WHAT IS ECOLOGICAL STABILITY ? In 2019 I posed this question informally to colleagues, using Twitter, a professional workshop that I lead, and a conference. Respondents on Twitter consisted mostly of ecological scientists, but the workshop included paleontologists, biologists, physicists, applied mathematicians, and an array of social scientists, including sociologists, anthropologists, economists, archaeologists, political scientists, historians and others. And this happened…
Numerous terms, with roots across multiple disciplines that deal with dynamic complex systems, are used interchangeably in the study of transitions to some extent because they are related by process and implication. But they do not necessarily always refer to the same phenomena, and it is useful to be explicit in one’s usage (maybe at the risk of usage elsewhere). Regime shift, critical transition and tipping point are three of the more commonly applied terms in the ecological literature. They form a useful general framework within which to explore the concept of multiple states and transitions, and into which more detailed concepts can be introduced. Regime shift is defined here as an abrupt or rapid, and statistically significant change in the state of a system, such as a change of population size (Fig. 1A). Transient deviations or excursions from previous values, e.g. those illustrated in Fig. 1B}, are not regime shifts. “Regime” implies that the system has been observed to have remained at a stationary mean or within a range of variation over a period of time, and to then have shifted to another mean and range of variation. Regimes can be maintained by external or intrinsic processes, or sets of interacting external parameters and internal variables, but the ways in which the processes are organized can vary. Sets of processes can be dominant, reinforcing the regime; understanding this simply requires one to associate a regime with our previous discussions of system states and attractors. Regime shifts occur then when sets of processes are re-organized, and dominance or reinforcement shifts to other parameters and variables.
Fig. 1A – Hypothetical regime shift
Fig. 1B – Two populations of red-wing blackbirds. See here for an explanation.
Regime shifts may be distinguishable from variation within a state, or continuous variation across a parameter range, by the time interval during which the transition occurs, if the interval is notably shorter than the durations of the alternative states. This of course potentially limits the confirmation of regime shifts as we can never be certain that observation times were sufficient to classify the system as being in an alternative state. The interpretation though is that the duration of the transition was relatively short because the system entered into a transient phase, i.e. moving from one stable state to another. The transition itself may be precipitated in several different ways, dependent on the type of perturbation and the response of the system. The perturbation could be a short-term excursion of a controlling parameter that pushes the system into another state, with the transition being reversed if the threshold is crossed again. More complicated situations arise, however, if internal variables of the system respond to parameter change without a measurable response of the state variable itself, and if the system can exist in multiple states within the same parameter range. These various characteristics of regime shifts serve to distinguish important processes and types of shifts that are more complex than simple and reversible responses to external drivers, such as “critical transitions” and “tipping points”.
We have already discussed several model systems with multiple states, one of those being a trivial state of population extinction (), and the other being an attractor when . Zero population size was classified as an unstable state, because the addition of any individuals to the population — — leads immediately to an increase of population size, and the system converges to a non-zero attractor. This is true regardless of the nature of the attractor (e.g. static equilibrium, oscillatory, chaotic), and makes intuitive sense — sprinkle a few individuals into the environment and the population begins to grow. This is not always the case, however, and there are situations where zero population size, or extinction, can be a stable attractor, or where converges to different attractors, dependent either on population size itself, or forcing by extrinsic parameters. The system is then understood to have multiple alternative states. I reserve this definition for circumstances where does not vary smoothly or continuously in response to parameter change (e.g. Fig. 1), but will instead remain in a state, or at an attractor, within a parameter range, and where the states are separated by a parameter value or range within which the system cannot remain, but will instead transition to one of the alternative states. Thus, the multiple states are separated in parameter or phase space by transient conditions.
We will explore a real-life example in the next post, and here is a teaser.
Cod in the North Atlantic.
Vocabulary Attractor – A compact subset of phase space to which system states will converge. Regime shift – An abrupt or rapid, and statistically significant change in the state of a system. System state – A non-transient set of biotic and abiotic conditions within which a system will remain unless acted upon by external forces. Transient state – The temporary condition or trajectory of a population as it transitions from one system state to another.
