A cautionary tale: leaf traits expressed on a mass basis

[note: I'll get to leaves in a bit, but first...] I remember as a grad student learning about the “density-dependence debate” and the somewhat amazing fact that a series of random positive numbers (which may represent a series of random population sizes) will show (arguably misleading) evidence of density-dependence. Take, for example, this randomly generated population series:

If I calculate the population rate of change as r = (population(t+1) - population(t))/(population(t)) and plot it against population size, I get:

Smaller populations tend to increase and larger populations tend to decrease. But isn’t this misleading? The population sizes were generated randomly from one generation to the next! There’s zero correlation between them!

The reason this happens can be viewed two ways: First, mathematically, the denominator of the y-axis variable is the x-axis variable (population(t)). Although it seemed like a sensible way to calculate and plot these quantities, random variation in population(t) will naturally make large x-axis values small y-axis values and vice versa, i.e. if x is big, then c/x is small and vice versa. Of course the same sort of thing can manifest in other ways whenever the axes of a plot contain functions of the same random variables. And it’s not just plots… if I had calculated a statistic or an additional variable based on the above plot, I’d be in the same boat. The upshot is that one must exercise extreme caution when relating quantities that are calculated using some of the same random variables.

The second explanation takes a more philosophical turn. The assertion is that a population that takes on a random size from generation to generation about some mean (e.g. 1500 in the plot above) actually is density-dependent because, on average, larger populations will be followed by smaller populations and vice versa. If instead I let the population randomly walk, where each generation’s population size is increased or decreased by some random amount relative to the previous generation, then large populations are not on average followed by small populations or vice versa:

and the supposed signature of density-dependence goes away

despite the fact that I’m still dividing the y-axis variable by the x-axis variable!

So what does this have to do with leaf traits? A lot, it turns out. The canonical leaf economics spectrum paper, Wright et al. 2004, includes plots and correlations of various leaf traits on mass- and area-bases. The leaf economics spectrum is a hugely influential framework for understanding leaf traits, and it’s fair to say that people all over the globe have reorganized their thinking and their research agendas based on it. How many talks have I sat in at ESA that made at least a passing reference to the leaf economics spectrum? Umpteen, at least. Likely more.

However, as recent papers by Osnas et al. (2013) and Lloyd et al. (2013) point out, a lot of the apparently strong mass-based correlations may actually reflect tremendous worldwide variation in leaf mass per area… and little more. Both groups of authors make a much more nuanced argument than I will make here, and you should read their papers! I may be biased, coming from the same lab, but I think the Osnas et al. (2013) contribution moves the ball forward to a greater extent by articulating the extent to which various leaf traits are mass- or area-related without falling into “statistical quicksand” (Lloyd et al.’s term for my bolded statement above).

Those papers are where it’s at, but I’ll add a nice blog-friendly contribution to the conversation… it’s a little exercise you (or your students) can perform in the comfort and safety of your own office, lab, or classroom. It’s easy and vividly illuminates the issue.

First, get your hands on the Glopnet dataset used in Wright et al. (2004) and Osnas et al. (2013). Wright et al. wisely made it public. You can download it here. Using your favorite statistics program, plot Log(Amass) against Log(Nmass). Amass is the maximum leaf level photosynthetic rate per leaf mass and Nmass is the leaf nitrogen content per leaf mass. Your graph should look something like this:

Sweet correlation, right? Let’s also graph Log(Aarea) against Log(Narea), which are equivalent measures on a leaf area basis:

Not so sweet a correlation. My natural (and as you can anticipate, naive) reaction to the above graphs is that there is a real relationship between A and N on a mass basis but not on an area basis. However, and this gets back to the population rate of change calculation above, NmassNarea/LMA and AmassAarea/LMA, where LMA is the leaf mass per area. So the mass-basis graph divides both axes by LMA. All else equal, large values of LMA will lead to small values of Nmass and Amass, and small values of LMA will lead to large values of Nmass and Amass.

