Systems Paleoecology – Chaotic Stability?

Tags

chaos, Lyapunov exponent, paleoecology, population growth, stability, strange attractor

The upcoming week is quite busy for me and I might not have time to write two posts. Therefore this post is a bit on the long side, but I am SO excited to have a busy week. Virtually. On the webcam. Okay…

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:

1. Welcome Back Video
2. Introduction
3. Malthusian Populations
4. Logistic Populations

5. Logistic Populations II
6. Deviations from Equilibrium
7. r, R, and Bifurcations
8. Quasiperiodicity and Chaos

“You can be wrong with math, but you can’t lie.” — Sabine Hossenfelder

IS A CHAOTIC POPULATION STABLE ? Quasiperiodicity and chaos are closely related, but whereas quasiperiodic systems are approximately periodic with approximately repetitive values, neither is true of chaos. Chaotic systems are not truly random, otherwise their strange attractors would drift and diffuse throughout the phase space. Yet, for all practical purposes a chaotic system is unpredictable, and distinguishing it from randomness can sometimes be very difficult. The sensitive dependence on initial conditions means that if one measured population size at a time t = a to be X(a), and at some later time t = b measured it to be X(b) = X(a) + ε, where ε is a small difference between X(a) and X(b), there would be absolutely no reason to expect the population dynamics during an equal interval of time after t = b to at all resemble the dynamics between times a and b (Fig. 1)! One’s record of population sizes would therefore be of little value to efforts to forecast future population sizes, or to infer past population trajectories. Thus, in spite of their intrinsically deterministic character, chaotic populations at short timescales cannot be classified as stable. However, if one could observe the system, or multiple iterations of the system at a timescale sufficiently long to characterize the strange attractor, then a chaotic system could be distinguished from a random one. The system could be considered stable at that timescale.

Chaotic populations best considered unstable, unless the relevant timescale encompasses enough time for the population to complete sufficient orbits that characterize the attractor as a strange one.

This is a very broad definition of stability, where we understand that the variability of the system is bounded, yet within those bounds the system might display very different characteristics.

Fig. 1: The divergence of two initially very similar population trajectories with starting sizes of X(t)=0.1 and 0.11. The populations diverge noticeably by the fifteenth generation, illustrating sensitive dependence on initial conditions.

Are real populations ever chaotic?

A reviewer once commented to me that “your model features a lot of chaos, but chaotic populations do not exist in nature”, to which I responded that I was not aware that any comprehensive surveys had yet been completed. The reviewer’s comment, however, does raise the interesting and important question of whether any natural populations are governed by intrinsic chaotic dynamics, and would therefore be intrinsically unstable. (Note that I specified “intrinsic”. We will discuss extrinsic abiotic and biological environments as potential sources of chaos much later on. For now, we are still dealing with single, isolated populations only).

That intrinsic chaos is a possibility results from the fact that all self replicating systems with positive feedback are capable of complex dynamics (e.g. Berryman and Millstein, 1989), including chaos. Hypothetically then, those dynamics should be intrinsic to biological populations, challenging a priori expectations that chaotic population dynamics are rare. May’s work stimulated a lot of interest in ecological chaos, yet the frequency of its occurrence in the wild remains unconvincing (Pool, 1989), despite demonstrations in the laboratory (e.g. Dennis et al., 2001; Costantino et al., 1997). There are two possible explanations, and they do lead to deeper thoughts on the evolution of ecology. First, perhaps chaos is common, but identifying chaotic populations is difficult. Alternatively, maybe chaotic population dynamics is indeed rare, in which case the seemingly legitimate mathematical prediction that it should be common raises the question of what suppresses it in the wild. The simple manipulation of R in the Ricker equation and logistic map (see previous posts) shows that populations are capable of far more interesting dynamics than simply remaining at equilibrium. We discussed other sources of population fluctuations in an earlier post, attributing those to non-demographic external drivers. Another common source of fluctuations must be the error associated with empirical censuses. One possible class of explanations for the rarity of chaos in observed population size time series therefore is that it is simply very difficult to distinguish among the various sources of fluctuations and deviations. Let us imagine though that we have managed such distinction for a population, and can control the non-intrinsic sources of variation. We could then use the series X(t) as data for the examination or estimation of chaotic features, such as strange attractors or Lyapunov exponents (we’ll have fun with those much later on). Unfortunately, substantially long series are required for robust estimation, and even long ecological series of a dozen or more observations fall short (beyond the typical duration of a grant, and hopefully longer than graduate school), and cannot convince skeptics of the existence (or non-) of chaos in natural populations. And rightfully so.

