Systems Paleoecology – Nonlinearity and Inequality

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inequality, nonlinear, nonlinearity, paleoecology, population dynamics, population growth, systems paleoecology

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

Previous posts in this series

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

6. Deviations from Equilibrium
7. r, R, and Bifurcations
8. Quasiperiodicity and Chaos
9. Chaotic Stability
10. Environmental Variation: Expectations and Averages

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)

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.

Counting biodiversity: Comparative species richnesses of coral reef communities

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biodiversity, coral reef, marine communities, nonlinear, species richness

How do you measure species richness? Our ongoing construction of model networks for Greater Antillean coral reefs relies heavily on data from the REEF project to document species occurrences on local scales. The overall picture, as reported in earlier postings on this blog, suggests that the Cayman Islands are richer in vertebrate (bony and cartilaginous fishes, and sea turtles) species than both Jamaica and Cuba. This would perhaps make some sense given an impression of lesser anthropogenic impacts, such as fishing, on Cayman reefs, as well as the number of protected areas there. The REEF database, however, also documents the number of times that a specific locality has been sampled, and a casual browse indicated that the Caymans have been sampled far more heavily than either Jamaica or Cuba. We therefore wanted to account for the effect of sampling effort.

A simple plot (see figure) of the number of species discovered per sampling event shows that indeed the Caymans (blue) have been sampled more heavily than Jamaica (red), and that the number of species (species richness; one measure of biodiversity) is greater in the Caymans. But what if Jamaica was sampled as much as the Caymans? There is no definitive way to answer this, of course, without actually increasing sampling effort in Jamaica. We can, however, ask if our conclusion of greater diversity in the Caymans is a fair one. The solid lines plotted through the data are nonlinear regression functions of the form

The pale-coloured ribbons behind each line represent 95% bootstrap confidence intervals. It should be fairly obvious that if sampling effort was equal between the two systems, that Jamaica would be at least as diverse as the Cayman Islands! I can’t explain why Jamaica is under-sampled relative to the Caymans without becoming speculative about socio-political matters and tourist tastes, so I’ll leave that up to others.

I did not include Cuba in this analysis for several reasons. First, Cuba is very clearly under-sampled when compared to the other two systems, and our data do not include data from important near-pristine areas such as Los Jardines de Reina. Second, Cuba is so much larger in area (coastal) than either other system, that perhaps some consideration of spatial coverage would be needed. And third, it is entirely possible that there are compositional differences, at least among invertebrates, between the northern and southern coasts of Cuba (see Roopnarine and Vermeij, 2000).

The simple message here is that mere counts of species within an area are insufficient measures of biodiversity. This has long been known among ecologists and has led to very interesting work on measuring biodiversity, the spatial distributions of species, and the composition of biological communities. But it’s a message worth repeating every now and then.

New paper – Networks, extinction and paleocommunity food webs

Tags

connectance, extinction, food webs, graph, link distribution, metanetwork, Network theory, networks, nonlinear, paleo-food web, power law, probability, real world networks, Robustness, simulations, trophic guild

Roopnarine, P. D. 2010. Networks, extinction and paleocommunity food webs in J. Alroy and G. Hunt, eds., Quantitative Methods in Paleobiology, The Paleontological Society Papers, 16: 143-161. (available here).

The paper is part of a volume, Quantitative Methods in Paleobiology, sponsored by The Paleontological Society. Full details are available here. The volume is also available for sale. Purchase one and support the Society!

New paper: Ecological modeling of paleocommunity food webs

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cascades, competition, connectance, edge strength, extinction, food webs, interaction strength, link strength, modeling, Network theory, networks, nonlinear, paleontology, power law, probability, real world networks, Robustness, Scientific models, simulations, small world networks, Tipping point, top-down cascade

Roopnarine, P. D. 2009. Ecological modeling of paleocommunity food webs. in G. Dietl and K. Flessa, eds., Conservation Paleobiology, The Paleontological Society Papers, 15: 195-220.

