# Systems Paleoecology – Nonlinearity and Inequality

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

There is always inequality in life — John F. Kennedy

John Lanchester, a British novelist and journalist, expresses a view that is becoming increasingly widespread in our increasingly stressed human socio-economic system: Inequality is not a law of nature. I disagree, but let me explain why before you form a judgement. As a scientist, one of my responsibilities, and one for which I have been trained, is to identify and explain laws of nature. In an earlier post I defined a natural law as “…a statement, based on multiple observations or experiments, that describes a natural phenomenon. Laws do not have absolute certainty and can be refuted by future observations.” In that sense, inequality would indeed seem to be a law given its persistence and pervasiveness. My disagreement with Lanchester, however, is based solely on the idea that laws do not have absolute certainty, and are not immutable. Laws are not necessarily fundamental, but instead arise from the unfolding of fundamental relationships and interactions over time (and consequently, in space). Lanchester goes on to state, “[inequality] is a consequence of political and economic arrangements, and those arrangements can be changed.” which is perfectly consistent with the nature of natural laws.

But where does inequality come from in biological systems? It can originate in the differences of rates between interacting processes, the context-dependent expressions of genes, the plasticity of behaviours, and so forth. One important net result of these variations is nonlinearity, a condition where the proportional relationship between an input and an output changes with the size of the input. We have seen nonlinearity already of course: exponential growth, and the logistic function where population size increases rapidly when it is small, but only very slowly when near carrying capacity. Those models incorporate nonlinear relationships to describe how we think populations grow. Remove them and your model of population growth is reduced to a simpler, linear, more boring, and less accurate description of real populations. Nonlinearity is more fundamental, however, than a mere ingredient for enhancing model accuracy, because inequality is an inescapable feature of the natural world. If that realization creates some discomfort, perhaps it’s because we commonly equate “fairness” or “equality” with equilibrium, a “balance of Nature”. There is no balance in nature, and that sort of static stability is neither necessary nor capable of explaining the complexity of ecological phenomena. In coming sections we will explore many types and consequences of nonlinearity in ecological systems, how those lead to complex ecological systems, why I (and many others) believe that those systems are often far from equilibrium, and how nonlinearity, broader concepts of equilibrium, and complexity, all generate and explain many aspects of ecological systems. Before I go there though, I’ll discuss a concise nonlinear concept, one that not only captures the essence of why understanding nonlinearity and its implications is rewarding, but also has broad implications in biology.

Jensen’s Inequality

The previous post discussed the potentially misleading outcome of treating population dynamics in mean environments when environmental variation is omitted. Another issue related to interpretations or forecasts involve environmental averages, and arises when the relationship between λ (growth rate independent of population size) and an environmental driver is nonlinear. We assumed in the previous example that the relationship was a simple linear one, e.g. higher temperatures drive a constant proportional increase of birth rate. But metabolic, physiologic and other phenotypic traits often respond nonlinearly to controlling or input factors based on nonlinear phenotypic relationships (e.g. surface area to volume ratios), or differences of response timescales of various organic systems, among other factors. In such cases, the mean performance in a variable environment is not the same as performance at the environmental mean. This is known in mathematics as Jensen’s inequality, and is one consequence of nonlinear averaging or, more generally, making linear approximations of nonlinear curves or surfaces (see Denny, 2017, for a very accessible review).

For example, examine the relationship between water temperature (T) and λ in species belonging to the marine copepod genus Arcatia (Fig. 1) (Huntley and Lopez, 1992; Drake, 2005). The relationship is exponential, and within the range of observed temperatures, incremental increases of temperature result in proportionally greater increases of population growth rates at higher temperatures. Now consider the case of two populations, one inhabiting a region where daily temperatures vary little, with a mean temperature of 20°C. The other population experiences daily temperature fluctuations between 15°C and 25°C, and also experiences a daily mean temperature of 20°C. The growth rate of the first population is the expected λ given the dependency on temperature, but λ of the second population is the mean λ experienced over the temperature range. Because the relationship between λ and T is a nonlinear concave up function, the average growth rate under variable conditions is greater than the growth rate at the average daily temperature:

Eq. 1: [(average growth rate) equals (average of function of temperature variation)] is greater than [(average growth rate) equals (function of average temperature)

$\left [ \bar\lambda = \overline{f(T)}\right ] >\left [ \bar\lambda = f(\bar T)\right ]$

Populations will grow faster for the copepods living under variable temperatures than for those living at the mean of that variability. How much greater depends on the shape of the function, and the range of
environmental variation. The opposite is true if the relationship is a concave downward function.