The proper way to demonstrate that most of the correlation in the mass-based graphs is due to variation in LMA is via the statistical machinery of Osnas et al. (2013) and Lloyd et al. (2013). But you can get a gut feel for it by fabricating fake mass-based data in Glopnet. Simply scramble the LMA column and recalculate mass-based values using the real Narea and Aarea values (which are still truly linked) together with the scrambled LMA data. Here’s the recipe:

  1. “Un-Log” the “log Aarea” column: Aarea = 10^(log Aarea).
  2. “Un-Log” the “log Narea” column: Narea = 10^(log Narea).
  3. “Un-Log” the “log LMA” column: LMA = 10^(log LMA).
  4. Create a new column, “LMA scrambled” that scrambles the LMA data (i.e. take each  LMA value and randomly associate it with a different row) – it’s up to you to figure out how to do this, but one possibility is to import the column into Microsoft Excel, create another column next to it with random numbers (use the formula “=Rand()”), and then sort both columns by the random number column. Bam! The LMA values have been scrambled!
  5. Create fake Amass data: log fake Amass = Log(Aarea/LMA scrambled).
  6. Create fake Nmass data: log fake Nmass = Log(Narea/LMA scrambled).
  7. Plot the fake data and compare it to the real data!

Sweet correlation, right? But this fake correlation is driven not by the relationship between photosynthesis and nitrogen, but by the the random variation in LMA, which divides both axes. The implication is that the correlation in the real data is similarly driven by random variation in LMA.

Now, there is a lot to be said here, and Westoby et al. (2013) have issued a rejoinder that deserves your careful consideration. With their improved method, Osnas et al. (2013) make the case that some leaf traits (e.g. nitrogen and dark respiration) are to some extent mass-based and to some extent area-based, whereas others (e.g. phosphorus and maximum net photosynthetic rate) are almost entirely area-based. Upshot, there’s a lot of important nuance here that needs to be understood by serious ecophysiologists. The leaf economics spectrum – far from dead – is being improved and refined. Science moves forward!



Game theory and plant ecology

Gord McNickle and I are pleased to announce that our review/idea paper on game theory and plant ecology is now officially published in Ecology Letters. I’m proud of the whole thing, start to finish, but if you read only one section, start with “Game theory essentials for the plant ecologist.” It may just change your life! …or at least your perspective on plant ecology:)

Elevated CO2 as steroids, fine roots as body builders

A lovely paper was published today! Authored by my good friend and colleague Caroline Farrior (with some help from me and a couple of literal wise guys), it presents a tractable model of competitive trees under conditions of water limitation. Although we use some different parameter names and combinations, it is really the sibling of our earlier paper that looked at competitive trees under conditions of nitrogen limitation. It’s chock full of savory ecological goodness (dig in!), but I’ll highlight only the most surprising result here.

Briefly, elevated CO2 improves plants’ leaf-level photosynthetic efficiency via two mechanisms. First, elevated CO2 increases the photosynthetic metabolism involved with carbon (which the plant turns into food) and decreases the photosynthetic metabolism involved with oxogen (which is a wasteful process and an evolutionary artifact that looks like a colossal mistake under current conditions). Second, elevated CO2 increases the net photosynthetic rate when a plant is water limited because more CO2 diffuses into the leaf for a given amount of water lost from the leaf. I’ll only focus on the second mechanism here (i.e. everything that follows pertains only to the second mechanism!).

You might expect elevated CO2 to allow plants to store greater amounts of carbon under water limited conditions because of that enhanced water-limited photosynthesis, and you’d be in good company. However, our results suggest that competitive plants will divert all of the net increase in photosynthate to fine roots. Of course, the plants could maintain competitive parity by keeping their original fine root investment and putting the additional photosynthate elsewhere. Similarly, a group of body builders could maintain competitive parity by agreeing not to take an undetectable steroid. In both cases however, “cheaters” would be at a such huge advantage that the only rational alternative is for everyone to take the steroids/ build up a greater fine root armament. But of course, after doing so, everyone is back where they started, at competitive parity! It’s a tragedy of the commons.

Because fine roots store carbon over vastly shorter timescales than wood, the net effect of elevated CO2 under water-limited conditions is a negligible change in carbon storage. While this is perhaps sad news for the fate of our changing climate, we have other papers in the works that suggest that in many other important scenarios (other than water limitation), competitive plants should respond to elevated CO2 in ways that tend to reduce atmospheric CO2.


The race is never over!