Find the paper here:
http://zeus.calacademy.org/roopnarine/Selected_Publications/Roopnarine_09.pdf
or here
http://zeus.calacademy.org/publications/

Prey dynamics

Tags

extinction, Network theory, networks, nonlinear, quasi-periodic, simulations, Tipping point

The figure here is very similar to the one in the previous post, but these results are for the guild of shallow infaunal suspension feeders (primarily clams). The main difference is the more regular increase in the number of species that become extinct as the perturbation magnitude () increases. Another interesting note is that this guild is not the only driver, or any driver at all, of the behaviours exhibited by the guild of predators. Those predators may or may not prey on members of this guild, and also have an array of prey in other guilds. So the oscillatory behaviour seen at higher perturbation levels is probably system-wide. And it is system-wide because of indirect effects via network links. One wonders what a summary of the results would look like, and what the implications are for individual species population dynamics.

1. For example, even at a very low perturbation level, maximum sustainable population sizes oscillate wildly before settling down to a new stable state (which can in fact be the initial one, or zero, indicating extinction). One would assume that population sizes would follow this trend, if the timescales of the perturbation and population growth were sufficiently close. What if they are not? How does this affect what one would actually observe for a given species?
2. What is the distribution of stable states over the perturbation range? Are the oscillations observed at high perturbation level convergent, i.e. if run long enough they would also settle to a new stable state? Or are they asymptotic, but never settle down, or settle to two alternative states? One way to find out would be to simply run the series for many additional steps. Another would be model the oscillations themselves, and see if the convergence is linear or asymptotic. And what is the perturbation range of the bifurcations? At what point do we begin to observe oscillation/bifurcation, and is it synchronous throughout the community? Only one way to find out, but I’ll probably have to write some Sed/Awk or Perl scripts to handle these large datafiles.

Tipping point II

Tags

cascades, extinction, food webs, Network theory, networks, nonlinear, simulations, Tipping point, top-down cascade

Red ellipses represent the two threshold points.

The story so far: We have a food web of a shallow coastal marine community from the Late Miocene of the Dominican republic. The metanetwork comprises 29 guilds, 139 guild-level links, and 130 species. A perturbation of the system, where all three primary producer guilds plus detritus were systematically and incrementally removed from derived species-level networks, results in the typical CEG result: that is, a relatively flat and low level of secondary extinction () over a broad range of perturbation magnitude (), succeeded by a rapid transition to a state of high secondary extinction. In fact, for this community, there are two transitions. The first occurs at , and represents a very minor but secular increase in . The second transition occurs at and represents a catastrophic increase in . Topological-only perturbation of the system makes it very clear that these transitions correspond exactly to two stages of the perturbation: First, the complete extinction or removal of benthic autotrophs and complete disruption of the particulate detritus supply. The second and greater transition occurs at the complete extinction of the benthic macroalgae and macrophytes. Accompanying the second transition is the complete extinction of the benthic herbivore guilds which specialize on the macroalgae and macrophytes (and derived detritus), comprising families such as the Phasianellidae, Cerithiidae, Vitrinellidae, Haminoeidae and Retusidae. This is accompanied by extinction of species in other more generalist guilds that include macroalgae in their diet.

Extinction of those heterotrophic taxa is not itself the cause of the major tipping point though. Simulations where the perturbation is specifically removal of these herbivores result in very low levels of secondary extinction, with no tipping point or threshold. The obvious question then is, why does extinction of the macroalgae drive the system to a new state? The qualitative answer is that the complete loss of this resource, and the bottom-up propagation to the herbivores, in turn cause intense top-down cascades of compensatory responses from higher level consumers. These cascades propagate throughout the network, even to the remaining source of production, the phytoplankton. The result is a tremendous loss of species. A very curious thing, however, is that phytoplankton productivity in the network is almost 3 times greater than macroalgal productivity, reflecting the much greater diversity of planktivores. So why does the collapse coincide with loss of the macroalgae?

Perturbation of macroalgae (top) and phytoplankton (bottom), and resulting loss of autotrophic productivity because of top-down effects (left), and secondary extinction of heterotrophic species (right).