If the relationship between population growth rate and an environmental factor is nonlinear, then the average growth rate under variable conditions does not equal the growth rate under average conditions.

As with environmental variation, these mathematical considerations take on increased significance under current global climate change conditions where both environmental means and variances are shifting (Drake, 2005; Pickett et al., 2015).

Vocabulary
Law — A scientific law is a statement, based on multiple observations or experiments, that describes a natural phenomenon. Laws do not have absolute certainty and can be refuted by future observations.
Nonlinear — A nonlinear system is a system in which the change of the output is not proportional to the change of the input (Wikipedia).

References

• Denny, M. (2017). The fallacy of the average: on the ubiquity, utility and continuing novelty of Jensen’s inequality. Journal of Experimental Biology, 220(2):139–146.
• Drake, J. M. (2005). Population effects of increased climate variation. Proceedings of the Royal Society B: Biological Sciences, 272(1574):1823–1827.
• Huntley, M. E. and Lopez, M. D. (1992). Temperature-dependent production of marine copepods: a global synthesis. The American Naturalist, 140(2):201–242.

# Systems Paleoecology – Environmental Variation: Expectations and Averages

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

Environmental Variation

The discussion to this point has treated the intrinsic demographic parameters, r, R and K as constants. However, it cannot be overstated that under the current conditions of climate change, and the interactions of multiple anthropogenic drivers of population change, great importance must be placed on understanding the potential environmental impacts on those parameters and population dynamics. The impact of direct environmental perturbations was considered briefly in an earlier post, but in reality the relationship can be, and usually is, more complicated. Our collection of environmental data is expanding at a rapid pace, enabling monitoring of variables such as air temperature, precipitation, etc. at scales ranging from meters to the entire planetary surface. Similarly, we are reconstructing environmental histories in ever more detail, ranging from sub-decadal to multimillenial timescales. Those data are revolutionizing views of the relationship between stability and the environment, but they bring their own challenges. The study and literature of that relationship are vast and growing though, and the discussions in this post and the next will therefore be limited to two important topics; the treatment of time in population dynamics, and the effect of nonlinear relationships between population dynamics and environmentally impactful factors. The topics capture two fundamentally important features of environmental variation: the inconstancy of the environment, and organismal responses to the variation.

Expectations and Averages

Imagine a population growing in a randomly varying, but stationary environment, v. We’ll call it a “noisy” environment. The environment varies from one interval of time to another, but the mean environment is constant. We refer to the mean environment as the expected environment, E(v), and its value is independent of time, i.e. E(v) is expected to have the same value no matter when the population exists. Let us assume that the population growth rate, λ, is at any given time a simple function of the value of the environment, say larger values of v increase birth rates and lower values increase death rates. We use λ to signify that we model growth rate here as independent of populations size. This model was introduced in an earlier post (Eq. 2). Then the long-term value of λ is also a simple expectation or function of the mean environment. We will express this as

Eq. 1: (expected or long-term population growth rate) = (function of the mean environment)

$E(r) = f(E(v))$

A crucial question is this: Is the observed population size, X(t) , equal to the expected size given E(v) and λ? Returning to one of our earlier and simplest models, the expected size of the population after elapsed time T , given an initial population size X(0) and a deterministic, average growth rate of $\langle\lambda\rangle$, is

Eq. 2: (deterministic population size) = (initial population size) x (the product of growth rate, multiplied by itself T times)

$X_T = X_0\langle\lambda\rangle^T$

Mathematical digression
X(0) is the initial population size, and $\langle\lambda\rangle$ is the factor by which the population increases during each interval of time. Therefore, after the first interval, population size is
$X_1 = X_0 \langle\lambda\rangle$
because $\langle\lambda\rangle$ is an average growth rate.
$\langle\lambda\rangle = \frac{1}{N}\sum\frac{X_n}{X_{n-1}}$
where N is the number of observations in your population size time series. After the second interval,
$X_2 = X_1 \langle\lambda\rangle = \left ( X_0 \langle\lambda\rangle\right )\langle\lambda\rangle = X_0\langle\lambda\rangle^2$
Thus, after T intervals,
$X_T = X_0 \langle\lambda\rangle^T$

