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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…

Previous posts in this series:

Phase spaces and attractors

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.
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)

X_{t+1} = (X_{t}-\delta X)e^{R\left [1-\frac{(X_{t}-\delta X)}{K}\right ]}

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 $5^{mathtt{th}}$ and $95^{mathtt{th}}$ percentiles.
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.