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…
The story of R does not end with bifurcations and oscillations. Increasing R beyond our explorations in the previous post yields continuing bifurcation, and reveals yet another type of dynamic where the system continues to oscillate between several values, but now only approximately. The cycle does not repeat precisely, only coming close to previous values. Such cycles are often termed “quasiperiodic”. The attractor of a quasiperiodic system is an apt visual descriptor of the system’s dynamics (Fig. 1). Long-term observations of a quasiperiodic system are unlikely to yield a precise repetition of values, but the attractor is nevertheless bound in phase space. This system can therefore be described sufficiently in a statistical manner, and is invariant to variation of the initial condition (X(0) ) of the system. The trajectory in phase space visits the attractor’s distinct regions in a repeating cycle termed an invariant loop (Fig. 1).
Fig. 1 — Transitions of a discrete logistic function with increasing R. R=2.7, K=100, and X(0)=1.0. The plot illustrates a quasiperiodic series. The series was iterated for 2,000 generations. Plot on the left shows population size for generations 1900-1930, while on the right is plotted the attractor for the entire series. Grey lines on the attractor plot traces the trajectory of the populations in phase space.
The system, however, is intrinsically noisy, and this raises two questions: (1) Can a noisy system be stable? (2) Can intrinsic noise be distinguished from noise generated in response to external factors? Answering the first question is difficult because our previous definition of stability no longer applies for the following technical reason: Population size X is measured as a real number. Given any two real numbers, there is an infinite count of real numbers of greater precision between them. Therefore, in the example figured below, although the quasiperiodic attractor consists of four visibly distinct regions, the population could cycle among those regions without ever precisely repeating itself! Deciding the stability of a system on this basis, however, would seem to be both an unnecessary mathematical technicality as well as impractically misleading to the scientist. The system is still bound by the attractor, for all “closed” situations, and the compactness of the attractor ensures statistical predictability given an adequate number of observations. I therefore choose to classify it as stable. There are two cautionary notes for practitioners though. First, apparent noise in this system is generated by an intrinsic, deterministic component, and is not due to external influences. Second, variability of a system’s dynamics is not necessarily an indication of instability. Let’s summarize this, because it becomes important in later discussions.
The intrinsic properties of a population may generate infinitely variable, but nevertheless deterministic and statistically predictable dynamics.
Quasiperiodicity is a well-documented phenomenon in climatic and oceanographic systems (e.g. McCabe et al., 2004), where processes such as El Niño and the Pacific Decadal Oscillation possess intrinsic oscillatory properties that are not completely overridden by external drivers (e.g. orbital dynamics), resulting in approximate and drifting semi-cycles.
Chaos
Increasing R even further yields a transition to a final and most complex type of dynamics. Figure 2 illustrates the dynamics when R = 3.3. The time series of X is a succession of apparently randomly varying population sizes, with X sometimes exceeding 2K (K = 100), and also coming perilously close to zero (extinction). Yet, the attractor shows that these values belong to a compact subset of phase space, in fact one that is similar to the quasiperiodic attractor, but where the dense regions of the latter attractor are now connected by intervening points. More significantly, X no longer traces a regular cyclic path or loop through the attractor, but instead jumps unpredictably from one region to another. This is chaos (Li and Yorke, 1975).
Fig. 2 — Transitions of a discrete logistic function with increasing R. R=3.3, K=100, and X(0)=1.0. The plot illustrates a chaotic series. The series was iterated for 2,000 generations. Plot on the left shows population size for generations 1900-1930, while on the right is plotted the attractor for the entire series. Grey lines on the attractor plot traces the trajectory of the populations in phase space.