Assume realistically, however, that the environment is a randomly varying one, though, and this environmental stochasticity means that there is no single value of v, but instead a distribution of values. Say that we characterize this distribution as a normal one, a mean and variance. Then, by Eq. 1, the population growth rate λ is also a distribution of values. If we assume here that Eq. 1 is a simple linear function, then r is also distributed normally, with mean $\langle\lambda\rangle$. A population living in that environment will have a value of r, during each interval of its history, drawn from the distribution. In that case, population size after time T is now given as

Eq. 3: (deterministic population size) = (initial population size) x (the product of observed growth rates)

$X_T = X_0\prod_{t=1}^{T}\lambda_t$

Surprisingly, actual population size given a varying environment is always smaller than that predicted by the mean environment (Fig. 1)!

The mean growth rate predicts the deterministic population size in the absence of environmental variation, but the series of observed λ(t) predicts otherwise. The environment, v, is a random variable with a mean and variance, and we therefore treat population growth as a resulting random variable. The reason for the discrepancy is subtle — growth rate is now a function of environmental variation from one interval of time to the next — but the implications are important. The mean or expected growth rate, $\langle\lambda\rangle$, is derived from the environmental mean, and is hence an arithmetic mean,

Eq. 4: (average growth rate) = function of (averageD environmentAL VARIATION)

$\langle\lambda\rangle=f[(1/T) \sum v(t)]$

(because there are T observations), treating population growth as an additive process. Population growth, however, is in actuality a multiplicative process, where future population size is a multiple of initial population size and a series of randomly varying growth rates (Eq. 3) (Lewontin and Cohen, 1969). The true mean growth rate is therefore the geometric mean of the observed growth rates,

Eq. 5: (average growth rate) = (geometric average of observed growth rates)

$\bar\lambda_G = \left (\prod \lambda_t\right )^{1/T}$

which are themselves functions of the varying environment during a given interval of time, and not the mean environment over time (use the same logic as in the box above). You should be able to convince yourself that the geometric mean is always equal to or less than the arithmetic mean. In this case the geometric mean reflects the fact that population growth is not based on the average environment, but instead on the population’s history, that in turn reflects the manner in which the environment varies. This distinction is an important one to remember whenever dealing with historical, path-dependent processes, such as paleo-population time series, and there are many ways of saying it, e.g. the performance of an individual over time does not equal the average performance of the group, because the system is non-ergodic, i.e., time matters. Or, the observed value of a random variable is not the same as averaging the expected value over time. For our purposes here, the following is the important implication in a world of changing and increasingly variable environments.

Population size in a varying environment is not equal to expected population size in the mean environment.

Vocabulary
Arithmetic mean — Sum of a set of numbers divided by the size of the set.
Environmental stochasticity — The impact of nearly continuous perturbations on individual birth and death rates.
Ergodic — Property of a dynamic system if the behaviour of the system during a sufficiently long interval of time is typical of the system’s behaviour during other intervals of similar duration.
Geometric mean — The $n^{\mathtt{th}}$ root of the product of a set of numbers, where n is the size of the set.
Stationary — A series is stationary if distributional parameters, such as the mean and variance, do not change over time.

References
Lewontin, R. C. and Cohen, D. (1969). On population growth in a randomly varying environment. Proceedings of the National Academy of sciences, 62(4):1056–1060.

# Systems Paleoecology – Chaotic Stability?

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:

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.

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

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

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).

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

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:

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).

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).

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.

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

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:

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.

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)

$x(t+1) = rx(t)[1-x(t)]$

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.

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.

# Systems Paleoecology – Deviations from Equilibrium

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.

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).

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.

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.

# Systems Paleoecology – Logistic Populations II

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:

The logistic equation, covered in the previous post, is a differential equation, where time is divided up into infinitesimal bits to model the growth and size of X (“infinitesimal” knots your stomach? I cannot recommend Steven Strogatz’s “Infinite Powers” enough!). We can also simulate the logistic model in discrete time to get a better feeling for it, where X(t + 1) is population size in the next “time step” or generation. This approach is instructive because anyone can play with the calculations using a calculator or spreadsheet! Here is an example of a discrete version of logistic growth, the Ricker difference equation (Ricker, 1954).

