Bamboo Mathematicians

In the late 1960s, a species of bamboo called Phyllostachys bambusoides–commonly known as the Chinese Mainland Bamboo or Japanese Timber Bamboo–burst into flower. The species originated in China, was introduced to Japan, and later into the United States and other countries. And when I say it flowered, I mean it flowered everywhere. Forests of the plant burst into bloom in synchrony, despite being separated by thousands of miles. If, like me, you missed it, you will probably not live to see it happen again. The flowers released pollen into the wind, and the fertilized plants then produced seeds that fell to the ground. The magnificent bamboo plants, which can grow 72 feet tall, then all promptly died. Their seeds later sprouted and sent up new plants. The new generation is now close to fifty years old and has yet to grow a single flower. They won’t flower till about 2090.

We can say this with certainty this because Chinese scholars have kept such careful records for such a long time. In 999 A.D. they recorded a flowering of Chinese Mainland Bamboo. It was probably an astonishing sight, since no one alive at the time had ever seen the species flower before. The bamboo plants died, their seeds sprouted, and the forests did not flower again till 1114. After the species was imported to Japan, the Japanese recorded flowers in the early 1700s, and then again in 1844 to 1847. The flowering in the late 1960s was just the next burst of a 120-year cycle.

An 1885 illustration of Chusquea abietifolia, with a 32-year flowering cycle. Gray Herbarium Library, Harvard University Herbaria
An 1885 illustration of Chusquea abietifolia, with a 32-year flowering cycle. Gray Herbarium Library, Harvard University Herbaria

This remarkable cycle would be fascinating enough on its own. But it turns out a number of other species of bamboo grow flowers on cycles lasting decades, too. A species called Bambusa bambos flowers every 32 years, for example. Phyllostachys nigra f. henonis takes 60 years.

Three biologists at Harvard got puzzled by these cycles and recently set out to find an explanation for how they evolved. In the journal Ecology Letters, they offer up a tantalizing hypothesis: bamboo cycles have reached their remarkable lengths through some simple arithmetic.

Like all scientists, these biologists (Carl Veller, Martin Nowak, and Charles Davis) stand on the shoulders of giants. Or one giant in particular–the ecologist Daniel Janzen, who over the years has cast off a huge number of creative, influential ideas with unsettling ease.

In the mid-1970s Janzen came up with an explanation for why bamboo plants would flower in synchrony. He noted that rats, birds, pigs, and other animals devour colossal numbers of bamboo seeds. Each gobbled-up seed represents the loss of a potential offspring. If there are enough seed-predators, and they are hungry enough, they can wipe out a bamboo plant’s entire set of seeds.

Bamboo plants might fare better, Janzen argued, if they flowered at the same time. They would overwhelm their enemies with food. Even if they gorged themselves to bursting, they would still leave some seeds untouched. Those surviving seeds would then have enough time to grow into plants that could defend themselves with tough fibers and bitter chemicals.

Once bamboos fell into flowering lockstep, it would be hard for them to slide out. If a few bamboo plants flowered a few years too early, animals would feast on their seeds, and their out-of-sync genes would fail to make it into future generations.

Other scientists have found support for Janzen’s idea. Swamping enemies with seeds really does lower the overall harm that seed-eaters cause to each individual plant. But Veller and his colleagues still had questions. How did the bamboo plants get into those beneficial flower cycles to begin with? And how did various species end up with such long–and such different–flowering rhythms?

The scientists developed a mathematical model based on what’s known about bamboo biology. They started out with a bamboo forest in which almost all the plants flower annually, as some bamboo species do.

But the population also contained some mutants. They had mutations in their flower-timing genes, so that they needed two years to flower instead of one. Some of the two-year mutants flowered in even years, while others flowered in odd years. Spending two years between flowering instead of one could have some advantages for bamboo plants. The plants could have more time to gather more energy from sunlight, which they could use to make more seeds, or give their seeds more defenses against predators.

As more of the forest becomes two-year plants, there are fewer plants releasing their seeds every year. Eventually, Veller and his colleagues found, a year arrives when the annual bamboo plants can’t produce enough seeds to survive the onslaught of animals. In one fell swoop, they’re wiped out. If it’s an odd year, then the odd-year two-year plants can get wiped out, too. If it’s an even year, the even-yeared plants take the fall. Either way, the whole forest gets abruptly synchronized into flowering every two years.