CHAOTIC SYSTEMS ARE EXERCISES IN CONTRASTS. For example, chaotic systems are deterministic, not random (see Strogatz, 2018). The specification of a dynamical law (here our function for population growth) and an initial condition (initial population size) will always produce precisely the same population dynamics. Furthermore, chaotic attractors occupy well-defined regions of the phase space. Those attractors, however, will encompass an infinite set of values, are generally not loops, and are therefore described as “strange attractors” (David and Floris, 1971). This is a consequence of one of the most important features of chaotic systems, their sensitive dependence on initial conditions. All the systems discussed so far have equilibria or attractors that could be described as convergent, meaning that if two populations obeying the same dynamic law were started at slightly different initial population sizes, they would either eventually converge to the same equilibrial size (single state and stable oscillatory dynamics), or remain close in value (quasiperiodic). Chaotic systems come with no such guarantees, and populations with very small differences in initial size will diverge away from each other, ultimately generating different dynamics. They will nevertheless be confined to the strange attractor.
The transitions of dynamics exhibited by our discrete logistic Ricker model (Eq. 1 here), and also the logistic map (Eq. 1 here), are driven entirely by increasing the population growth rate R. The full set of transitions can be mapped with a bifurcation diagram which plots all the values that population size will attain for a particular value of R after an initial period of transient growth (Fig. 3). Thus, for R < 2.0, X(t) = K as t goes to infinity, but when R ≥ 2.0 the system undergoes its first bifurcation to a stable oscillation between two values. This is the first branch point on the diagram. The divergence of the branches as R increases reflects the increasing amplitude of oscillations around K. The transition to chaos at R = 2.692 for the discrete logistic model is obvious, as X now takes on a multitude of values, yet is bound within a range.
Bifurcation map of the Ricker function. Points show all population sizes at a given value of R for the range t=1900 to 2000. X(n) is population size relative to carrying capacity K=1). Stable bifurcations are obvious, beginning at R=2.0, and chaotic regions are identifiable as being occupied by numerous points.
Vocabulary Bifurcation — The point at which a (nonlinear) dynamic system develops twice the number of solutions that it had prior to that point. Real number — A real number is one that can be written as an infinite decimal expansion. The set of real numbers, R, includes the negative and positive integers, fractions, and the irrational numbers.
References David, R. and Floris, T. (1971). On the nature of turbulence. Communications in Mathematical Physics, 20:167–92. Li, T.-Y. and Yorke, J. A. (1975). Period three implies chaos. The American Mathematical Monthly, 82(10):985–992. McCabe, G. J., Palecki, M. A., and Betancourt, J. L. (2004). Pacific and Atlantic Ocean influences on multidecadal drought frequency in the United States. Proceedings of the National Academy of Sciences, 101(12):4136–4141. Strogatz, S. H. (2018). Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering. CRC Press.
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…
The previous post outlined the circumstances in which an intrinsically stable logistic population can deviate from equilibrium, or its attractor, when perturbed by the external environment. Those deviations are brought about by either direct perturbation of the population, or an alteration of the environment’s carrying capacity (for that species). There is a third parameter, however, that determines dynamics in our models, and that is the rate of increase (r or R). It is a life-history trait determined by the evolutionary history of the species (and population), and interaction of that trait with the environment. Its influence on X(t) is generally to accelerate (or decelerate) the overall rate of population growth, with higher values causing higher overall rates. We can see this by repeating the earlier perturbation example, but with a smaller value of r (Fig. 1). The rate at which a population recovers from a disturbance is therefore determined by its intrinsic rate of increase. This is one measure of resilience. Resilience has (too many) varied meanings in ecology and other sciences, and is typically applied to communities or ecosystems, i.e. multi-population systems. In this instance, however, resilience means specifically the time taken for the system to return to equilibrium, and can therefore be applied to our population. Holling (Holling, 1973) has termed this type of resilience engineering resilience, as the concept has broad application in physics and engineering. Under this definition, populations that recover more quickly are considered to be more resilient. Thus, the rate at which a population recovers from a negative perturbation is directly proportional to its intrinsic rate of increase.
Two populations with different intrinsic rates (blue, R=0.25; orange, R=0.5; K=100) recovering from simultaneous and numerically equal direct perturbations. The population with the higher r recovers faster to equilibrium, and thus has greater engineering resilience.
