "the equilibrium cannot be measured," the concept is "devoid of any practical ecological content" and "tests for population regulation cannot be expected to produce the expected results, making them next to useless. " Wolda then goes on to recommend the concept of stabilization (Reddingius and Den Boer 1989), though he does not explain why it is better. I will not belabor the semantic arguments surrounding the meaning of regulation, stabilization and densitydependence which have burdened the ecological literature for decades, and which I have addressed in another paper (Berryman 1989), but instead concentrate on several misconceptions in Wolda's commentary. 1. Wolda claims that "regulationist" arguments rest on the a priori assumption that a stable equilibrium (i.e., a point attractor1) exists. This implies the logic that stable equilibria are a necessary condition for densitydependence, or that equilibria give rise to density-dependence. Regulationists may have made this statement, but it is nevertheless illogical. As I have noted previously (Berryman 1987), equilibria are properties that may emerge from the operation of a dynamic system but they are not the causes of the dynamics. For example, models of predator-prey interactions may or may not exhibit a two-species community equilibrium, depending on the values of their parameters (e.g., see Arditi and Berryman 1991). Although Wolda cites my paper (1987) as a "mere dismissal" of the equilibrium concept, he entirely misses this point, or chooses to ignore it. Thus, the presence of an equilibrium is not a prerequisite to a theory of population dynamics nor to the analysis of population data. Regression, time-series, and key-factor analyses do not require that equilibria exist, but equilibrium structures may emerge as a result of these analyses. For example, attractors emerge from the analyses shown in figure 1 because the parameters have particular values; other values can produce no equilibria or unstable (repelling) equilibria. 2. Wolda implies that, because populations are observed to fluctuate with considerable amplitude, an equilibrium, if it exists at all, must also be changing with time and therefore cannot be estimated empirically. This misconception is due to a narrow perception of equilibrium as a stable node or point attractor. The modern theory of non-linear dynamics, which has been greatly influenced by population ecology, makes it clear that very diverse dynamic behaviors are possible from non-linear feedback systems, including periodic, quasiperiodic and pseudoperiodic cycles (cyclical attractors) and aperiodic chaos (strange attractors) (May 1974; Schaffer and Kot 1986; Berryman and Millstein 1989). Indeed, it may not be possible to determine if the observed fluctuations are due to random buffeting or to deterministic mutually causal (feedback) relationships between variables (e.g., inter-species interactions) (Ellner 1991). If this is not enough, we now know that populations can be influenced by more than one domain of attraction, giving rise to very complicated metastable behavior in varying environments (Ludwig etal. 1978; Berryman etal. 1984). Thus, wide fluctuations in population dynamics can be caused by many things including, but not necessarily limited to, endogenous periodicity and chaos, multiple domains of attraction, time-varying equilibria, stochastic buffeting, or any combination of the above. In addition, the fact that equilibrium structures can be influenced by time-varying forces may make their estimation difficult but it does not invalidate the concept of 1 The term attractor, meaning a region that attracts nearby trajectories of a dynamic variable, is introduced to help clarify the concepts of regulation and stability : Point attractor is a stable equilibrium that attracts trajectories to a fixed point. Cyclical attractor brings trajectories into a stable periodic, quasiperiodic or pseudoperiodic orbit; e.g., a predator-prey cycle. Strange attractor is a stable aperiodic orbit that may have a multiple equilibrium structure; i.e., a complex basin of attraction