Blog Author
Stella Uiterwaal
Citations
Volterra, V., 1926. Fluctuations in the Abundance of a Species considered Mathematically. Nature118, 558–560. https://doi.org/10.1038/118558a0
Scheuerl, T., Stelzer, C.-P., 2019. Asexual reproduction changes predator population dynamics in a life predator–prey system. Population Ecology61, 210–216. https://doi.org/10.1002/1438-390X.1017
Author background:
Vito Volterra was an Italian mathematician who worked developed new mathematical analysis methods, applying them to physics and biological mathematics. He was also in the Italian air force during World War I and fought against fascism in the 1930s – the last years of his life. In ecology, he is best known for his work on predator-prey models.
Thomas Scheuerl is an evolutionary biologist with ties to the Imperial College London in the UK and the Unniversity of Innsbruck in Austria. He works primarily with bacteria and rotifers. His work typically focuses on questions around adaptation and the benefits of sex.
Volterra
Volterra’s system has a resource and a consumer. The resource reproduces indefinitely by itself. The consumer would die by itself but can feed on the resource. As the consumer populations increases, resources are more likely to get eaten. As the resource population increases, resources are less likely to get eaten. Populations of the resource and consumer can be described by two equations, which are periodic functions of time with equal periods.
Volterra describes two types of systems: conservative and dissipative. A conservative system neglects the actions between conspecific individuals while in a dissipative system population increases are dependent on population size. He uses a three species system (resource, herbivore, predator) to illustrate these ideas. If that system is conservative, the predator cannot exist and the resource and herbivore cycle. If that system is dissipative, either the predator goes extinct and the herbivore and resource cycle, both animals die and the resource is left, or all three species coexist and cycle.
Scheuerl & Stelzer
The rotifer Brachionus calyciflorus can either reproduce sexually (here called a cyclical parthenogen (CP)) or asexually (obligate parthenogens (OP)). The two clones differ in sexual propensity and therefore possibly also differ in population growth rates, since reproduction is not limited by sexual induction in the asexual clones. The authors studied how this may affect predator-prey population dynamics at three different nutrient levels using the algae Chlamydomonas vulgarisas prey.
To do this, the authors set up chemostat populations containing rotifers, algae, and nutrients. An automated sampling system counted predator populations and algal populations were measured using a photometer. They found that asexual populations had greater variance around the mean (that is, higher amplitudes), but other parameters were similar between the two clones. They further found that differences between population dynamics of sexual and asexual rotifers was highest at low nutrient levels.
My thoughts
I enjoyed reading Volterra’s classic paper after reading many other papers that are based on his work. I also found the additional complexity of sexual versus asexual reproduction to be a thought-provoking twist to the rather basic population models presented by Volterra. Volterra’s work appeals to me because, although it is much too simplified to accurately represent real populations, it is a great way to visualize the forces that play into population fluctuations. It is also easy to add complexities (and therefore reality) to the model, such as the inclusion of sexuality.
Volterra’s paper described the population dynamics and species interactions in 2 simplified models: conservative associations and dispassive associations. The 3 different laws in the beginning of the paper that he used to describe fluctuations are reasonable, but I didn’t get what he showed in the figures. Also, the equation he used to calculate the fluctuation period right beneath figure 1 was not intuitive to me (maybe due to a lack of context in this very short paper). Nevertheless, I got what he said when I went into the examples at the end of this paper, and it was impressive to me given it was wrote about 100 years ago.
ReplyDeleteScheuerl’s paper extended the study of population dynamics into the system where species had both sexual and asexual reproduction. A question that puzzles me, and I don’t think the author explained it well, is why the asexual reproduction was higher than sexual reproduction under resource limited environment. In my understanding, some other species like Hydra spp. would have asexual reproduction (to increase the population quickly) when food is abundant and sexual reproduction (to adapt to the environment) when resource is limited.
Honestly, I found the Volterra paper a bit confusing, which is disappointing because it is the shortest paper so far. Stella's breakdown of the paper helped a lot in clearing up the main concepts. The paper included an equation which Volterra did not elaborate on. I was not clear on what variables were included. In saying this, and after reading Stella's description, it seems Volterra is 100% arguing in a logical way rather than in a mathematical way. Still, Volterra seems to point to his theory as obvious (at least this is my personal impression) when it is not. An example of this in a natural system would have been useful.
ReplyDeleteScheuerl and Stolzer's paper was a good fit with the Volterra paper. It seemed to provide a missing variable in Volterra's original paper, which was different modes of reproduction and their impact on population dynamics. I enjoyed the paper.
Although Volterra's work has faced some criticism for simplification and assumptions made, I found it interesting to read a paper that laid the groundwork for a lot of the modern work that has been done in modeling species interactions.
ReplyDeleteI found it interesting to see Scheuerl & Stelzrr apply these ideas to a system with both sexual and asexual reproduction life histories. Similar to Yuguo , I was a bit surprised and the results indicating heightened asexual reproduction under limited resources. While I do not work with this sort of system in my own research and may simply be missing something as a result, it would have been nice to get a bit more thorough explanation of this phenomenon in the discussion section.
