Comparing mass extinctions


The plant-eating pareiasaurs were preyed on by sabre-toothed gorgonopsians. Both groups died out during the end-Permian mass extinction, or “The Great Dying.” CREDIT: © Xiaochong Guo

Yuangeng Huang was a recent post-doctoral researcher in my lab at the California Academy of Sciences. Several years ago, while conducting his Ph.D. research at the China University of Geosciences (CUG) in Wuhan, Yuangeng trekked over to the United States and spent about six months in my lab, where he learned a great deal about the sorts of paleoecological modeling that we do. Then after completing his Ph.D., he joined me in my lab in the fall of 2019 as a post-doc. This was ideal, as part of Yuangeng’s dissertation research was on late Paleozoic and early Mesozoic terrestrial faunas from the Xinjiang region of China, which coincided with a large collaborative project in the Integrated Earth Systems program at the US NSF, of which I am a part. Yuangeng and I had great plans for the coming year, but of course 2020 was anything but a normal year. Unfortunately Yuangeng spent much of his time in the COVID-19 lockdown, along with most residents of the California Bay Area. Nevertheless, we adjusted as had so many other people around the world, and we did manage to do some very nice work, in my opinion. Yuangeng headed back to Wuhan in the fall of 2020 to take up a new position at CUG.

The first product of our time together is a recent paper published in the Royal Society Proceedings B, entitled “Ecological dynamics of terrestrial and freshwater ecosystems across three mid-Phanerozoic mass extinctions from northwest China“. In the paper we examined a span of 121 million years, comparing three mass extinctions: the end-Guadalupian event, the end Permian, and the Triassic-Jurassic transition. The end Permian mass extinction is by far the most severe recorded in the geological record. We compared the stabilities of terrestrial-aquatic communities within this span by reconstructing functional networks and food webs of the communities, and using mathematical models to subject them to disruptions of primary productivity (photosynthesis). The results of such perturbations are disruptions of populations, and eventually extinctions as the magnitude of the perturbations increases. What we found is that the two smaller events struck at times when communities were at relatively lower stability compared to the end Permian, but that recovery after the latter event was significantly more prolonged. This is yet a bit more evidence of the uniqueness of the end Permian extinction and the Permian-Triassic transition. Interestingly our results here are consistent with results for similar examinations of coeval communities from the Karoo Basin of South Africa, despite those communities being both taxonomically and ecologically different from the Xinjiang communities.

Comparing the stabilities of late Palezoic and early Mesozoic communities from northwest China (Huang et al., 2021). “Collapse threshold” is the point at which a community collapses when subjected to disruptions of primary productivity.

The paper is the result of a great collaboration, led by Yuangeng, involving researchers from China (including Yuangeng’s dissertation advisor, Zhong-Qiang Chen), the United States (including Wan Yang and myself), and the United Kingdom (Michael Benton). There are important communities from these times preserved elsewhere in the world (Russia, Brazil, Australia), giving us the potential to eventually truly understand the dynamics underlying these important events in biodiversity’s history.

And here is the link to a very nice commentary from the great science communications team at the Academy, “How Life on Land Recovered After “The Great Dying” Mass Extinction Event.”

Economic Cascades and the Costs of a Business-as-Usual Approach to COVID-19


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We have a new preprint available. This is a deviation from the usual business of paleo- and ecological communities, in that we examine cascading unemployment in socio-economic systems when subjected to outbreaks of COVID-19. It is of course inspired by our eco-evolutionary modelling, where we’ve shown that the manner in which systems are structured, particularly how species are partitioned among functions and how those functions interact are critical determinants of system stability and persistence. Below is a plain-language summary, and the preprint may be found here. The figure above shows forecasts of workers lost in the city of Fresno, California, over a 151 day interval from the beginning of an outbreak. R_0 is the infamous initial transmission rate, and we modeled scenarios of R_0 ranging from 0.9 to 6.0. The upper grey surface is the fraction of initial employment (February, 2020) lost to severe illness and death, and the lower surface is the total fraction of employment lost as the initial losses cascade through the economic systems and are amplified by inter-industry dependencies.