I’m not generally a grumpy guy, and years ago I made peace with the popular misuse of the phrase “begs the question.” (See? I’m easy-going, through and through!) But when I read Robert Krulwich’s NPR blog entry “New Superhero, 3,200 Years Old, Turns Air Into Wood Superfast : Krulwich Wonders… : NPR” I blew up in a self-righeous huff. As I endeavor to impress upon my audiences whenever I give a public talk, our default caricature of how forests work leads otherwise good scientists to make bad assumptions. Krulwich summarized that bad caricature nicely:

“Young trees, we thought, suck in great gulps of CO2 and then, with a series of chemical tricks, turn that air into wood, adding bulk to the trunk, thickening the branches, spitting out the oxygen, and they keep at it for years, fighting for sunshine, until, if they’re lucky, they end up taller than their neighbors, so at last they can relax, and eventually, slow their growth rate, weaken and die. That’s what we’d call a normal tree cycle. But giant sequoias, apparently, do it differently.”

Yes, giant sequoias do it differently… just like every other tree. The race is never over, and a canopy tree continues to jockey for position, gaining or losing space to its (adversarial) neighbors until the day disease or wind finally ends its life. Height growth is always paramount to a forest tree, no matter its stage in life. They never “relax.”

How did the caricature of a static canopy tree come to be? I think there are three ingredients. First, we have ground-level familiarity with myriad herbaceous species that do have a somewhat fixed maximum height. Second, we’re short and so we can’t have the same first-hand knowledge of the goings on in the canopy. Third, a tree’s height growth scales as the square root of its diameter, which means that as trees grow very large, their height growth – though still paramount – becomes harder and harder to perceive. Ancillary to the third point is the observation that disturbance regimes are somewhat regular for any given eco-region, which tends to maintain a characteristic canopy height across a landscape as a balance between young and old forests.



Time-lapse of a very distant storm

In summer 2011, we enjoyed a wicked storm several times a week and a brilliant sunset almost every evening. I kept wishing I had the means to make nice time-lapse movies. That wish came true last winter, but this past summer was remarkably devoid of sky-shattering storms and stunning sunsets:(

I did manage to get one excellent movie of a very distant storm (~150 miles away) while I was in northern Wisconsin. Ironically, I only had the camera and none of my time-lapse gear… so it’s a little shaky. (I like to think it adds character:) It’s worth realizing that the thunderhead I captured is several miles high…impressive stuff.


Counterfactual adaptation: branch razors

Ah, the first counterfactual adaptation: branch razors. Why shouldn’t the most distal branches on a tree have little razor-sharp appendages out in front that would tear up neighboring trees’ leaves and branches? This seems like a more direct route to maintaining a light-harvesting territory than the one employed by extant trees: ceaseless height growth and overtopping…

Counterfactual adaptations

Allow me to introduce a new “category” for my posts: Counterfactual adaptations. In order to describe what a “counterfactual adaptation” is, I first have some give you some preamble. Roughgarden, May, and Levin wrote in their introduction to Perspectives in Ecological Theory:

Whether or not in mathematical form, theoretical ideas serve many purposes. They suggest observational protocols and manipulative experiments both in the field and in the laboratory. They provide frameworks around which curricula can be organized, so that the body of facts can be given coherence rather than presented as a jumble. Most important, perhaps, theory can imagine and explore a wider range of worlds than the unique one we inhabit, and by so doing can lead to fresh perceptions and new questions about why our actual world came to be as it is…

I love this quote, and as much as I appreciate ecological theory’s ability to explain why the world is the way it is, I appreciate even more its ability to explain why the world isn’t some other way. When I stare at the analytical results of a mathematical model, there is often parameter space that bears no resemblance to the unique world that we inhabit. This is part of the “wider range of worlds” of which Roughgarden, May, and Levin wrote. Commonly, consideration of parameter values or of other results from the model suggest why that parameter space is biologically unrealistic. These insights are deeply satisfying to me.

In my day to day musings, I sometimes imagine an adaptation that seems like a “good idea,”* but that doesn’t exist in our world. It is a counterfactual adaptation and is part of the “wider range of worlds than the unique one we inhabit.” Perhaps there is no evolutionary path from our current biota to a particular counterfactual adaptation. But the more exiting possibility is that there is a good, but non-obvious reason why a particular counterfactual adaptation would in fact be a “bad idea.”* Relevant theory should permit the counterfactual adaptation, but then explain why it is biologically unrealistic.