EQ. 1: (FUTURE POPULATION SIZE) = (CURRENT POPULATION SIZE) x (EXPONENTIAL REPRODUCTION LIMITED BY CARRYING CAPACITY AS IN THE LOGISTIC MODEL)

$X_{t+1} = X_{t}e^{R\left (1-\frac{X_{t}}{K}\right )}$

r has been replaced by R, the main difference between the logistic equation and Eq. 1 being that, because we no longer measure time as continuous but instead step discretely from one generation to the next, we measure the intrinsic rate of increase as the “net population replacement rate”. The population again, after an initial interval of near-exponential growth, settles down to a fixed value at K (Fig. 1). Both the continuous and discrete logistic patterns of population growth are, by all definitions, stable populations in equilibrium. They are stable because, at least within the scope of the models, once a population attains its carrying capacity there is no more variation of population size. This brings us to our first definition of stability.

Stability: An absence of change.

In the following sections we will cover a real-world example of logistic growth, and then go through the derivation of the logistic function itself.

An Example of Logistic Growth

The state of Washington in the United States employed a program of harbor seal (Phoca vitulina) culling during the first half of the twentieth century. The seals were considered to be direct competitors to commercial and sport fishermen. The state sponsored monetary bounties for the killing of seals until 1960, by which time seal populations must have been reduced significantly below historical levels. Additional relief arrived for the seals in 1972 with passage of the United States Marine Mammal Protection Act. Monitoring of seal populations along the coast, estuaries and inlets of Washington, primarily by the Washington Department of Fish and Wildlife, and the National Marine Mammal Laboratory provided a time series of seal population size, spanning the beginning of recovery in the 1970’s to the end of the century (Jeffries et al., 2003). Population sizes from one region of the coastal stock, the “Coastal Estuaries”, show a logistic pattern of growth (Fig. 2). The function fitted to the data (using a nonlinear least squares regression) is y = 7511.541/[1 + exp[−0.265(x − 1980.63)]] (r-squared = 0.98; p < 0.0001; note that “r-squared” is the coefficient of correlation, not our intrinsic rate of increase). Given an initial population size of X(0) = 1,694 in year 1975, the function yields estimates of r = 0.265 and K = 7,511. This excellent example of logistic growth in the wild, or recovery in this case, was unfortunately brought to us courtesy of the ill-informed belief that the success of human commercial pursuits necessitate, or even benefit from, the destruction of wild species.

Deriving the logistic equation
Equation 1 in the previous post is the logistic growth rate of the population, but it is not the logistic function itself. That function is obtained by integrating the growth rate dX/dt, and the process is instructive because, as illustrated in later sections, our ability to do so with more complicated dynamic equations is quite limited.

The logistic growth rate is first re-written to eliminate the X/K ratio (makes it easier to proceed)
$\frac{dX}{dt} = rX\left ( 1-\frac{X}{K}\right )$
$\Rightarrow K\frac{dX}{dt} = rX\left ( K-X\right )$
and then re-arranged to separate variables,
$\frac{K\, dX}{X\left ( K-X\right)} = r\,dt$
The logistic function is derived by integrating both sides, but doing so with the left hand side (LHS) requires simplification using partial fractions (some of you might remember those from high school math; or not).
$\frac{K}{X\left ( K-X\right)} = \frac{A}{X} + \frac{B}{K-X}$
$\Rightarrow K = X\left ( K-X\right ) \left [ \frac{A}{X} + \frac{B}{K-X} \right ]$
$\Rightarrow K = A(K-X) + BX$
$\Rightarrow K = AK - X(B-A)$
The solutions to the final equation are A=1 and B-A=0, yielding B=1. Therefore
$\frac{K}{X\left ( K-X\right)} = \frac{1}{X} + \frac{1}{K-X}$
Now if we wish to integrate our logistic differential equation,
$\int \frac{K\, dX}{X\left ( K-X\right)} = \int r\,dt$,
we can substitute our partial fractions solution and proceed as follows.
$\Rightarrow \int \left ( \frac{1}{X} + \frac{1}{K-X}\right ) dX = \int r\,dt$
$\Rightarrow \int \frac{1}{X}\, dX + \int \frac{1}{K-X}\, dX = \int r\,dt$
And if you recall our integration of the Malthusian Equation, the solution is
$\Rightarrow \ln{\vert X\vert} - \ln{\vert K-X\vert} = rt + C$
$\Rightarrow \frac{K-X}{X} = e^{-rt-C}$
Let $A=e^{-C}$, a constant. Then
$\frac{K}{X} - 1 = Ae^{-rt}$
$\Rightarrow \frac{K}{X} = Ae^{-rt} + 1$
$\Rightarrow X(t) = \frac{K}{Ae^{-rt}+1}$
which is the equation for logistic population growth! Whew.

