Systems Paleoecology – Regime Shifts II

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WHAT IS ECOLOGICAL STABILITY ? In 2019 I posed this question informally to colleagues, using Twitter, a professional workshop that I lead, and a conference. Respondents on Twitter consisted mostly of ecological scientists, but the workshop included paleontologists, biologists, physicists, applied mathematicians, and an array of social scientists, including sociologists, anthropologists, economists, archaeologists, political scientists, historians and others. And this happened…

Previous posts in this series

The Grand Banks of Newfoundland.

The Atlantic cod, Gadus morhua, has been the foundation of one of the world’s major fisheries for several centuries, with the Grand Banks off Newfoundland (Fig. 1) being amongst the most productive fisheries in the world. Figure 2 (below) shows the estimated population size of the Atlantic cod on the southern Grand Banks from the years 1959 to 2005 (Power et al., 2010), where G. morhua has historically been the most prominent component of that productive fishery. Cod populations, however, declined across the North Atlantic during the second half of the 20th century, most notably in the northwestern Atlantic. Several factors might have played roles, including warming ocean temperatures and a resurgence of the predatory grey seal across its historic range after implementation of protection from hunting (Neuenhoff et al., 2019). There is little doubt, however, that over-exploitation by commercial fisheries has been one of, if not the most effective driver of the decline. The Grand Banks population initially increased steadily during the monitored period, reaching a maximum in 1966. Thereafter it declined significantly until 1976, after which it appeared to stabilize, before beginning a steady decline in 1985. Fishing mortality of older individuals (> 6 years old) meanwhile fluctuated, peaking in 1975 and again in 1991. Change point analysis, which is used to detect changes in the statistical distribution of data within a time series, suggests that the population underwent significant shifts in 1971, 1985 and 1993. Each time, the mean population sizes of the intervals defined by those shifts descended through transient phases to significantly lower sizes. The first transition, around 1971, was preceded by an interval of highly variable population size, yet those sizes all exceeded population size from 1971 onward. The period from 1975-1985 was characterized by reduced population size and low variability, but another change occurred in 1985, and by 1993 the population size had stabilized at abysmally low numbers, marking the end of the commercially viable fishery.

Estimated population size (connected line) and fishery catch size (red) for the period 1959-2005. Arrows show times of presumed regime shifts, as identified by change point analysis.

It seems reasonable to hypothesize that each interval between the change point transitions represents a stable state or regime of the population, with fishing mortality thus driving or contributing to transitions between multiple alternative states (Fig. 2). The relationship between fishing and population size is relatively straightforward. Initially, an increase of total catch in the late 1960’s followed an increase of population size, but then declined as population size itself began a steep decline in 1967. However, although population size continued to decline, catch size again increased in 1971, coincident with the time marked by the change point analysis as the first transition to a state of smaller population size. Catch size subsequently followed the decline of population size, reaching a minimum during 1978, after which it began to increase again, presumably in response to the relatively “stable” population. Population size increased after 1980, reaching a new maximum in 1985, but then after a sharp increase of catch size, began its decline to the low numbers at the turn of the century.

A phase map (Fig. 3) captures this journey of potential alternative stable states and transitions, and reveals two general types of regime shifts. First, the initial transition to the second state that persisted from the late 1970s to 1985 was likely both precipitated and maintained by fishing pressure. The continued decline of catch size between 1971 and 1978 might have in fact facilitated some recovery of population size. This is the first type of regime shift and the separation of states—an external driver is capable of moving the system between states, and of maintaining the system in at least one of those states. The basic dynamics of this type of regime shift can be understood in terms of external parameters. The second type of shift, however, is less transparent, because it involves intrinsic properties and dynamics of the system. Look closely at the population trajectory from 1995 on, the last transition of the series. Catch size is negligible over a period of ten years, yet there is no sign of population rebound. If, as claimed earlier, population size is driven by catch size, why didn’t the population recover, and what kept it in the final, most recent attractor or state?

Phase map of cod population size, plotting consecutive years against each other. Blue trajectories show intervals of presumed population stability, i.e., stable states. Red trajectories show the pathways of transition between those states.

References
Neuenhoff, R. D., Swain, D. P., Cox, S. P., McAllister, M. K., Trites, A. W., Walters, C. J., and Ham-mill, M. O. (2019). Continued decline of a collapsed population of Atlantic cod (Gadus morhua) due to predation-driven Allee effects. Canadian Journal of Fisheries and Aquatic Sciences, 76(1):168–184.