Another tactic would be to estimate the parameters of the population’s dynamical law (i.e. its growth function) directly from the time series. One could then solve the equation analytically, and settle without dispute whether the function is chaotic. In the case of single populations, this amounts to estimating R (single population models based on r, such as the logistic equation are not self-replicating, lack positive feedback, and are incapable of being chaotic; more on that in the future). R can in theory be estimated for natural populations, and is an important exercise in conservation biology and natural resource management. Say then that the underlying R is known, and the resulting dynamical law is capable of yielding chaos. If one accepts that the models are reasonable, albeit simplistic approximations of real-world stationary, periodic, or quasiperiodic dynamics, then why not real-world chaos? The strongest arguments presented against this point to natural processes and mechanisms that suppress the emergence of chaos in otherwise suitable natural systems.

Several biological factors have been shown to prohibit the emergence of chaos in simple population dynamic models. First and foremost is the fact that many values of R are too low to generate chaos in the models. But current surveys of R in real populations might not constitute a set sufficiently large, or phylogenetically broad enough to actually test this argument. Other mechanisms that have been shown to dampen emergent chaotic dynamics in models include sexual reproduction, intraspecific genetic variation, and metapopulation (emigration/immigration) dynamics (e.g. Scheuring, 2002). Incorporating these features of added realism can delay or completely prevent the onset of chaos in discrete models, and it has therefore been argued that given their frequent or near-ubiquitous occurrences, chaos is most likely absent from, or impossible in the dynamics of real populations. Far from dismissing the importance of chaos, however, for me these arguments simply deepen the mystery. They imply that chaos lurks just behind those preventative “barriers”, and that it could erupt into reality should barriers either fail to evolve in a species, or are removed from ecological settings. For example, in my opinion the following questions remain unsettled.

1. Do populations with chaotic dynamics perform more poorly, or have higher rates of extinction, compared to those with non-chaotic dynamics?
2. Have any life-history traits, such as intrinsic population growth rates or sexual reproduction, evolved in response to population dynamics? If there is differential performance between chaotic and non-chaotic populations that affects organismal fitness, then does population dynamics act as a mechanism of natural selection on traits that affect those dynamics?
3. What happens to populations in which the effectiveness of chaos-suppressing factors such as population genetic variance or metapopulation dynamics, decline?

Descent into Chaos

Both genetic variance and metapopulation connectivity are impacted during times of ecological crisis, such as extinction events in the fossil record and the current anthropogenically-driven environmental crisis. For example, imagine the landscape of the Karoo Basin, or southern Africa in general, during the end of the Permian, where climatic drying fractioned and fragmented previously widespread or highly connected populations (Sidor et al., 2013; Smith and Botha-Brink, 2014). Does the likelihood or frequency of chaos increase under such conditions? The answer must be yes, if those mechanisms were in the first place responsible for the absence or suppression of chaos. As an example, consider a simple metapopulation model that has been used to argue for the rarity of intrinsic chaos in natural populations (adapted from Rohani and Miramontes, 1995; Ruxton and Rohani, 1998). The dynamical law is the Ricker model with an added factor

EQ. 1: (FUTURE POPULATION SIZE) = [(CURRENT POPULATION SIZE) x (EXPONENTIAL GROWTH LIMITED BY CARRYING CAPACITY )] + IMMIGRANTS

where i is an immigration rate from other populations in the metapopulation. The addition of i each generation acts as a “floor” of X(t) below which it cannot fall. Thus, i prevents the extinction of X. i also dampens or eliminates chaos in Eq. 1. For example, when i = 0.1, that is 10% of K individuals migrate into the community each generation, chaos disappears completely, leaving only bifurcations as R is increased (Fig. 2).

