Posted by: oikosasa | May 9, 2014

Linking theory with empirical studies of competition

“Nature is much more complicated” than predicted in theoretical  models. In the Early View Paper “Matrix models for quantifying competitive intransitivity from species abundance data”, Werner Ulrich and colleagues try to fill a gap between models and mother nature when it comes to competition interactions.

Below is their summary of the study:

Ecologists have devoted much effort to inferring competitive processes from observed patterns of species abundances, morphology, and particularly from changes in the spatio-temporal distribution, (i.e. species co-occurrences). Classic assembly rules models, derived from the principle of competitive exclusion, predict that differences in competitive abilities should cause non-random patterns of species occurrences among sites and generate inequalities in species abundances within sites. Competitively inferior species are predicted to occur less frequently and at lower abundance, and an important and largely unresolved question is how such species can persist in a community over long time periods (also known as Darwin’s paradox and nicely explained in .

Simple competition models assume that species can be ranked unequivocally (A>B>C…>Z) according to their competitive strength. However, nature is much more complicated and intransitive competitive networks can generate loops in the competitive hierarchy (e.g. the rock-scissors-paper game, in which A>B>C>A). Importantly, such loops allow weak competitors to coexist with strong ones. Additionally, the structure of such loops might be modulated by environmental factors. Experimentally competitive strength can be tested with simple two-species systems. Testing competitive interactions in many-species systems requires an increasing number of species exclusion experiments.


Up to now no comprehensive theoretical framework existed to infer competitive loops from observed patterns of species abundances. Existing mathematical models based on presences and absences of species (on perfect competitive exclusion) have rarely been applied to empirical data. Our new paper in Oikos seeks to fill this gap in our knowledge. We introduce a statistical framework for evaluating the contribution of intransitivity to community structure using species abundance matrices that are commonly generated from replicated sampling of species assemblages. We use a stochastic back-engineering procedure to find a transition probability matrix that predicts best observed distributions of abundances in an ordinary Markov chain approach. We then use a probabilistic argument to convert this transition matrix into a pairwise competition matrix that contains the information of competitive strength between all species in the community. Our approach can be used for abundance data, time series, and abundances in combination with environmental data. Our case study on necrophagous flies and their hymenopteran parasitoids revealed strong hints towards instable competitive hierarchies. In other words the competitive outcome in these communities (having at least five species) strongly depended on environmental conditions but also on the spatial structure of fly and parasitoid occurrence.


We hope that our new approach sparks a fresh new look at competitive interactions in ecological communities and helps to assess and appreciate the importance of intransitivity for the coexistence of species in natural communities. We fell the need for joint approaches that link existing methods using the temporal and spatial co-variation in species abundances and occurrences with methods (like our) that reconstruct competitive hierarchies. 

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  1. I have a few comments. First, I think that we do need new statistical approaches for such problems. I’m glad to see that this is being investigated. I will need to look more closely at the new method and its implementation. Second, I’m posting this to help cross-connect some different areas of investigation.

    I used time series data on a data set in which coexistence was rather limited and there were only three competitors – but they were in a non-transitive loop – and optimized the parameters of a Markov model to estimate the transition probabilities among the competitors. Many of the software tools I might use now were not widely available then; the effort was crude but did bear fruit. Another statistical strategy was applied for publication (Kirkup and Riley 2004).

    Since then, several collaborators (Wouters, Valayer, Wang) have been quietly working on some additional statistical models to identify non-transitive relationships in similar data sets. One aspect we are interested in is recognizing system stability. Other work has also demonstrated that bacterial antagonisms play key roles in defining communities (Cordero et al 2012, Pérez-Gutiérrez et al 2013).

    All this falls under the rubric of ‘frequency dependent selection’ – Hartl and Clark say that in the general case of frequency dependent selection, almost anything can happen (Principles of Population Genetics). However, many of these direct competition genes (antagonism, contact dependent inhibition, self-non-self/greenbeard, etc) are mediated by ‘low frequency’ genes and alleles (Shapiro and Polz 2014, Cordero, Ventouras, DeLong and Polz 2014). .

  2. This paper addresses an interesting problem. Thanks for blogging about it.

    Putting aside the environmental and temporal data for the moment, it seems like this method tries to reconstruct all m-choose-two competition coefficients based on their contributions to a single vector of length m describing equilibrium abundance. For 25 species, that would entail estimating 300 coefficients from 25 data points.

    What am I missing?

    • Hi Dave

      It’s not so simple. We are trying to reconstruct that competition matrix that best fits a series of abundance distributions. Having only one vector this is impossible. Of course there are many such competition matrices but we found that the best fitting ones have common structures. Basically we ask if any competition matrix is able to generate observed patterns of abundance and how strong is the effect of competitive intransitivity. These are rather qualitative aspects. In many case no competitive strength matrix is able to generate observed abundances. This is then a clear indication that competition is of minor importance.


      • Hi Werner, thanks for your reply.

        I think it’s unfortunate that you didn’t make this point clearer in your paper. I expect that many readers will take phrases like “using abundance matrices to estimate species competition and patch transition matrices” at face value.

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