WHAT IS ECOLOGICAL STABILITY? In 2019 I posed this question informally to colleagues, using Twitter, a professional workshop that I lead, and a conference. Respondents on Twitter consisted mostly of ecological scientists, but the workshop included paleontologists, biologists, physicists, applied mathematicians, and an array of social scientists, including sociologists, anthropologists, economists, archaeologists, political scientists, historians and others. And this happened…
LET US NOW FAMILIARIZE OURSELVES WITH TWO CONCEPTS THAT WILL BE USEFUL AS WE CONTINUE : “PHASE SPACE” AND “ATTRACTOR”. A phase space is the set of all possible states in which a system can exist. In the examples from earlier posts, our system was the population and the phase space was the set of values that population size could take, ranging hypothetically between 0 (extinction) and ∞ (but remember finite planetary sizes!). An attractor is a subset of the phase space, i.e. a subset of the possible states. This subset is called an attractor because, given sufficient time, the system will move (be attracted) from an initial location somewhere in phase space (its initial population size), toward the attractor. Technically, the attractor is described as compact, because it is a defined subset or region of the phase space, and it is asymptotically stable, meaning that the system will approach it asymptotically over time. The attractor in the earlier single species logistic examples is a single number, the carrying capacity K. Asymptotic attraction is illustrated in Fig. 1 for two populations, one of which begins below carrying capacity (X(0)<K) and therefore increases toward K, and another where initial population size exceeds K (X(0)>K) but subsequently declines. The figure also illustrates the phase space trajectories of the two populations as they converge on their common attractor. The phase space is the set of values that population size could possibly take, whereas the attractor is where you expect to find the population when it is in equilibrium.
Left – Two populations with different initial sizes (X(0)) both converge toward the attractor at K=100. Right – Phase space of the trajectories showing convergence to attractor. In this case the phase space is visualized by plotting population size against itself at two successive time intervals.
Deviations from equilibrium
ARE MODEL POPULATIONS WITH SIMPLE EQUILIBRIA AND ATTRACTORS REASONABLE REPRESENTATIONS OF REALITY? Population X is a model of stability because once it attains its fixed value, the equilibrium attractor, it will remain there. Is this, however, a realistic expectation for a real population? We can imagine the population growing or shrinking because of external disturbances. For example, a storm could kill a number of individuals, driving the population below K, or a wet season could result in a greater than expected number of births, driving the population above K (Fig. 1). The remarkable thing about this stable equilibrium system, however, is that it will always return asymptotically to K, if the population does not become extinct. It is attracted to K after displacement from its fixed point.
How often do we observe this condition in natural populations? I would argue very infrequently, perhaps hardly ever. There are several reasons for this, some of which stem from the possibility that simple equilibria might be relatively rare in nature. Discussing those reasons will occupy a good deal of later sections. But even if simple equilibrium state dynamics were common, observing them could be rare because real populations are not closed, isolated systems. Populations must be open because living organisms, and hence their populations, require the passage of energy through their systems to remain alive. That energy comes ultimately from the Sun or geochemical reactions, and hence all living systems are open and exposed. Therefore, we can think about what happens to X when we remove it from its model box and expose it to the environment: The population will be driven away from equilibrium in direct response to environmental disturbances, its guaranteed return to equilibrium being dictated by r (or R) and K. Disturbances that affect the population directly, so-called direct perturbations, were illustrated in Fig. 1. A simple model could be written as
EQ. 6: (FUTURE POPULATION SIZE) = [(CURRENT POPULATION SIZE) – (MORTALITY DUE TO DIRECT PERTURBATION)] x (EXPONENTIAL REPRODUCTION LIMITED BY CARRYING CAPACITY)
where δX is mortality due to a direct perturbation. The dynamics of a return to equilibrium after perturbation are termed transient, because they exist temporarily between times when the system is in equilibrium.