It’s also possible that the forest wouldn’t just have two-year mutants, but mutants that took three years or more to flower. Veller and his colleagues found that in their mathematical model, bamboo plants with longer flowering cycles could also take over. Exactly which cycle won out was partly a matter of chance, because how many seeds bamboo plants successfully produce in a given year can fluctuate due to the weather and other unpredictable conditions. Whichever cycle emerges as the dominant one, the whole forest then evolves to stay synchronized. Any outliers flowering out of sync get wiped out, just as Janzen had proposed.

There is one exception, though: a mutant bamboo plant can evolve a new cycle that’s a multiple of the original one. Imagine that a two-year bamboo turns into a four-year one. Every year it flowers, it’s protected by the two-year plants flowering at the same time. And it’s got an advantage over them: it can spend the extra time making more seeds.

Even though the four-year flowers need twice as long to produce their seeds, the scientists found, under some conditions they can still become increasingly common over a few centuries. Eventually, the whole forest will lock onto the four-year cycle.

But bamboo can’t evolve the other way, the scientists found. If a four-year forest produces a two-year mutant, it will flower half the time in years when it has no protection from predators. The only direction it can go is towards longer cycles. If a four-year forest produces an eight-year mutant, it can have the same advantage that the four-year plants originally had: well-protected time.

Veller and his colleagues realized that they could test this model. Over millions of years, they reasoned, species should have multiplied their flowering cycles. It’s likely that they could only multiply the cycles by a small number rather than a big one. Shifting from a two-year cycle to a two-thousand-year cycle would require some drastic changes to a bamboo plant’s biology. Therefore, the years in a bamboo’s cycle should be the product of small numbers multiplied together.

The mathematics of bamboo offers some promising support. Phyllostachys bambusoides has a cycle of 120 years, for example, which equals 5 x 3 x 2 x 2 x 2. Phyllostachys nigra f. henonis takes 60 years, which is 5 x 3 x 2 x 2. And the 32 year cycle of Bambusa bambos equals 2 x 2 x 2 x 2 x 2.

Veller et al 2015 Ecology Letters
Veller et al 2015 Ecology Letters

The scientists found more support when they looked at how bamboo species have evolved. Here’s an evolutionary tree of Phyllostachys bambusoides and its close relatives. It’s possible that their common ancestor had a five-year cycle, and then the cycle multiplied by small factors along each branch of the tree.

But could this just be a kind of meaningless bamboo numerology? Is it just a coincidence that these species display such elegant multiplications? Veller and his colleagues carried out a statistical test on bamboo species with well-documented flowering cycles. They found that the cycles are tightly clustered around numbers that can be factored into small prime numbers. It’s a pattern that you would not expect from chance. In fact, they argue, this test offers very strong evidence for multiplication (for stat junkies: p=0.0041).

There is plenty of opportunity to put this model to the test. A lot of species of bamboo have long flowering cycles that no one has measured very carefully. Scientists could see how newly studied cycles fit into Veller’s model. If scientists find a new species of Phyllostachys that’s got a 23 year cycle, for example, it would be mathematically impossible for it to have evolved from a five-year ancestor. One thing’s for sure, though. If this model requires scientists to sit around watching bamboo, waiting for it to flower, this is going to take a few generations of scientists to settle.

26 thoughts on “Bamboo Mathematicians

  1. interesting and well reasoned

    What’s known about the mechanism that controls the flowering cycle?

    Similarly interesting is that some fauna have cycles that increase their survivability over time — but cycles that are mathematically different from those of bamboo – and cycles that arose for (only) a (slightly) different reason!

    eg: Cicada:

    They tend to have cycles that are a PRIME(!) number of years.

    That’s to keep potential predators from synchronising their cycles.
    (If they can’t do that, they can’t ~mass up on~ the cicadas – and so, wipe them out.)

    Raises the question of how cicadas manage to accomplish larger-integer cycle evolution!

    (One possibility might be larger (DNA transcription) variation: Some gestate on a yearly or small-prime cycle. They tend to be more likely to get eaten up. Whereas ones that have a longer cycle have to invest too much ~energy~ in accomplishing that to compete. (I guess those with 5, 7, 11, 13, 17, 19 -year cycles don’t have enough time (eg: for feeding as immatures) to
    do a good-enough job at reproducing.)

  2. This sounds analogous to the cicadas emerging in synchrony in a 13 yr or 17 yr cycles. The cicadas seem to have settled on prime number cycles to avoid being a predictable source of food for their predators.