References
Jeffries, S., Huber, H., Calambokidis, J., and Laake, J. (2003). Trends and status of harbor seals in Washington State: 1978-1999. The Journal of Wildlife Management, 67:207–218.
Ricker, W. E. (1954). Stock and recruitment. Journal of the Fisheries Board of Canada, 11:559–623.

# Systems Paleoecology – Logistic Populations

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

We were introduced to the logistic equation (Eq. 1) in the previous post,

EQ. 1: (POPULATION GROWTH RATE) = (INTRINSIC RATE OF INCREASE) x (POPULATION SIZE) x (GROWTH REGULATION)

$\frac{dX}{dt} = rX\left( 1-\frac{X}{K}\right )$

and then explored Malthusian populations, where populations grow according to the first term of the function, rX. Examining Eq. 1, however, we note that the second term on the RHS includes K, the carrying capacity of the environment. K is a bit of an odd parameter because it is never something that we can determine a priori, but instead must be deduced or inferred from observations (preferably long-term observations). It is basically our assessment of the number of individuals of the species that can be accommodated in the environment, whatever that assessment might be based on. Knowing, however, that all resources on the planet are finite, we infer that all species must have finite carrying capacities. In that second term, we note that as X increases and approaches K, the entire second term approaches zero. K puts the brakes on growth.

The sizes of all populations are ultimately limited on a finite planet.

The logistic model captures this intuitive fact.

Figure 1 shows our population from the previous post, but this time with a carrying capacity set at 100 individuals. The population grows from a small initial size and asymptotes toward K. The growth rate is initially very rapid, as the population expands to “fill the environment”, but slows down as fewer additional individuals can be accommodated. There are two equilibria for this population, i.e. points at which, once attained, X will remain. The first is trivial, being zero, or an extinct population, and the other is K. “Zero” is an unstable equilibrium, because the population will always increase away from zero if X > 0, i.e. if reproducing individuals are added to the environment.

Parameter r

Like K, r is also an interesting parameter. What do we mean when we say “intrinsic rate of increase”? r is the instantaneous rate of change of population size, and can be estimated over an interval of time as the per capita number of births minus the per capita number of deaths. It is not a population growth rate, the actual population growth rate, dX/dt, being a function of population size (X), r, and, in the case of limitation by carrying capacity, K. The proper way to interpret r, and other such modifiers of growth rate (dX/dt), is as the net reproduction per individual in the population which, being the number of individuals produced per individual per unit of time, yields units of 1/time . Population growth rate is a function of this individual productivity and standing population size, or the number of individuals produced, in total, per unit of time. Contrast what happens when r is positive versus when it is negative. If r > 0, then dX/dt is positive and the population increases. If r < 0 then population growth rate is negative, and the population shrinks.

Deriving r is instructive, and helps us to understand this widely-estimated and applied parameter. We begin with a simple, density-independent model (Lewontin and Cohen, 1969), i.e. a population whose growth is not limited by the number of individuals in the environment.

EQ. 2: (FUTURE POPULATION SIZE) = (POPULATION GROWTH RATE) x (CURRENT POPULATION SIZE)

$X_{t+1} = \lambda X_{t}$

where λ is the proportional change in population size between times t and t+1, sometimes termed the “deterministic population growth rate”. Re-write population growth, letting growth in a single time interval be a function of the total numbers of births (B) and deaths (D) during the interval: X(t+τ) = X(t) + BD, where τ is the amount of time elapsed. The growth rate during the interval then is given as the increase (decrease) of population size, divided by the duration of the interval.

GROWTH RATE WITHIN A SINGLE INTERVAL IS THE NUMBER OF BIRTHS MINUS THE NUMBER OF DEATHS DIVIDED BY THE DURATION OF THE INTERVAL

$\frac{\Delta X}{\tau} = B-D$

If we now express B and D as results of the production and losses of all individuals in the population, then B = bX and D = dX, where b and d are the per capita birth and death rates. It should then be easy to see
that if we shrink the duration of the interval to be infinitesimally small (i.e. τ tends toward zero, the basis of calculus; nice connection to Isaac Newton and the pandemic of his time), we derive the differential formula

$\frac{dX}{dt} = bX - dX = (b-d)X = rX$

which if you recall is both the basis of the Malthusian exponential growth formula, and also Eq. 1 without the limitation of carrying capacity!