Power, D., Morgan, J., Murphy, E., Brattey, J., and Healey, B. (2010). An assessment of the cod stock in NAFO divisions 3NO. Northwest Atlantic Fisheries Organization SCR Doc, 10:42.

More Ecosystems, Epidemics and Economies

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I tag-teamed with my dean, the Academy‘s Chief Scientist, Dr. Shannon Bennett last night to give a couple of presentations for NightLife’s “NightSchool” series. Shannon is a virologist who specializes in emergent zoonotic diseases. These are shorter talks, and mine has an update of the past week of work on our CASES (Complex Adaptive Socio-Economic Systems) project. Here’s the summary from the NightSchool crew:

“What does the COVID-19 pandemic look like through the lens of scientists who study viruses and evolutionary ecology? Join Dr. Shannon Bennett, virologist and the Academy’s Chief of Science, and Dr. Peter Roopnarine, Curator of Geology and Paleontology, for a fascinating dive into the past—and the future?—of our current pandemic.

Set off on an evolutionary journey of the virus behind it all, SARS-CoV2, with Dr. Bennett as your guide. She’ll talk about its origins and where it’s going, and also bust a few myths, expand on what scientists and society are most concerned about today, and what you can do about it.

Can fossils help us understand the impact of epidemics on the economy? Dr. Roopnarine, who creates models of complex food webs that existed millions of years ago, will talk about the complex systems that dominate our lives today and what happens when ecosystems, epidemics, and economies collide. He’ll explore why ecosystems rise and fall and the lessons learned there, and apply them to present-day economies and the current work being done to understand the dynamic impact of COVID-19 on these systems.”

Systems Paleoecology – Regime Shifts I

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WHAT IS ECOLOGICAL STABILITY ? In 2019 I posed this question informally to colleagues, using Twitter, a professional workshop that I lead, and a conference. Respondents on Twitter consisted mostly of ecological scientists, but the workshop included paleontologists, biologists, physicists, applied mathematicians, and an array of social scientists, including sociologists, anthropologists, economists, archaeologists, political scientists, historians and others. And this happened…

Previous posts in this series

Numerous terms, with roots across multiple disciplines that deal with dynamic complex systems, are used interchangeably in the study of transitions to some extent because they are related by process and implication. But they do not necessarily always refer to the same phenomena, and it is useful to be explicit in one’s usage (maybe at the risk of usage elsewhere). Regime shift, critical transition and tipping point are three of the more commonly applied terms in the ecological literature. They form a useful general framework within which to explore the concept of multiple states and transitions, and into which more detailed concepts can be introduced. Regime shift is defined here as an abrupt or rapid, and statistically significant change in the state of a system, such as a change of population size (Fig. 1A). Transient deviations or excursions from previous values, e.g. those illustrated in Fig. 1B}, are not regime shifts. “Regime” implies that the system has been observed to have remained at a stationary mean or within a range of variation over a period of time, and to then have shifted to another mean and range of variation. Regimes can be maintained by external or intrinsic processes, or sets of interacting external parameters and internal variables, but the ways in which the processes are organized can vary. Sets of processes can be dominant, reinforcing the regime; understanding this simply requires one to associate a regime with our previous discussions of system states and attractors. Regime shifts occur then when sets of processes are re-organized, and dominance or reinforcement shifts to other parameters and variables.

Fig. 1A – Hypothetical regime shift
Sizes of two populations of the Red-Winged Blackbird,Agelaius phoeniceus, from the Gulf of Mexico. Left - Texas; right - Florida. Thick horizontal red lines show series medians, and thinner lines the $5^{mathtt{th}}$ and $95^{mathtt{th}}$ percentiles.
Fig. 1B – Two populations of red-wing blackbirds. See here for an explanation.

Regime shifts may be distinguishable from variation within a state, or continuous variation across a parameter range, by the time interval during which the transition occurs, if the interval is notably shorter than the durations of the alternative states. This of course potentially limits the confirmation of regime shifts as we can never be certain that observation times were sufficient to classify the system as being in an alternative state. The interpretation though is that the duration of the transition was relatively short because the system entered into a transient phase, i.e. moving from one stable state to another. The transition itself may be precipitated in several different ways, dependent on the type of perturbation and the response of the system. The perturbation could be a short-term excursion of a controlling parameter that pushes the system into another state, with the transition being reversed if the threshold is crossed again. More complicated situations arise, however, if internal variables of the system respond to parameter change without a measurable response of the state variable itself, and if the system can exist in multiple states within the same parameter range. These various characteristics of regime shifts serve to distinguish important processes and types of shifts that are more complex than simple and reversible responses to external drivers, such as “critical transitions” and “tipping points”.