  3. It’s so beautiful how this is essentially the opposite of the cicadan defense mechanism of using prime cycles.

  4. I think there would also be the matter of competition with mature plants. A bamboo plant flowering out of sync will have to compete with mature bamboo plants whereas those that flower together face little competition(because all the mature plants died at the same time). The longer cycles also means that they have more time to crowd out the shorter cycles. As bamboo groves tend to creep, they may quickly overtake the areas covered by out-of-sync plants that died before the seeds have a chance to grow.

  5. Isn’t the evolution of cicadas, with speciation into 13-year, 17-year, etc., driven for similar reasons? Interesting.

  6. Yes for cicadas, but Magicicada emerges at prime-number intervals [13 or 17 years], so no factorial increments. It’s been suggested that the prime numbers avoids parasites/predators that may key to shorter-period [say, 3, 4, or 5-year] species co-occurring, then “pump up” when the Magicicada emerge…

  7. (Not sure why my previous comment never posted, but)
    I love the contrast between the highly prime cicada life cycles and the highly composite bamboo life cycles. Why the fundamental difference when they’re both doing the same thing? I’d imagine it’s because bamboo plants flower multiple times during their lives, while cicadas only live for one emergence. Perhaps that means cicadas have a much greater advantage in making a large prime jump like 13 or 17, whereas bamboo will still have older short-cycled flowers keeping the short-cycled predators alive, making a large jump less worthwhile.

  8. I left the following comment, here, at … well, just before the following:

    “confirmed Follow” by email @ 4:02 PM yesterday (the 15th)

    The comment was up when I went on to something else – but then not up when I sent some friends the URL – then ~pending review~ when I went to check – then up there just like a perfectly good comment.

    Now it’s gone. WHat’s up?

    (URL: http://phenomena.nationalgeographic.com/2015/05/15/bamboo-mathematicians/#comment-2445953)

    interesting and well reasoned

    What’s known about the mechanism that controls the flowering cycle?

    Similarly interesting is that some fauna have cycles that increase their survivability over time — but cycles that are mathematically different from those of bamboo – and cycles that arose for (only) a (slightly) different reason!

    eg: Cicada:

    They tend to have cycles that are a PRIME(!) number of years.

    That’s to keep potential predators from synchronising their cycles.
    (If they can’t do that, they can’t ~mass up on~ the cicadas – and so, wipe them out.)

    Raises the question of how cicadas manage to accomplish larger-integer cycle evolution!

    (One possibility might be larger (DNA transcription) variation: Some gestate on a yearly or small-prime cycle. They tend to be more likely to get eaten up. Whereas ones that have a longer cycle have to invest too much ~energy~ in accomplishing that to compete. (I guess those with 5, 7, 11, 13, 17, 19 -year cycles don’t have enough time (eg: for feeding as immatures) to
    do a good-enough job at reproducing.)

    1. Sorry that your comments did not go through. No one deleted your comment; we are having a problem with some comments appearing so that the bloggers can approve them and are working to get it fixed. Best, Erika Engelhaupt

  9. Dear Mr. Zimmer:

    This is not alwaya true. Fifty some years ago, when I was in high school, as habit, inplant about two hacters bamboos that were implanted from my aunt’s farm at other town in Taiwan. Well, when I was in the college, all my bamboos were flowering and dead., However, my aunt’s bamboos didn’t flower.
    When I asked her why?
    The farmer’s answer was they don’t like your soil.
    Could not get a science answer.

  10. There is the competing effect that the longer a bamboo plant waits to seed, the less chance it will get to seed and pass on it’s genes, due to being eaten. Therefore, the drive to extend to longer cycles would be balanced against this effect by the population of predators.

  11. My husband wanted to join in the discussion a few days ago but his comment seems to have gotten lost in the ether. I thought I would try to post it. Here it is:

    Interesting and well reasoned.

    What’s known about the mechanism that controls the flowering cycle?

    Similarly interesting is that some fauna have cycles that increase their survivability over time — but cycles that are mathematically different from those of bamboo – and cycles that arose for (only) a (slightly) different reason!

    eg: Cicada:

    They tend to have cycles that are a PRIME(!) number of years.

    That’s to keep potential predators from synchronising their cycles.
    (If they can’t do that, they can’t ~mass up on~ the cicadas – and so, wipe them out.)

    Raises the question of how cicadas manage to accomplish larger-integer cycle evolution!