Finally, it should be noted that the simulation illustrated in Fig. 1 is not a plot of Eq. 1. The formula is for the logistic growth rate, whereas the plot is of population size over time — the integral of Eq. 1. In the next post I will show you how to derive the logistic equation itself,

$X(t) = \frac{K}{Ae^{-rt}+1}\, \mathtt{,}$

discuss a discrete and accessible parallel to the continuous logistic function, and give a real-world example of logistic population growth.

Vocabulary
Carrying capacity The maximum population size that can be supported by the environment.
Intrinsic rate of increase Intrinsic birth rate minus intrinsic death rate, or the per capita productivity of an individual per unit of time.

References
Lewontin, R. C. and Cohen, D. (1969). On population growth in a randomly varying environment. Proceedings of the National Academy of Sciences, 62(4):1056–1060.

# Systems Paleoecology – Malthusian Populations

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

THE SIMPLEST CONCEPT OF ECOLOGICAL STABILITY IS A POPULATION OF STATIC, UNCHANGING SIZE. For example, if we are considering the population size of a species, X, then X in static equilibrium has the same, unchanging value over time. Or, population size might be variable but over time the net change of population size is zero, in which case the system could be in dynamic equilibrium. I will refer to the entire population as a system, by which I mean simply a collection of potentially interacting objects. In this case the interacting objects are conspecific members of the same population.

The basic questions of stability to be answered for any ecological or related (e.g. socio-ecological) systems are:

1. Is the system static or dynamic?
2. If the system is dynamic, can we formulate a dynamical law that predicts the next state of the system given the current state?
3. Given a dynamical law, can we determine the driver or drivers of the system’s dynamism? Is the driver intrinsic? That is, if the system was a closed system, would it remain dynamic, or, how would the dynamics change?

Note that in closing a system by isolating a population, one can include anything in “the box” to the exclusion of other potential drivers, e.g. a population and the fishing pressure on it, excluding climate change. Such an approach is a primary motivation and guide to building scientific models.

LET US BEGIN WITH A VERY SIMPLE MODEL OF POPULATION GROWTH, the logistic model. Here we describe the growth of a population as regulated by two factors: the intrinsic, or population-specific rate of increase of the species, e.g. measured as average birth rate minus average death rate; and the carrying capacity of the environment, i.e. the maximum population size of the species in this place and time based on limiting factors, such as biological resources and the abiotic environment. In mathematical terms, we write this as

EQ. 1: (POPULATION GROWTH RATE) = (INTRINSIC RATE OF INCREASE) x (POPULATION SIZE) x (GROWTH REGULATION)

$\frac{dX}{dt} = rX\left( 1-\frac{X}{K}\right )$

where dX/dt is the population’s growth rate (change in X per unit of time), r is the intrinsic rate of increase of the species, and K is the population’s carrying capacity.

This differential equation expresses a logistic growth rate, where growth is exponentially fast at small population size, but slows toward zero as the carrying capacity is approached. The first product on the right hand side (RHS) of Eq. 1, rX, would if solved for X result in exponential or Malthusian growth,

EQ. 2: (POPULATION SIZE AFTER TIME t) = (INITIAL POPULATION SIZE) x (EXPONENTIAL REPRODUCTION DURING TIME INTERVAL t)

$X_t = X_0 e^{rt}$

where $X_0$ is the initial population size (Fig. 1). The growth rate is the Malthusian growth rate (r) (Malthus, 1798) multiplied by the number of individuals already in the population, and thus accelerates over time, yielding an exponential relationship.

Any population with a positive growth rate will increase exponentially if left unchecked.

The growth rate rX makes sense, as it can be interpreted as the number of offspring that an individual will produce per unit of time, multiplied by the number of individuals in the population. But why does that yield an exponentially increasing population? The answer is given below, but first a brief Malthusian digression.

(From Wikipedia) Thomas Robert Malthus FRS (February 1766 – December 1834) was an English cleric, scholar and influential economist in the fields of political economy and demography. In his 1798 book An Essay on the Principle of Population, Malthus observed that an increase in a nation’s food production improved the well-being of the populace, but the improvement was temporary because it led to population growth, which in turn restored the original per capita production level.