Plain language summary

We use a coupled epidemiological-economic model to predict the unemployment that would be incurred by major Californian socio-economic systems if outbreaks of COVID-19 were permitted to run their courses. This is a baseline against which it is important to compare contrasting approaches that prioritize either non-pharmaceutical actions intended to disrupt spread of the disease, or safeguards to uninterrupted economic activity. We find that high unemployment would be unavoidable as the effects of worker death or debilitating illness cascade through the economic network. While predicted unemployment is lower than actual unemployment during the pandemic, that benefit comes at the cost of greatly elevated mortality. The impact would also be disproportionately more severe among smaller, goods producing, and typically inland socio-
economic systems.

Erratum The figures for economic losses at the end of the third paragraph, pg. 9, should read $168.25 billion and $1.7 trillion respectively.

Systems Paleoecology – Allee Effects 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

In the previous post, we discussed the dramatic decline of the Atlantic cod (Gadus morhua) off Newfoundland over the past 60 years. I left us with the question of why, given the very limited catch sizes since the 1990’s, there was little evidence of population recovery (at least up until 2005). An Allee effect is a likely explanation for the failure of the population to recover during that extended period of reduced fishing pressure.

Beginning around 1994, the population may have become limited by an Allee phenomenon, or more appropriately mechanism, where a population’s size is limited far below the presumed carrying capacity, or observed maximum population size, because of reduced population size itself. Analogous to carrying capacity, where an upper limit is set on population growth rate by the effects of a relatively large population size, an Allee effect is an upper limit set by relatively small population size. Intuitive examples are easy to find, e.g. (1) species that require sufficient numbers for successful defense against predators will be increasingly limited by predation at low population size; (2) species for which habitat engineering by a sufficient number of individuals is necessary for offspring success; (3) species that depend on a minimum number of participants for the formation of successful mating assemblages. G. morhua, in which individual fecundity increases with age and body size (to a limit) (Fudge and Rose, 2008), is known to form, or have formed, large pelagic assemblages during spawning. Allee effects, therefore, describe situations where individual fitness depends on the presence of conspecifics, and is positively correlated with population size.

One vulnerability of populations subject to Allee effects is that small population size becomes an inescapable trap, with the likelihood of extinction increasing as population size declines. The reasons for this are twofold. First, if growth rates decline to zero or even become negative below an Allee threshold, then the state of zero population size becomes a stable state and extinction is assured. If you recall, our earlier models of population growth considered X= 0 (extinction) to be an unstable steady state; unstable because the addition of reproducing individuals to the population would result in divergence away from the zero state —population growth. Second, even if growth rate never becomes negative below the Allee threshold, a sufficiently large or sustained decline of population size increases the probability of extinction due to random events, a phenomenon termed stochastic extinction. Stochastic extinction, the probability of which could increase with deteriorating environmental conditions, is of interest to anyone studying extinction, including paleontologists, and will be discussed in a later section. Here, however, we will first explore several simple models of Allee effects.

Models of Allee effects

In the logistic model (Eq. 1 here), mortality rate increases as population size, X, approaches carrying capacity K, and population growth rate subsequently declines. The logistic model has two alternative steady states, X=K and X= 0, the latter of which is unstable as discussed above. The extinct state is a stable attractor, however, in the presence of an Allee effect. There are several simple models that demonstrate the effect, but to appreciate them, and the Allee effect itself, let us first examine the relationship between population size and growth rate under the logistic model. If we plot growth rate (dX/dt) against population size in the logistic model (Fig. 1), we see that the rate increases steadily at small population size, reaches a maximum when population size is half of the carrying capacity —X(t) =K/2— and declines steadily thereafter, reaching zero at carrying capacity. This value can be arrived at analytically because what we are visualizing is the rate of change of growth rate itself, technically the second derivative of the logistic growth equation. If we expand the logistic growth rate equation
\frac{dX}{dt} = rX\left ( 1-\frac{X}{K}\right )
\Rightarrow \frac{dX}{dt} = rX - \frac{rX^2}{K}
and take the derivative, we derive the acceleration (or deceleration) of the rate of change of population size as a function of population size itself.
\frac{d^2X}{dt^2} = r - \frac{2rX}{K}
Setting d2X/dt equal to zero —the point at which growth rate is neither accelerating nor decelerating— we get the maximum that is illustrated in Fig. 1.
\frac{d^2X}{dt^2} = r - \frac{2rX}{K} = 0
\Rightarrow X = \frac{K}{2}
The important thing to note here is that growth rate is always positive when 0<X(t)<K, that is, when population size lies between zero and the carrying capacity.