We have already discussed several model systems with multiple states, one of those being a trivial state of population extinction (X=0), and the other being an attractor when X>0. Zero population size was classified as an unstable state, because the addition of any individuals to the population — X_1>X_0=0— leads immediately to an increase of population size, and the system converges to a non-zero attractor. This is true regardless of the nature of the attractor (e.g. static equilibrium, oscillatory, chaotic), and makes intuitive sense — sprinkle a few individuals into the environment and the population begins to grow. This is not always the case, however, and there are situations where zero population size, or extinction, can be a stable attractor, or where X converges to different attractors, dependent either on population size itself, or forcing by extrinsic parameters. The system is then understood to have multiple alternative states. I reserve this definition for circumstances where X does not vary smoothly or continuously in response to parameter change (e.g. Fig. 1), but will instead remain in a state, or at an attractor, within a parameter range, and where the states are separated by a parameter value or range within which the system cannot remain, but will instead transition to one of the alternative states. Thus, the multiple states are separated in parameter or phase space by transient conditions.

We will explore a real-life example in the next post, and here is a teaser.

Cod in the North Atlantic.

Vocabulary
Attractor – A compact subset of phase space to which system states will converge.
Regime shift – An abrupt or rapid, and statistically significant change in the state of a system.
System state – A non-transient set of biotic and abiotic conditions within which a system will remain unless acted upon by external forces.
Transient state – The temporary condition or trajectory of a population as it transitions from one system state to another.

Ecosystems, Epidemics, and Economies

Yesterday I gave a talk for the “Breakfast Club” series at the Academy (California Academy of Sciences). The club is a twice weekly series of online talks started by the Academy in response to the widespread shelter-in-place and shutdown orders. It’s intended to bring a bit of our science and other activities to those interested who, like so many of us, find ourselves mostly limited these days to online interaction.

My talk focused on some new work that we are doing in the lab, related to the COVID-19 pandemic, but inspired by and built partly on our paleoecological and modelling work. I hope that you find it interesting! Oh, and while you’re there, check out the other talks in the series (link above)!

Systems Paleoecology – States, Transitions, and Extinctions

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WHAT IS ECOLOGICAL STABILITY ? In 2019 I posed this question informally to colleagues, using Twitter, a professional workshop that I lead, and a conference. Respondents on Twitter consisted mostly of ecological scientists, but the workshop included paleontologists, biologists, physicists, applied mathematicians, and an array of social scientists, including sociologists, anthropologists, economists, archaeologists, political scientists, historians and others. And this happened…

Previous posts in this series

The state of a population, as discussed to this point, is the result of intrinsic control exerted by internal variables (e.g. a life-history influenced trait such as R), the impacts of external parameters (e.g. water temperature), and often the response of internal variables to those parameters. These three factors, coupled with preservational conditions, underlie all the stratigraphic dynamics of an idealistically isolated fossil species. Even the dynamics of an isolated population will vary over time, though, because of evolutionary change and environmental variation and change. Thus the state of the population is expected to vary temporally. The states that we have so far considered have been either steady, or vary predictably with parameter changes (e.g. Fig. 1). It is now broadly recognized, however, that dynamic systems often behave or respond in non-smooth ways, where a system may transition discontinuously, and often unexpectedly, from one state to another. The surprises are twofold in nature: first, single systems may possess multiple states —multiple attractors. Second, the transitions between states are often abrupt. Such transitions bear various names that have entered into conventional ecological literature and everyday conversation, including tipping point, critical transition, and regime shift.

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.
Two populations with different intrinsic rates responding to and recovering from a sudden loss of individuals. See here for an explanation.