I performed two separate perturbations to answer this question. First, I perturbed the system by removing macroalgae only, and second by removing phytoplankton only. The top row of the second figure shows the results of the first experiment. Secondary loss of autotrophic resources (left column) as a result of top-down effects is effectively zero. Secondary extinction of heterotrophs (right column) is significant but not dramatic. There is a mild increase in the region of , which represents the loss of the specialized herbivore guilds. Removing phytoplankton had a more dramatic impact, reflecting the greater overall dependence of the community on phyloplankton resources. There is a clear threshold, occurring at approximately . At this point, resource loss to the community is great enough to trigger the catastrophic top-down cascades and feedback within the network. Therefore, it seems that in the previous experiment, where all resource guilds were perturbed, the complete loss of macroalgae triggers the top-down cascades and compensatory feedback that in turn deplete phytoplankton resources to the point where the system transitions to a higher state of secondary extinction. This conclusion is supported by the fact that when all producer guilds are perturbed, the contribution or perturbation of phytoplankton at the tipping point is 38%, whereas when only phytoplankton are perturbedm the tipping point occurs at 50%.

Some closing observations:

1. Topological analyses of network vulnerabilities are likely to underestimate the severity of link losses when those links have variable interaction strengths, and the nodes have varying properties. In the case of a biological community, species could and are likely to alter interaction strengths to compensate for lost resources (i.e. links). Topological vulnerability analyses should be well suited for networks with static properties, perhaps such as power grids and the internet (though I’m no expert here!), but are ill-suited for dynamic networks, such as those describing transportation, metabolic/physiologic and ecologic systems.
2. An hierarchically structured, directed network such as an ecological community should be resistant to a broad array of random perturbations. This is a function of both the underlying link distributions (as already understood in the case of static networks or graphs), as well as the compensatory abilities of consumer species, and the variance of dietary breadth. The network is, however, vulnerable to the loss of highly linked nodes. Here I am referring specifically to basal, autotrophic nodes, and not necessarily keystone consumer species. Not all autotrophic nodes are equal, however, as shown in the above results. Nevertheless, because of the complexity of the species interactions and the hierarchical divisions of ecological functions, there should be strong nonlinearities in the network responses. This is borne out by the differences between the topological-only and fully dynamic simulation results. The nonlinearities are expressed as two or more alternative states of secondary extinction, separated by rather sharply defined thresholds of perturbation. I can think of no way in which to analytically predict the threshold points, but heuristically I would argue that they should exist in every ecological community.
3. Perturbation of top-level consumers are observed in nature to often result in top-down cascading effects, compatible with such notions as keystone predators. I will show in later results that the CEG model captures all this. The results will also show, however, that while top-down effects can be locally catastrophic, i.e. for individual species or groups of closely linked species, they are never globally catastrophic in the manner in which bottom-up perturbations are. This conclusion has implications for understanding the role of ecological collapse in large scale extinctions observed in the fossil record. It also has implications for the ongoing biodiversity crisis, where species far removed from the “tops” of food webs are increasingly threatened by climate change and habitat destruction.
4. An close examination of many of the results presented in this blog will show apparent “bifurcation” of the results, e.g. beyond the threshold point in the lower right graph above. These observations suggest that there is more than one type of species-level network that can be derived from the same metanetwork. So, while the higher-level organization of the community is the same, networks are being generated that vary enough in their interspecific link topologies to yield very different responses to the same level of perturbation. I believe that this is a statistical property of the underlying trophic link distributions and the resulting multinomial probabilities from which the species-level networks are drawn stochastically. In the case of the above results, where one set of networks is significantly more resistant than the other (i.e. they have a much higher tipping point), this mathematical feature of the model is not likely to be of great relevance ecologically. That is because the lower threshold is already so high, in this case, 50% shutdown of primary productivity. Those are catastrophic environmental conditions and would occur with very low frequency in nature on a large scale. There are cases, however, such as the Early Triassic Lystrosaurus zone community, where there seem to be multiple alternative states at very low perturbation levels. Those communities would very likely have experienced frequent low-level perturbations, and then one has to consider whether: (1) this feature of the model is a mathematical artifact, in which case one wonders about the constraints necessary to prohibit it in nature, or (2) the feature is real, and then one wonders how species within a community cope with such a situation.

Three paths to network collapse

Tags

extinction, networks, nonlinear, simulations

There are three responses to bottom-up perturbation that have been observed in both the models and simulations of real communities. First, there is the “typical” CEG response (described in an earlier post). The low levels of secondary extinction at low levels of perturbation are basically as expected and predicted theoretically by the topological collapse of the network as the perturbation propagates through the network.