Fig. 1: The relationship between population growth rate and population size under a logistic model. In this example carrying capacity K=100.

There are several ways in which an Allee effect can be modelled in a logistically growing population. For example, if the Allee threshold is represented as a specific population size A, then the effect can be incorporated into the logistic formula as
\frac{dX}{dt} = rX\left( 1-\frac{X}{K}\right ) \left( \frac{X-A}{K}\right )
(Lewis and Kareiva, 1993; Boukal and Berec, 2002). The first term on the RHS of the equation is the logistic function, where growth declines to zero as X approaches K. The second term introduces the threshold, A, with growth rate declining if X < A, and increasing when X > A. Here, the effect is treated as the difference between population size and the threshold, taken as a fraction of carrying capacity, or maximum population size. Note that if A=0 —there is no Allee effect— the model reduces to the logistic growth model. A more nuanced model, where A must be greater than zero —an Allee effect always exists— treats the Allee threshold as equivalent yet opposite to K, representing a lower bound on growth rate (Courchamp et al., 1999).
\frac{dX}{dt} = rX\left( 1-\frac{X}{K}\right ) \left( \frac{X}{A}-1\right )
If A=1 —in which a population comprising a single individual is compromised under all circumstances— then the strength of the Allee effect depends on the size of the population. In both models, growth rate becomes negative below the threshold A, effectively dooming the population to extinction (Fig. 2). This condition is often termed a “strong” Allee effect.

Negative growth rates, a feature that is common to many models of the Allee effect, can be somewhat problematic from a conceptual viewpoint because of their determinism. We’ll pick this point up in the next post, and also discuss why paleontologists might care about both Allee effects, and model determinism.

Fig. 2: Two models of strong Allee effects illustrates as plots of population growth rate vs. population size. K=100. Red shows the first model where growth rate is relative to the Allee threshold A as a function of K. Blue shows the second model where growth rate is relative to the threshold A itself.

Allee effect — A positive correlation between individual fitness, or population growth rate, and population size. This means that fitness and/or growth rates decrease with declining population size.
Second derivative — The derivative of a function’s derivative (the first derivative), thus the acceleration (deceleration) of a rate. E.g. the first derivative of a body in motion, described by position and time, is velocity or speed. The second derivative is acceleration, or the rate at which the speed is changing.
Stochastic extinction — A relationship between the probability of a population’s extinction, and population size and/or environmental variability. In general, the risk of extinction increases due to random fluctuations of either factor.
Strong Allee effect — Population growth rate becomes negative below some threshold of population size.

Boukal, D. S. and Berec, L. (2002). Single-species models of the Allee effect: extinction boundaries, sex ratios and mate encounters. Journal of Theoretical Biology, 218(3):375–394.
Courchamp, F., Clutton-Brock, T., and Grenfell, B. (1999). Inverse density dependence and the Allee effect. Trends in Ecology & Evolution, 14(10):405–410
Fudge, S. B. and Rose, G. A. (2008). Changes in fecundity in a stressed population: Northern cod (Gadus morhua) off Newfoundland. Resiliency of gadid stocks to fishing and climate change. Alaska Sea Grant, University of Alaska Fairbanks.
Lewis, M. and Kareiva, P. (1993). Allee dynamics and the spread of invading organisms.Theoretical Population Biology, 43(2):141–158

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.

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.

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).

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).


  • 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.

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.

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.