Discussions of multiple states generally reference communities and ecosystems, e.g. clear vs. turbid lakes, forests vs. grasslands, and coral-dominated vs. algal-dominated tropical reefs. Transitions and multiple states in such multispecies systems are facilitated by nonlinear relationships among species, enhancing and balancing (positive and negative) feedback mechanisms among demographic variables and environmental parameters, and asynchronicity (or synchronicity) of driving and response processes. Can transitions and multiple states occur in the single species population systems on which we have focused so far? Hypothetically, it is possible, but we will have to re-examine and re-think some of the simpler models of environmental shifts and responses outlined in earlier posts. When the community to which a population belongs undergoes a transition between states, it is probable that the population will also change states, but not necessarily so. A species could persist within the multiple states of a community and yet maintain a stable population size or remain within a single attractor. Shifts and responses, however, may also yield a population with distinct stable states separated by a parameter threshold, or parameter range that is much shorter than the ranges within which the population would remain stable — an abrupt transition. “Abrupt” need not refer to time only, but instead more properly refers to the relatively narrow parameter range separating different system states. The state of the population within the transitional parameter range is transient, and we can therefore describe the dynamics of the population as comprising multiple stable states, separated by transient transitional conditions. And, whereas most work in this are has focused on communities and ecosystems, there are situations where transitions can be understood within the framework of single populations. Furthermore, such transitions often have implications for the persistence or extinction of the population. Those transitions and what they imply about population growth and extinction will be the focus of the remainder of this series.

However, before digging into the dirt that I love best, I will offer a rather random assortment of readings and other resources. State transitions, particularly those occurring within complex systems, are all the rage these days. This is the area, in my opinion, where systems science truly serves as a unifying concept across multiple parts of the real world, ranging from universal to microscopic scales, and across boundaries of the physical, biological, and human worlds. I wish that I could reach behind me right now and pull my favourite books off the shelves and list them for you, but, alas, I cannot. Why? Because here in the San Francisco Bay Area my institution remains closed (with most of my library) because of the awful intersection of complex little bundles of viral proteins and nucleic acids and complex human systems, including the biological, sociological, and economic. So, if you the reader is a fellow resident of the United States, I will leave you with a polite and humble request: Please wear your damned mask. Okay, now a few resources.

Systems Paleoecology – Nonlinearity and Inequality

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WHAT IS ECOLOGICAL STABILITY ? In 2019 I posed this question informally to colleagues, using Twitter, a professional workshop that I lead, and a conference. Respondents on Twitter consisted mostly of ecological scientists, but the workshop included paleontologists, biologists, physicists, applied mathematicians, and an array of social scientists, including sociologists, anthropologists, economists, archaeologists, political scientists, historians and others. And this happened…

Previous posts in this series

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).

Fig. 1: Relationship between water temperature and population growth rate of the marine copepod genus Arcatia (Drake, 2005; Huntley and Lopez, 1992) (orange circles). The expected relationship (blue line) is exponential (fitted with an iterative least squares analysis). Red circles show expected growth rates at 15°and 25°C, while red squares show the expected growth rates for a population living at the mean temperature of 20°C (lower square), and in the range of 15-25°C.

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

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WHAT IS ECOLOGICAL STABILITY? In 2019 I posed this question informally to colleagues, using Twitter, a professional workshop that I lead, and a conference. Respondents on Twitter consisted mostly of ecological scientists, but the workshop included paleontologists, biologists, physicists, applied mathematicians, and an array of social scientists, including sociologists, anthropologists, economists, archaeologists, political scientists, historians and others. And this happened…

Previous posts in this series

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 difference between deterministic, or expected growth rate given an environmental mean (red line), versus actual growth trajectories based on variation about the same mean (using Eqs. 4 and 5).
Fig. 1: The difference between deterministic, or expected growth rate given an environmental mean (red line), versus actual growth trajectories based on variation about the same mean (using Eqs. 4 and 5).

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?

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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.

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

Are real populations ever chaotic?

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

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

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

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

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

Descent into Chaos

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

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

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).

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

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

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

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

References

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

Systems Paleoecology – Quasiperiodicity and Chaos

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WHAT IS ECOLOGICAL STABILITY? In 2019 I posed this question informally to colleagues, using Twitter, a professional workshop that I lead, and a conference. Respondents on Twitter consisted mostly of ecological scientists, but the workshop included paleontologists, biologists, physicists, applied mathematicians, and an array of social scientists, including sociologists, anthropologists, economists, archaeologists, political scientists, historians and others. And this happened…

Previous posts in this series:

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).

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.
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).

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.
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 (Eq.~ref{eq:discrete_logistic}). 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.
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

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WHAT IS ECOLOGICAL STABILITY? In 2019 I posed this question informally to colleagues, using Twitter, a professional workshop that I lead, and a conference. Respondents on Twitter consisted mostly of ecological scientists, but the workshop included paleontologists, biologists, physicists, applied mathematicians, and an array of social scientists, including sociologists, anthropologists, economists, archaeologists, political scientists, historians and others. And this happened…

Previous posts in this series:

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.
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)

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.

Transitions of a discrete logistic function with increasing $R$. Values of $R$, from upper plot to lower: 1.9, 2.0,
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.