My plan is to collect my counterfactual adaptations here. I’d love to hear yours!

*yes, yes… adaptations are not “ideas”:)

The next billion years

A few months ago, I placed this geologic timeline on my desktop. I wanted to see it again and again and again, so that the majesty of its arc would sink into the pit of my brain.

I think the mission has been successful, and now I’m excited to create my own version of a geologic timeline. In addition to the information contained here, I’d like to include graphs that depict changes in the atmosphere, percentage of land, ice cover, etc. Look for it as soon as I can catch my breath from all these job applications and revisions:)

In addition to adding new information, I’ll make mine linear. There’s something strange about the circular depiction above, with the zero hour and now both sitting at twelve o’clock… it doesn’t provide any room to imagine the future. I had been planning to leave about five billion years of blank space (at which point my high school education had taught me the sun will go red giant), but then I learned from Wikipedia (that’s right) that the sun is expected to heat up within the next one billion years such that no liquid water will exist on Earth.

As adaptable as our flavor of carbon-based life seems, I’m just not optimistic that it can persist without liquid water. So it seems that life on Earth is in the autumn of its life, so to speak. Like an island that rises from the sea, hosts eons of riotous life, and then abandons that life as it sinks back into the sea, so too is life’s finite flourishing on island Earth.

Addressing the comments from a rejected paper

A little while back, I reviewed a paper that had a fundamental flaw. I won’t go into details, but suffice it to say that the flaw was incontrovertible; the claim they were trying to make did not follow from their results. I detailed this in my review and even suggested some ways that they might correct the mistake …or I suggested that they could at least qualify their claim. Based on my review and another not-so-glowing review, the paper was rejected.

Now, after all that effort, I am dismayed to discover that the authors simply resubmitted the exact same manuscript [EXACT SAME!!!] to another journal and that a random throw of the reviewer dice evidently turned up a pair of “accepts” this time. The paper is now published, fundamental flaw and all.

And the irony is that I discovered this as I was working hard to address reviewer comments  from my own rejected paper before I resubmit elsewhere! Yeeeeesh……

The tremendous intermittency of selection pressure on wood construction: an object lesson for the temporally-challenged ecologist?

Think of the most wicked storm you experienced this past summer… think about the power and fury of the gust front that announced that storm. For me, it was a storm a few weeks ago where the winds in Chicago where I live were knocking everything around with a violence that truly scared me as I walked home. The previous summer a tree had gone down a block away under similar circumstances, stopping a traveling minivan in its tracks (not the minivan on the left, but the one hiding on the right). Miraculously, its massive branches fell just in front and just behind the vehicle… a moment sooner or later and the driver would have surely been killed. But I digress…


As awe-inspiring as it is to witness a massive tree felled by the wind, it is perhaps more awe-inspiring that most trees stand strong against such tremendous forces! Thus, it is undoubtedly true that at any given moment, a tree’s investment in wood far (far!) outstrips the demands placed upon it by gravity and the gentle breezes that prevail most of the time. Here, the relevant selection pressure for wood construction and investment occurs with great infrequency.

So here’s a thought experiment: Imagine for a moment that humans only live for a few days, but that we still possess technological innovation, cultural evolution, and scientific curiosity (i.e. suspend your disbelieve and imagine that everything else about humans is the same:). How might the perspective of an ecologist studying the ecophysiology of wood in her few short days differ from ours? I suggest that although she knows that “wind events” happen and that they often kill trees, she is unlikely to ever experience one of these events in her short lifetime and that their importance will take a back seat to the sorts of things that she can measure and experience. I suggest that she may either be perplexed by an apparent over-investment in wood or, worse, she may convince herself of other reasons why that level of investment “makes sense.”

You see where I’m going with this… As humans who spend at best 100 years on this planet, we are unlikely to experience the sorts of cataclysms (multi-decadal droughts, glaciation, epidemics, periods of volcanism, etc.) that we know occur with some regularity, geologically-speaking, and that exert profound selection pressure on organisms. Might this cause ecologists to be perplexed by the phenomena they study or, worse, convince themselves of other reasons why those phenomena “make sense?”