Fig. 2: The dampening effect of a steady immigration rate on bifurcation and chaos in the Ricker model. Reading from left to right, top to bottom, immigration rate i = 0.1, 0.05, 0.01. 0.005, 0.001, 0.0.

This simple model and related versions have been used to demonstrate the suppression of chaos by first applying a small value of i, e.g. i = 0.001, which yields a mild but obvious reduction of chaos (Fig. 2, bottom left plot). i is then increased incrementally, showing the progressive reduction and eventual elimination of chaos. Metapopulation dynamics are not, however, intrinsic features of the populations involved, but instead depend on both intrinsic features and external variables, such as the frequency and distribution of suitable habitat patches, the carrying capacities of those patches, and their biogeographic connectivities. All those external variables may be affected during times of environmental stress. Therefore, in presenting the results of the model, I reversed the sequence of i in Fig. 2 to be a decreasing one, showing the expected outcome as the environment deteriorates, thereby disrupting spatial connections and perhaps rendering populations less capable of acting as sources of immigrants. The result is obvious — as habitat connectivity becomes increasingly fragmented, we should expect chaos to erupt in various ranges of R where it was previously suppressed by immigration.

Reductions of diversity and simplification of systems during times of ecological crisis do not necessarily lead to simplified dynamics.

Vocabulary
Lyapunov exponent — The divergence rate of two initially close trajectories of a dynamic system.

References

• Berryman, A. and Millstein, J. (1989). Are ecological systems chaotic—and if not, why not? Trends in Ecology & Evolution, 4(1):26–28.
• Costantino, R. F., Desharnais, R., Cushing, J. M., and Dennis, B. (1997). Chaotic dynamics in an insect population. Science, 275(5298):389–391.
• Dennis, B., Desharnais, R. A., Cushing, J. M., Henson, S. M., and Costantino, R. F. (2001). Estimating chaos and complex dynamics in an insect population. Ecological Monographs, 71(2):277–303.
• Rohani, P. and Miramontes, O. (1995). Immigration and the persistence of chaos in population models. Journal of Theoretical Biology, 175(2):203–206.
• Ruxton, G. D. and Rohani, P. (1998). Population floors and the persistence of chaos in ecological models. Theoretical Population Biology, 53(3):175–183.
• Scheuring, I. (2002). Is chaos due to over-simplification in models of population dynamics? Selection, 2(1-2):179–191.
• Sidor, C. A., Vilhena, D. A., Angielczyk, K. D., Huttenlocker, A. K., Nesbitt, S. J., Peecook, B. R., Sébastien Steyer, J., Smith, R. M. H. & Tsuji, L. A. (2013). Provincialization of terrestrial faunas following the end-Permian mass extinction. Proceedings of the National Academy of Sciences, 110(20), 8129-8133.
• Smith, R. M., & Botha-Brink, J. (2014). Anatomy of a mass extinction: sedimentological and taphonomic evidence for drought-induced die-offs at the Permo-Triassic boundary in the main Karoo Basin, South Africa. Palaeogeography, Palaeoclimatology, Palaeoecology, 396, 99-118.

Systems Paleoecology – Quasiperiodicity and Chaos

Tags

attractor, bifurcation, chaos, paleoecology, population growth, quasiperiodicity, strange attractor

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:

1. Welcome Back Video
2. Introduction
3. Malthusian Populations
4. Logistic Populations

5. Logistic Populations II
6. Deviations from Equilibrium
7. Systems Paleoecology – r, R, and Bifurcations

Quasiperiodicity

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.