Deriving the Malthusian Equation
Let the Malthusian growth rate be
$\frac{dX}{dt} = rX$
Here the growth of X is a function of X, that is, the rate of population growth depends on the size of the population. In order to calculate the size of the population at any given time, however, we first re-arrange the equation to make it a function of time elapsed, dt.
$\frac{dX}{X} = r \, dt$
Integration to solve for X is then straightforward.
$\int \frac{1}{X}dX = \int r \, dt$
The integral of 1/X is the logarithm of X.
$\Rightarrow \ln{X} = rt + C_1$
$\Rightarrow e^{\ln{X}} = e^{rt + C_1}$
$\Rightarrow X = e^{rt} \cdot e^{C_1}$
$\Rightarrow X = C_{2}e^{rt}$
C 1 and C 2 are simply constants. And of course, e, or Euler’s number, raised to any positive number yields a relationship where the output increases faster as the input increases.

Vocabulary
Carrying capacity — The maximum population size that can be supported by the environment.
Closed system — A system where the passage of specified materials, objects or information into or out of the system is prohibited or controlled. The system is isolated from the rest of the Universe, behaving as if nothing else existed.
Dynamics — The study of how systems change or develop over time.
Dynamic system — A system that varies or changes over time. Dynamic systems can be in equilibrium, if net change is zero.
Equilibrium — A condition where the state of a system either does not change, or experiences no net change over time.
Law A scientific law is a statement, based on multiple observations or experiments, that describes a natural phenomenon. Laws do not have absolute certainty and can be refuted by future observations.
Population A system of conspecific individuals occupying overlapping space at the same time.
System A collection of potentially interacting objects.

References
Malthus, T. R. (1798). An essay on the principle of population. Library of Economics and Liberty.

# Systems Paleoecology – Introduction

Tags

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…

We have a capacity for imagining situations that are not implied by the data. . . Lee Smolin

The concept of “stability” in science is an evolving one, partly because of the advent of systems approaches to multiple disciplines. To the extent that the 20th century was the century of the small (the atom, the gene, the bit), we can claim the 21st century to be the century of systems: ecological, genomic, socio-eco-economic, information, and so on. In the end I don’t think that we yet have a complete understanding of stability, or perhaps we do not yet fully know what it is that we need to understand.

The workshop took place at the Leibniz Center in Berlin, during May, 2019.

In this blog series I will outline my own current views on what stability means in paleocology — the study of the ecological aspects of the history of life. Although stability is a multi-disciplinary concept, my discussion will be biased heavily toward ecological and paleoecological systems as those are my areas of expertise. However, the concepts and discussion are hopefully general enough to be of multi- and trans-disciplinary interest. In instances where they are not, or fall short of being applicable in another discipline, I urge others working in those areas to formulate terms and definitions as needed so that in the end we have a comprehensible and comprehensive terminology, and can truly understand what stability means in all the dynamic systems that we are dealing with today.

Paleoecological concepts

Ecology, including paleoecology, is a fundamentally observational discipline for which a large and broad array of explanatory principles and theories has been developed, e.g. the principle of competitive exclusion (Gause’s law, Grinnell’s principle), the Theory of Island Biogeography, and Hubbell’s Unified Neutral Theory of Biodiversity. These laws, principles and theories differ from foundational theories in other scientific disciplines, such as General Relativity, quantum mechanics, evolution by natural selection, and population genetics, in being limited in the numerical capabilities or precision of their predictions. E.g., many species that compete for resources will coexist in the wild without exclusion, and assemblages of species competing for resources often do not behave neutrally. Despite this, there is an underlying strength to predictive ecological theories and models when they are based on sound inductive reasoning, for the limits of their applicabilities to the real world or inconsistencies with empirical data expose the sheer complexity and high dimensionality of ecological systems — competitors may coexist because of differing life history traits (e.g. dynamics of birth-death rates), incomplete or intermittent resource overlap, spatial and temporal refuges from superior competitors, pressure from predators, and so forth. This complexity of ecological systems is in turn driven by four main factors: the geosphere, evolution on short timescales, history on long timescales, and emergent properties.