The second response is also part of the typical CEG response, and is the nearly discontinuous and significant increase in secondary extinction, implicating a threshold response. It isn’t clear at this point exactly what precipitates the threshold change, that is, is it the loss of highly linked species? Or is it threshold responses in compensatory increases of link strength? Or something else? But, at the very least I know that it does not involve topological effects, because the threshold is neither predicted mathematically, nor is it present in the results of topological extinction-only simulations. It is therefore definitely a feature of the demographic (Lotka-Volterra-type) interspecific interactions.

Finally, there is the occasionally observed high level of secondary extinction at low levels of perturbation. Surprisingly, I think that this is not a demographic effect, as we’ve long hypothesized. It is not likely to be, since it occurs in the topological extinction simulations of the 3-guild model! Those results point to the presence of at least two distinctly different topological classes of species-level networks. One class responds according to the mathematical predictions, and the other exhibits the anomalous response. I believe that this response is essentially a result of constructing networks that have very low linkages in the critical places. The probability of this occurring must lie in the multinomial probabilities of species linkages based on guild trophic link distributions, and the uncertainty of these probabilities, or the entropy of the metanetwork, increases under certain combinations of guild species richnesses.

Nonlinear cascades

Tags

cascades, connectance, extinction, food webs, networks, nonlinear, simulations

All the simulations described so far are of bottom-up perturbations to the basal level producer guilds. Network theory predicts that given networks such as food webs, with power law-like link distributions, the networks should be robust against random removal (extinction) of nodes, while being highly vulnerable to the removal (perhaps targeted) of highly linked (hub) nodes. This of course is a topological prediction, since it in no way incorporates dynamics of link strengths, compensatory modification of link strengths, or extinction thresholds (e.g. those Allee effects). Then, why do the CEG predictions and simulations of topological effects have a gradually, mildly exponential, rate of increase of secondary extinctions as the number of nodes removed is increased? The answer is two-part:

1. The probability of link loss increases with the in-degree of the consumer. Therefore most of the links being lost at any given level of perturbation are lost by highly linked species. But those species are also the most resistant to extinction.
2. The probability of secondary extinction increases almost linearly for consumers of very low degree, but almost not at all for the most highly linked species, until levels of perturbation are very high. Therefore most of the extinctions that occur at low to mid- levels of perturbation are of poorly connected species.

The nonlinear increase seen in the CEG simulations is therefore likely a response to a threshold being reached where highly linked consumers, though still robust to topological extinction, initiate significantly devastating top-down cascades because of compensatory increases of link or interaction strengths.

It is also therefore reasonable to hypothesize that very high levels of secondary extinction at low perturbation levels is the result of having a few highly connected, upper-level consumers. This could explain the great difference, at those perturbation levels, between the topological expectations and the simulations. Should low diversity communities, or communities with low diversities of high trophic level consumers, then be limited to those consumers being very specialized?

First topological extinction results

Tags

extinction, food webs, networks, nonlinear, paleontology, simulations

Program topo_CEG needed a bit of re-writing. The adjacency matrices generated from real communities are very large, due to high species richnesses. The matrices are so large that they cannot be initialized as simple arrays in C++, at least not on the stack. Had to use the Boost MultiArray function.

Comparison of full CEG (red) and topological-only results.

I ran 10 simulations of the Dicynodon Assemblage Zone (DAZ) community. Topological secondary extinction increases slowly, and then somewhat exponentially, as a result of increasing bottom-up perturbation. This is very encouraging in that the results are similar to the analytical results that can be obtained by using the combinatoric version of the model presented in Roopnarine (2006); results of application of this model to the DAZ were presented in Roopnarine et al. (2007). The main difference is that the simulations capture the effect of the stochasticity of the perturbation vector. Now, if we compare these results to those obtained with the full CEG simulation model applied to the DAZ, there are two obvious differences:

1. First, the full model yields higher levels of secondary extinction.
2. Second, the full model yields the “typical” CEG result, which means that there is, at some level of perturbation, a rapid increase (threshold) in the level of secondary extinction.

Therefore, topological extinction cannot account for the CEG results.

What else is there? The obvious missing feature are the top-down cascades that are initiated as a result of compensation for lost links/resources. And there is also link strength variance. These two features apparently generate a lot of the nonlinearity of the model. Exactly how much can be measured by basically subtracting the topological results from the full model results. This will require combining the full simulation program and topo-CEG. Going to need a bit of parallelization here!