New Publication in Nature: Long-term cyclic persistence in an experimental predator-prey system.

We just published a new study in the journal Nature (see also the favourable coverage in a Nature News and Views by Alan Hastings).

The story behind the paper

The story goes back to my time as a junior research group leader at the University of Potsdam. At this time I had just received a scholarship from German VW-Stiftung to investigate synchronization in ecological systems, with one major objective to explore the possibility of syncing experimental predator-prey cycles in the laboratory. For this we collaborated with the lab of Ursula Gaedke in Potsdam (just 10 min walk from my former office in the Institute of Physics) where we build-up a chemostat system consisting of microalgae as prey and rotifers as predator. We were lucky that at this time Gregor Fussmann, a proven expert in experimental predator-prey cycles, was joining Ursula's group - so with Ursula, Gregor, Guntram Weithoff (another member of Ursula's group), and Lars Rudolf (my PhD student working in this project) we just had the critical mass of theoreticians and experimentalist in one place (as well as the enthusiasm) to tackle this project - but we soon found out that this endeavour was much more challenging than expected.

As it turned out, our first co-cultures of predator and prey organisms didn't exhibit the expected regular population cycles. Instead, population densities were fluctuating rather irregularly in time. Which meant that we had to set back our original objective of connecting multiple oscillating chemostats systems in intricate coupling topologies. Instead, we focused on the more modest goal to realize at least a single-patch predator-prey cycle. Being the only pure theoretician in the team, here I had to rely on the lucky hand (and the experience) of my team members to optimize the experimental conditions. We systematically varied all kinds of experimental parameters, such as the identity of the algal species, abiotic conditions, and measurement devices (e.g., by establishing a non-invasive instrument to obtain algal densities from optical measurement of the medium turbidity). To better understand our organisms we even reduced our experiments to single-species algal systems, which turned out to have surprisingly complex dynamics on their own - leading to some interesting ‘side-product’ studies (PNAS 2010, Nature Comm. 2015).

A world-record time series

When we turned back to our predator-prey experiment, which had been going on all the time, we had a pleasant surprise: In our undisturbed experimental system the predator and prey species had adjusted on a regular population cycle, cycling along over more than one year and having accumulated in the mean over 50 cycles - in other words, we had realized the longest (in terms of number of cycles) predator-prey time series ever seen.

This observation, which we coined as ‘long-term cyclic persistence’, addresses a decade-old fundamental question in ecology, namely whether predator and prey populations are able to coexist indefinitely. Even though such cycles are predicted by ecological theory, they are deviously difficult to realize in a controlled experimental system. In all previous attempts that we know of, either predators or prey were eventually going extinct or reached a stationary state. Our experimental results give strong evidence that intrinsically generated predator–prey cycles are not a mere construct of ecological theory but really do occur in living communities.

The arduous journey to accumulate the evidence

We found these results interestingly enough that we submitted our findings as a Letter to Nature. This was in 2008. As so often, our manuscript was (at first) rejected. Basically our referees agreed that this was a cool story, but the verdict was that something was still missing to put the paper ‘over the top’. This started the long painstaking process of collecting sufficient evidence to proof that our results were robust.

This process included most notably the replication of the experiments in independent chemostat runs, to demonstrate that the occurrence of long-term cycles was no lucky coincidence. Furthermore we tested the predator-prey cycles in environments with changing conditions and under temporal forcing with pulsed nutrients. In total, we produced time series of more than 2000 measurement days (and this does not include the countless failed experiments, where the chemostat runs had to be stopped due to technical failures or contamination by bacteria).

Wavelet analysis

We supplemented our experiments with sophisticated statistical analysis. This was necessary because on close inspection our measured time series showed a remarkable duality of regular and irregular behavior. Even though predator and prey were oscillating throughout the experimental duration, these oscillations were characterized by a huge variability. This included strong variations in amplitude and also in the period length of the cycle, which could vary between 4 to 10 days. To tease out these characteristics of the cycle we resolved to bivariate wavelet analysis, allowing us to extract the phase relationships from our noisy time series. While wavelet analysis today is a standard tool to study periodic time series, here we developed a special technique, where at every time instance we estimated the period length of maximal co-oscillating power, and subsequently took the phase evolution of all signals from the wavelet cross-spectrum at this instantaneous period length (see the corresponding source code on my GitHub page).