    (One possibility might be larger (DNA transcription) variation: Some gestate on a yearly or small-prime cycle. They tend to be more likely to get eaten up. Whereas ones that have a longer cycle have to invest too much ~energy~ in accomplishing that to compete. (I guess those with 5, 7, 11, 13, 17, 19 -year cycles don’t have enough time to do a good-enough job at reproducing.) — PMH

  12. The proposed model is quite fascinating but it does not cover all the real cases that can be found in nature. For example: it is a rather common fact that a few culms in a bamboo clump of, say, Chusquea and Guadua (two Southamerican endemic bamboo genera) will flower occasionally. In some cases, the flowering culm dies out but in others it does not. In most cases that I have seen, no seed is produced in these odd flowering episodes. Even more intriging is the fact that distinct populations of a given species have distinct flowering cycles. This fact suggests that these populations, although they belong to the same species, are genetically distinct.
    The monocarpic bamboos are all woody species (ex.: Bambusa, Phyllostachys) and the polycarpic are all herbaceous (ex.: Olyra, Raddia, Raddiella, etc.).

  13. very interesting!

    …but I’m still wondering to what extent what subsets of all bamboos truly flower at the same time

  14. releasing seeds every few decades or centuries might have another advantage because in between those flowering cycles the pests don’t have a regular supply of food therefore there wouldn’t be many pests around when the bamboo germinate- thus a higher success rate?

  15. Math in Biology is not unusual at all, I’d actually say it is the norm. All DNA alignments are done using mathematical algorithms of statistical inferences.

    I would also like to point out that all the talk about the cicadas is looking at it in the wrong direction. It is not that they are on prime numbers to avoid potential predators, it is that the ones that were not prime year cycle individuals did not avoid predators—thus leaving only more rare prime numbers. It the predation level was constant year to year and low enough, there would not be enough of a difference in the fitness between the different year cicadas to result in such a clear pattern.

    Keep in mind that there needs to be a selection of some sort to result in a directional change in the trait (# of years in cycle for cicadas). Potential predators do nothing to cause such a selection. There must be a realized predation. Put it this way, say you are extremely attracted to and would reproduce with anyone that has to blue eyes. Well, if no one has blue eyes you will never reproduce with some one with blue eyes so it doesn’t matter.

    Finally, to the person that made the comment about bamboo reproduction. If seeds are not produced at all, even just simple clonal seeds of the mother plant, the new “individuals” you see are simply shoots that are connected by stolons. So, in a way, all those shoots are a single individual.

  16. Mr. Doldrum, several:

    o Thanks for the reply to my wife’s question about math in Biology (although it was largely to probe what was going on with the comments in general).

    o Re your saying that I have is backwards: Well, it looked that way because I was using the colloquial construction, X evolved Y /in order to/ Z. That means that Y evolved in the Xs because it gave them a competitive advantage (which latter, I imagine we could both refine further down into the grass).

    That said, I would’ve probably said the same this that you did in response to the way I put it. 😉

  17. So Mr. Doldrum, the reason why a whole bamboo genus flowered at the same time is because they were connected (originally) by stolons! They were essentialy the same plant. But then the stolons get broken, plants exported, but still they flower at the same time…..? Something missing in the original paper?

  18. @PatO’C I have not bothered to look into the particulars of those species, but it is certainly possible all the seeds in the forest are identical. If the entire forest dies after going to seed and the seeds are so rapidly devoured, it seems likely that from the seeds that survive, stoliniferous growth occurs to rapidly reform the bamboo forest. This would mean that the seeds are all identical or that at least there are very few groups of unique seeds. Unique in the sense that maybe of only 1% of the the total individual bamboo plants represent genetically unique individuals and the remaining 99% are clones of the individuals in the 1%.

    If this is the case, yes, once the stolons are broken you and the plant is moved you are still transferring a living organisms with the same genetics as the parent, that has no clue it has been moved out of its forest. This seems to me to indicate an internal “clock” that is able to use some mechanism to keep track of time. Testing should be done by storing seeds of the same species and start them at different years to see if they still keep the same flowering cycle.

    I also think the mathematically impossible bit of a 32 year cycle evolving from a 5 year cycle is a bit of BS. I don’t doubt it appears mathematically impossible, but unless they fully understand the genetic and epigenetic factors leading to the flowering cycle and how random mutations could affect such a cycle I doubt that it is biologically impossible as well.

  19. It is quite interesting and we should work on it. we will see what can do about it. Probably all those who are interested may have to come together and work on it.

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