The geosphere, atmosphere and hydrosphere, including tectonic, oceanographic and atmospheric processes, affect ecological systems on multiple spatio-temporal scales. Geospheric dynamics determine the appearance and disappearance of islands, the erection and removal of barriers to dispersal and isolation, patterns and rates of ocean circulation and mixing, climate, and weather. The mechanisms of genetic variation and natural selection determine whether, how and how quickly populations of organisms can acclimatize or adapt to their ever-changing, dynamic environments. Those accommodations in turn feedback to their abiotic and biotic environments. No ecological system, however, is solely or even largely a product of processes occurring on generational, ecological, or contemporaneous timescales, for the collection of species that occupy a particular place and time — a community — arrived at that point via path-dependent histories. What you see now depends very much on what came before. Those histories are themselves a cumulative set of past responses of populations, species and communities to their abiotic and biotic environments. And those populations of multiple species, when interacting, are complex systems with emergent properties such as stability. Emergent properties can act as additional drivers of population and community dynamics in feedback loops that both expand and contract the scope within which ecological dynamics deviate from the pure predictions of principle-based theories and models.

Models

The following work will make extensive use of mathematical models, because I believe that they are useful and somewhat underutilized in paleoecology, and because I like them. One guide to understanding the utility of model-based approaches in ecology and paleoecology is to question the soundness of their underlying assumptions, and to explore why those assumptions might appear to be inaccurate when a particular approach is applied to the real world. And both ecological and paleoecological theories are laden with assumptions, sometimes explicit, but often implicit. Ask yourself the following questions: Do real populations ever attain carrying capacity? Are the sometimes complex dynamics predicted by intrinsic rates of population growth ever realized in nature? Are populations ever in equilibrium? What are the relative contributions of intrinsic and extrinsic processes to a population’s dynamics? Are communities stable? If they are, is stability a function of species properties, or of community structure, and if the latter, where did that structure come from? Is community stability always a result of a well-defined set of general properties, or is the set wide-ranging, variable, and idiosyncratic? And, are the answers to these questions based on laws that have remained immutable throughout the history of life on our planet, or have the laws themselves evolved or varied in response to a dynamic and evolving biogeosphere? In the posts that follow, I will introduce basic concepts that are essential to understanding ecological stability, and to equip us to further explore more extensive and sophisticated models that are beyond the scope of the blog. I will attempt to build the concept of stability along steps of hierarchical levels of ecological organization, and to relate each of those steps to paleoecological settings, concepts and studies. This will not be a series on analytical methods. It is about concepts and conceptual models. There are already rich resources and texts for paleoecological methodologies.

The Series

The posts will be divided into parts, each successive part building on the previous one by expanding the complexity of the systems and the levels of organization under consideration. Part I deals with isolated populations, an unrealistic situation perhaps, but an idealization fundamental to understanding systems of multiple species. And, it is populations that become extinct. This part contains a lot of introductory material, but it is essential for laying groundwork for later sections that both deal with more advanced and original material. Advanced readers might wish to skip over these posts, but there is original matter in there, and I welcome feedback! Part II addresses community stability, with an emphasis on paleoecological models and applications. Part III explores the evolutionary and historical roots of ecological stability, including the origination of hierarchical structure and community complexity, stability as an agent of natural selection, and the selection and evolution of communities and ecosystems.

Caveat lector!

The discussion will be technical in some areas, because “systems” is a technical concept. Mathematical models are used extensively because I have found them to be a more accessible way to understand the necessary ecological concepts, sometimes in contrast to actual ecological narratives. Ecological systems are complicated and complex, and models offer a way for us to focus on specific
questions, distilling features of interest. Useful models are in my opinion simple, and they can serve as essential guides to constructing narratives and theories of larger and more complete systems. I will therefore taken great care to outline and explain basic concepts and models (Do not fear the equations! But feel free to ignore them..). Examples of real-world data and analyses will be included in many sections. Additionally, code for many of the models will also included. I use the Julia programming language exclusively (but I have also used C++, Octave and Mathematica extensively in the past, and recommend them highly). I regard R‘s power with awe, but I am not a fan of its syntax.

My hope is that the series will successfully build on concepts and details progressively, and that at no point will readers find themselves unable to continue. I don’t think that a technical mastery is at all necessary, but it can deepen one’s qualitative grasp significantly. And one should never underestimate the power to impress at a party if you can explain mathematical attractors and chaos!

And finally, what follows is unlikely to comprise my final opinions on this topic.