Another way in which the external environment may perturb X is by raising or lowering the carrying capacity. K encompasses many factors which share in common the fact that they limit population growth increasingly as population size approaches K. A relaxation or tightening of any of those constraints would therefore be manifested as a change of K. For example, the transition to a wetter climate could result in a landscape capable of supporting more individuals of a tree species, or an expansion of dysoxic waters could reduce the habitable area on a lake bed. In either case, the underlying dynamics of the population remain unchanged except for a simple response to the change of K, and a shift of the attractor in phase space (Fig. 2).
Left- The population, after reaching its equilibrium at K=100, is perturbed directly. It subsequently recovers, but the carrying capacity has now been reduced by the environment to K=80. Right – The population’s trajectory in phase space. Blue dots represent the two attractors.
But what if the perturbed population is disturbed again before it reaches equilibrium? In that case the population remains in a transient state, and one can imagine a situation where the frequency of environmental disturbance is greater than the time required for the population to reach equilibrium after being displaced from it. The population would be in a constant state of transience, fluctuating as a function of the direct perturbations and its intrinsic equilibrium dynamics. Some workers have suggested that many populations may in fact exist in a perpetual state of transience and rarely or never reach their equilibria (Hastings, 2004).
Thus the external environment can keep a population away from its intrinsic equilibrium. Under such circumstances, is the population stable? This is a situation where I would argue that the answer depends on the perspective of the observer, and the purpose(s) for which the population is being assessed. One could argue for or against stability in the following ways:
• The population is intrinsically stable because it grows logistically, has a simple equilibrium, and if left alone would settle to its attractor. • The population is not stable because it responds to a variable in the external environment, and is predictable only to the extent to which external drivers can be predicted.
In either case, the simple dynamics of the system allow us to choose and communicate the perspective from which the system is being approached. Unfortunately, reality is rarely so simple. Examine Figure 3, which illustrates population size trajectories for two local populations of the Red-Winged Blackbird (Agelaius phoeniceus) in the southeastern United States (USGS, 2014; Dornelas et al., 2018). The population from the Gulf Coast of Texas had several dramatic deviations from the median size. The first of these occurred in 1988, when the population increased by an order of magnitude in a single year. This was followed by an incremental decline over the next two years to levels below the median population size. The second excursion, in 2002, was almost twice as large as the previous, this time followed by a decline to precipitously low levels. The smaller Floridian population in contrast exhibited a single significant excursion during 1995, followed by an immediate return to a more expected size. Could either of these populations be described as being in stable equilibrium? Perhaps this is the case for the Floridian population, with the 1995 excursion being an environmentally-driven transient increase. The constant population fluctuations could similarly be attributed to an extrinsic driver of smaller magnitude, and censusing errors. The larger excursion of the Texan population was likely driven by an increase of food resources, but was the subsequent decline driven by overpopulation and a return to typical food levels? Or was there a coincidental occurrence of a very negative external event? Either explanation is possible, but there is a third type of mechanism, which is the production of complex dynamics by intrinsic factors, in this case interacting with external drivers. The next few posts will address complex dynamics, intrinsic and externally driven, as well as transitions of population states, and contribute to our interpretation of complicated population size trajectories and stability.
Sizes of two populations of the Red-Winged Blackbird, Agelaius phoeniceus, from the Gulf of Mexico. Left – Texas; right – Florida. Thick horizontal red lines show series medians, and thinner lines the 5th and 95th percentiles.
Vocabulary Attractor A compact subset of phase space to which system states will converge. Equilibrium A condition where the state of a system either does not change, or experiences no net change over time. Phase space The set of all states in which a system can exist. Transient dynamic A transient dynamic describes a population’s trajectory as it returns to equilibrium after displacement, or transitions from one equilibrium state to another.
References Dornelas, M., Antão, L. H., Moyes, F., Bates, A. E., Magurran, A. E., and et al. (2018). BioTIME: A database of biodiversity time series for the Anthropocene. Global Ecology and Biogeography, 27:760– 786. Hastings, A. (2004). Transients: the key to long-term ecological understanding?. Trends in Ecology & Evolution, 19(1), 39-45. USGS, P. W. R. C. (2014). North American Breeding Bird Survey ftp data set, version 2014.0.
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