The wavelet analysis revealed some unusual properties: On the one hand, the time delay between all characteristic measurements in our cultures appeared to be remarkably constant across all trials. Thereby predator and prey showed the typical phase lag of 90 degrees (a quarter of the cycle), whereas the egg-ratio (that is, the ratio of eggs by the total number of rotifers) unexpectedly preceded that of its algal food. (This is not immediately obvious as egg production is directly connected to food availability; we were later able to explain this as an outcome of the stage structure of the rotifer population). On the other hand, these rigid phase relationships were not always present. Without any external disturbances they frequently gave way to irregular regimes in which predator and prey oscillated independently without a fixed phase relationship. The origin of these sudden shifts between regimes of strongly phase-locked and independent oscillations is still mysterious. It gives the impression as if the predator-prey cycle is not able to operate at an intermediate level of irregularity - enforcing either a tight sync of predator and prey or none at all.

Note, this spontaneous switching between two different oscillation modes has an amusing ‘analogy’ in the so called Dzhanibekov Effect, where a rotating body regularly switches between to different rotation modes.

A stage structured model

To get more insight into the system we also developed a mathematical model. The necessity of modelling is motivated by the observation that our ability to understand the dynamic behaviour of complex (ecological) systems relies to a large part on the correctness of very simple theoretical assumptions – including those about how predator and prey interact to produce cyclic dynamics. Replicating our experimental system in a model gave us the assurance that we were using the right building blocks when attempting to understand what is happening in the real system.

Our best idea to explain the unusual phase signature in our predator-prey system was based on the stage structure of the animal population. It thus became clear very fast that the basic framework of the predator-prey model should be that of a time-delayed ordinary differential equation (for an introduction into the state of the art of stage-structured population models I highly recommend the monograph by de Roos and Perrson). And indeed, our model simulations showed that with a sufficient long maturation delay of juvenile rotifers such models naturally predict the experimentally observed phase signature, where the phase of the egg-ratio preceded that of the prey.

However, since stage-structured population models are able to produce rich dynamic outcomes, we had the hope to do even better: Maybe the mysterious shifts between regular and irregular oscillating regimes could be mechanistically explained as noise-induced transitions between different (large- and small-amplitude) cyclic attractors? From this moment on, modeling turned into the proverbial search for the needle in the haystack. While most model parameters were basically known and didn't give much leverage for model variation, it was not obvious which dynamic processes might be missing in our model to produce the desired outcome. In the end, all our attempts in this direction failed. We were not able to find a model variant that exhibits such dynamic transitions between different oscillation regimes, but at the same time is also consistent with our experimental setting and reproduces the other characteristics of our measured time series (e.g. the phase relations).

Thus, finally we had to ‘accept defeat’ and reluctantly changed our modelling strategy. Our new paradigm was to present the most parsimonious model that only included the operative parameters of the chemostat system, the essential organismal variables (algae and rotifer life stages), and only those processes needed to capture the experimental results. Using the resulting model we managed the balancing act of simultaneously describing the observed range and variability of state variables (in the forced and unforced scenarios), the emergence of long-term oscillatory persistence (with the correct period length), the correct phase relationships between all state variables, power and wavelet spectra, and the typical durations and shifts between different oscillatory regimes (in our final model explained as streaks of unusual stochastic disturbances).

While the final model looks inconspicuously simple, in fact it is the end-result of a long process, accumulating a tremendous amount of model development and testing - iteratively increasing our understanding about the functioning of the system. This is reminiscent to the creative process in painting, where quite similar the final result with which the artist was pleased can never tell the observer what the artist really did, how he made decisions, his failed attempts, and dead ends of the creative process. (But note that artificial intelligence now makes it possible to reconstruct underlying hidden layers in paintings).

Another world record?

In total, after our first submission in 2008 it took us nearly 11 years of further experiments, data analysis and modelling until we had sufficient confidence for a new submission. We finally resubmitted our paper to Nature in 2019, when our resubmission was directly accepted for publication. With such a delay between first and revised submission, with this publication we may have set a ‘world record’ also in a very different sense.

Bernd Blasius
Professor for Mathematical Modelling

I am interested in the theoretical description of complex living systems at the interface of theoretical ecology and applied mathematics