- Elizabeth
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ReplyDeleteI was surprised based on the title of Volterra's paper that there was such little math in the actual text. Compared to previous papers like Nicholson and Cole, Volterra seemed to use more of mathematical logic to explain his reasoning rather than explicit math equations. Volterra's ideas are simplified and do not account for many variables, which makes his model easy to apply in basic predator-prey models but more complicated when interactions between multiple species in an ecosystem are considered.
DeleteI liked that the companion paper directly compared their results to the predictions from the theoretical model. The results that sexual reproduction limited maximum population densities were consistent with the model, but the experimental results differed from the model in regards to rotifer density in high nitrogen levels. Although models can be useful tools for prediction, the variations in biological systems can sometimes be difficult to predict and account for in the models that we use.
The sheer number of different ideas packed into Volterra's paper is stunning. In one paper, he manages to talk about stable and unstable equilibria, predator-prey limit cycles, human-induced trophic cascades, and biomass overcompensation. So it's probably not shocking that the paper is a bit hard to read, especially since it doesn't use any of the recognizable modern terms for these concepts. I really enjoyed unpacking this paper, and I think it's a prime example of the kind of writing we don't see anymore, both for better and for worse. The free-flowing association of diverse ideas is kind of amazing, but it's also very challenging when the author doesn't provide sufficient background.
ReplyDeleteThe Scheuerl and Stelzer paper is also fascinating, and presents a useful modern contrast to Volterra. It addresses a classically framed hypothesis in a clear, linear story line and elegantly demonstrates that underlying life history (i.e. reproductive mode) may influence the period and amplitude of population cycles. This is a fascinating extension to the classic theory of population dynamics and serves as excellent example of how to incorporate biological realism into mathematical models.
For the first, possibly in my entire life, the Volterra left me actually wanting more equations. I feel its would have added to the text and clarified some of his points.
ReplyDeleteAs for the newer paper, the complexity of sexual vs, asexual reproduction is vast. It includes consideration into things like parasite presence (the Red Queen Hypothesis), accumulating mutations (Muller's ratchet), the cost and likelihood of finding a partner, etc. So just looking into population fluctuations, though interesting, is far from complete.
-Miranda
I agree with Lyndsie that there are a ton of ideas in Volterra 1926. I hadn’t read it before and was surprised by how prescient he was. I agree with the other comments that Volterra’s paper could have included more of the mathematical analysis. Since I’ve taken classes on differential equations, some of the analysis was clear, but some parts were still opaque to me (especially the section on food chains).
ReplyDeleteI thought the Scheurl and Stelzter paper was a great companion to Volterra. The two papers are a good demonstration of how “tests” of theory have changed. In the past many “tests” of theory were done by qualitative comparison of observation data to theoretical predictions, whereas now we are able to more quantitatively confront model predictions with data. It’s not clear to me why they didn’t try to fit their data to the published model that they referred to, I wonder if that may have been helpful in figuring out why the theory and experiment had contradictory results about the effect of N addition. Also, if the differences they found between obligate and cyclical parthenogens are a direct result of reproduction mode, or if the difference in growth rate is sufficient to explain the difference.
- David
I really enjoyed the way in which Volterra’s figure and mathematical expressions accounted for the changes we might expect during predator-prey fluctuations, also (subtly) factoring in the role of between-predator competition (in addition to just prey ability) in affecting those cycles. I think he could have been a bit more explicit in describing the feedbacks that might affect the populations of both (based on his equation), but it was neat that he borrowed from Darwin’s descriptions of such feedbacks to bolster his point, and then discussed how this simple model is also applicable to host-parasite interactions. I thought that the conservative vs. dissipative types of associations he introduced at the end were interesting, but I would have liked to see more examples of how population distributions might change.
ReplyDeleteThe system in the Scheuerl and Stelzer paper was interesting to learn about – it seems to be a fairly common pair of model organisms, but I had never read about them. I liked that their experimental design was a nice, simple block design (3 treatments and 2 reproduction modes), but I found the fact that they excluded algae data (the PREY) from an analysis focused on predator-prey dynamics to be a non-trivial limitation of the study. Although they did account for the role of algae/algal growth by varying the nitrogen levels (which inherently affects algal growth) and showing the optical density of algae, I would have liked to see more discussion of the variation in algae availability for the rotifers and its affects among the treatments. As a side comment, I really disliked their format/presentation of Figures 2 and 3 (ugly and poorly constructed!).
As others already mentioned, it was annoying that Volterra never included his actual equation in the paper. If I had not already come across his work in other classes, I would have been hopelessly lost. I was also unhappy that he neglected to define the Greek letters he used and did not describe his graphs well nor define the axes clearly.
ReplyDeleteThe Scheurl and Stelzter paper also neglected to include their mathematical model in the paper, which I think all ecological papers should include. I did like how the discussion brought back the broader implications to the advantages and disadvantages of cyclical and obligate parthenogensis.