Importantly, however, a population isolated from conspecific populations can never grow faster than its intrinsic rate of increase.
Bifurcations
The intrinsic rate of increase can also be a source of dynamics more complex than those presented so far. This is particularly acute in the discrete time, or difference, models because of the recursive feedback loop present in those models (i.e. X(t + 1) is a direct function of X(t)). May (1976) highlighted this using a discrete logistic model.
EQ. 1: (future population size) = [(intrinsic growth rate) x (current population size)] x (growth limited by carrying capacity)
where x is population size standardized to a carrying capacity of 1 and is restricted to the interval 0 < x < 1, and r is the intrinsic growth rate.
May showed that very complex dynamics, such as chaos, can emerge from this very simple model of population growth with non-overlapping generations, as r is increased. The same holds true for the discrete Ricker logistic model presented earlier (Eq. 1). In that model, values of R < 2.0 yield the expected equilibrium logistic growth, but even at values as low as 1.8 < R ≤ 1.9, interesting behaviours begin to emerge — approaching the carrying capacity, population size will overshoot K very slightly before converging to it (Fig. 2A). This is a transient, pre-equilibrium excursion. At R = 2.0 the system undergoes a dramatic shift from the single-valued equilibrium point to an oscillation between two values around the carrying capacity (Fig. 2B). You will notice that the transient overshoot is preserved, and in fact the amplitude of the oscillation is initially large, but the system eventually converges to two fixed values. Those values represent a new attractor, because the system will always converge to an oscillation between them. The value R = 2.0 is a critical point at which the system is said to undergo a bifurcation, with the equilibrium now consisting of two population sizes.
Transitions of a discrete logistic function with increasing R. Values of R, from upper plot to lower: 1.9, 2.0. K=100, and X(0)=1.0. The upper plot illustrates a quasiperiodic series, while the lower plot is chaotic. Each series was iterated for 30 generations. Plots on the left show population size, while on the right they plot the attractor for the entire series.
The amplitude of the oscillations grows as R increases, and the system eventually undergoes further bifurcations, e.g. where the population oscillates between four fixed points. Is the population still stable? The determination of stability now depends on two factors, the first of which is the timescale at which the population is observed. Population sizes and the attractor are repeating cycles, with X(t) cycling (or “orbiting”) between an ordered set of points. Therefore, if the length of time over which X is observed exceeds the period of the attractor, one will observe the system repeating itself, but if it is shorter, the question of stability remains open unless the underlying dynamical law is known. Second, the observation of multiple cycles allows a complete description of the system’s dynamics, and one could then conclude that the system is confined to a compact subset of the phase space. Most importantly, one would conclude that the system is deterministic and predictable. Recall that deterministic means that the entire future trajectory of the system is knowable, given the law by which the system evolves or unfolds over time, i.e. the dynamic equation and the initial condition of the system (X(0)). This is a very Newtonian system which will continue in this manner unless or until acted upon by an external force. The system is as stable as it was when it possessed a simple equilibrium, the only difference being that the attractor now traces a fixed trajectory in phase space comprising multiple values rather than occupying a single point. We can therefore refine our definition of stability.
Intrinsic stability: An intrinsically stable population expresses a finite set of infinitely repeating values.
This definition encompasses both our earlier simple equilibrium, and our newer oscillatory equilibria. It also encompasses further bifurcations that the system undergoes as R is increased, e.g. to a four point attractor.
Vocabulary Bifurcation — The point at which a (nonlinear) dynamic system develops twice the number of solutions that it had prior to that point. Engineering resilience — The time taken for a system displaced from equilibrium to return to equilibrium.
References Holling, C. S. (1973). Resilience and stability of ecological systems. Annual review of Ecology and Systematics, 4(1):1–23. May, R. M. (1976). Simple mathematical models with very complicated dynamics. Nature, 261(5560):459–467.
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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