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I would like to have an example of application and a brief description/definition (two or three lines) of the following fields. I would like to understand the exact differences between them. Other related informations are also extremely appreciated e.g. if these are fields in growth, the name of the best researcher in that field, best institution etc.

  • Computational Ecology;
  • Mathematical Ecology;
  • Numerical Ecology;
  • Statistical Ecology;
  • Ecoinformatics.
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I will think more about the Q, but generally these are fairly vague descriptions and I think that different researchers use them in different ways. Also common that researchers working on the same topic/field use different labels on their activities. –  fileunderwater Nov 8 '13 at 15:40
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Although work in those areas is definitely all related, there are some kind of general differences in how those labels tend to be used, and I'll take a stab at defining them:

Mathematical ecology - this is typically the creation of theoretical models composed purely from math (i.e. not stochastic computer models). Examples include a lot of classics from ecological theory, such as Lotka-Volterra models of predation and competition, but there are also people who still do this kind of research, such as Peter Chesson in his research on the storage effect.

Numerical ecology - this is not a term that I've heard used very frequently, but I think it tends to cluster pretty closely with mathematical ecology, perhaps with more of an emphasis on also including and analyzing empirical data. I hear it used more to refer to techniques than actual ongoing areas of research.

Statistical ecology - I think this term gets used more loosely because obviously statistics are important to most ecology research. It can either refer to research that makes heavy use of advanced statistics because the questions that its getting at are hard to untangle from other effects, or research that uses a statistical model to make its point (note that statistical models are generally different from the mathematical models mentioned above in that their goal is generally to determine how likely something is to happen rather than to untangle some fundamental principle of ecology). A lot of studies on neutral vs. niche-based community assembly would probably fall into the category of statistical ecology. Here is one of many examples.

Ecoinformatics - maybe this is just because I study ecoinformatics, but I tend to think that this is the most distinct of the fields that you mentioned, although it certainly still has a great deal of overlap with the others, especially since it's a pretty diverse subfield. In general, ecoinformatics focuses on making use of large datasets in ecology. This involves both research on developing methods for doing this (for example, I study ways to efficiently collect information from ecological wireless sensor networks. Another good example is the development of GIS tools) and research that uses these methods (e.g. a study that uses GIS to synthesize sensor data and observations of a given organism to determine what that organism's range will be as climate change progresses, or that otherwise seeks to pull ecological concepts out of the data).

Computational ecology - I left this until last, because I think it's the vaguest of the labels you listed. It definitely includes ecoinformatics, but it probably also includes a lot of statistical ecology (and perhaps numerical and mathematical ecology too). I think that the main reason that it's a useful label is that it can catch areas of research that don't really fit in to the other labels. For instance, I once saw a study on quantifying niche space using support vector machines. It was really interesting, and maybe it could be classified as ecoinformatics or statistical ecology, but it didn't really quite fit into either of those. So I guess I tend to see computational ecology as a catch-all term.

These are just the impressions that I have gotten - if others chime in too we can probably build a better consensus definition of these labels.

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"Numerical Ecology" is the title of a classic text by Legendre and Legendre, with a recent variation applied to R; but it is not clear how the field is different –  David Nov 18 '13 at 4:16
    
Well, this goes to show how differently they are perceived. I find Comp ecol, Stat ecol and Math ecol most common. Don't remember seing Numerical ecol, but can imagine that it can be used for e.g. simulation studies of theoretical models. For me Ecoinformatics is fairly vague (but overlapping with Comp ecol), but often dealing with Big data, next-gen sequencing and remote sensing. –  fileunderwater Nov 19 '13 at 11:37
    
I agree that those are the most common labels - just perhaps not the most descriptive. And I also agree about Ecoinformatics involve big data and remote sensing. My one disagreement would be on next-gen sequencing being part of ecoinformatics. I would definitely classify it as bioinformatics, or else systems biology if it's done in a more ecological context. –  seaotternerd Nov 19 '13 at 23:51
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