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.

Systems Paleoecology – r, R, and Bifurcations

Tags

attractor, bifurcation, ecology, mathematical model, paleoecology, population growth, resilience, theoretical ecology

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

In chaos, there is fertility. Anais Nin

The importance of r (and R)

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.

Lighting up an ecosystem

Tags

bifurcation, chaos, extinction, Network theory, networks, quasi-periodic, simulations, Tipping point

One of the final pieces needed to explain the critical/threshold point in a bottom-up CEG perturbation is an understanding of which species become extinct, and what the species dynamics look like during the cascade. Therefore, what I’ve done is to modify the basic simulation to capture the demographic properties (technically, the carrying capacities) of each species; results in HUGE output files. Shown here in this figure are the species dynamics for the guild of shallow infaunal carnivores (e.g. naticid snails) at three different perturbation levels. Note that the levels correspond to a low secondary extinction response, and the two critical points identified earlier. The top row of figures plot the dynamics of surviving species, and lower show those of the species which become extinct. The community dynamics were recorded for 250 steps beyond the initial perturbation. The first thing to note is that species become extinct very quickly. Beginning K for each species is standardized at 1, and the species that become extinct have, on average, lower in-degrees, i.e. lower numbers of prey, than do surviving species (statistical tests to follow later). That result matches expectations of the CEG combinatoric model.

The other thing to note is that at the low and mid-perturbation levels ( 0.2 and 0.55), species’ K respond immediately to a perturbation of the producer guilds, oscillate for several steps, but eventually settle down to a new stable K. This is a transition to new stable states for the species populations. At the perturbation level which coincides with the major critical point of secondary extinction, however, there is no indication that the species ever settle to a new state. Instead, there seems to be bifurcation and subsequent alternation between two alternative stable states; the species are lit up by the disturbance (each species is given a different colour in the corresponding plot for easy distinction). The series is quasi-periodic though, in that the system never returns to quite the same point on alternating steps. It is possible that the series eventually converge to a single, or two, stable points, but the current data cannot address that. Therefore, I’ll next repeat this simulation, but extend the data capture to 1,000 steps (2Gb file). Hopefully that will give some indication of whether the series are converging, diverging, are stable, or perhaps chaotic. Also, the results for the other guilds need to be examined.