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To me, these tags are used almost interchangeably at the moment, and I think any attempt to separate them would be awkward. The tag wikis are almost circular as well, which is another indication that they overlap.

Wiki for :

Theoretical models that explain a system using mathematical equations.

Wiki for :

The modeling of biological processes using the tools of applied mathematics.

My suggestion is to merge them, using as the main tag, since this is a bit more general (could e.g. include non-formalized theoretical biology as well). I have already suggested this tag synonym, but not much has happened since the voting on tag synonyms is at a standstill.

I do see that there might/could be a slight difference in emphasis in these tags, where could be used in a more technical sense (any application using advanced mathematics) even if the focus isn't theoretical development of the subject. However, the current use do not match this and the number of questions that use these tags aren't that many. Therefore, I think it is more useful to merge them.

EDIT

In either case, and after seing the answer from @Corvus, I have now tried to separate the two tags by editing their tag wikis:

  • Theoretical biology
    "Questions relating to biological theory or theoretical developments that deals with biological processes."
    full wiki entry

  • Mathematical models
    "Models that analyse biological systems or processes using the tools of applied mathematics."
    full wiki entry

Please comment or edit if you don't find them suitable.

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  • $\begingroup$ i voted for merging $\endgroup$
    – WYSIWYG
    Jan 13, 2015 at 9:43
  • $\begingroup$ What do moderators think - should these suggested synonyms be left alone to see if they are eventually resolved by voting or could you fix them manually? @MadScientist $\endgroup$
    – fileunderwater
    Jan 20, 2015 at 19:03

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In principle, these are very different things.

Plenty of theoretical biology contains little or no mathematics; the theory is verbal and conceptual. R.A. Fisher's original model of why there is an even sex ratio provides just one of many examples of elegant work in this vein. George Williams's 1957 paper on the evolution of senescence does include a little model, but it is really tangential to the main thrust of the paper. What could be a greater theoretical breakthrough than Darwin's mathematics-free Origin of Species? There's even a journal that focuses on this sort of thing: Biological Theory

Meanwhile, there is mathematical biology that is barely theoretical. Particularly in inference-heavy fields such as bioinformatics we find papers that make intense use of mathematics and/or statistics without advancing biological theory in the least.

In practice, as the tags are used on this site, they do seem hard to separate. There may be a slight difference in focus: theoretical biology focusing on simple conceptual models while mathematical biology focuses on analytic and computational techniques. But maybe I'm just imagining things. In any event, I might suggest waiting to see if the distinctions start to sort themselves out.

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  • $\begingroup$ It sounds like the description of the theoretical-biology tag should be changed. $\endgroup$
    – canadianer
    Jan 21, 2015 at 8:35
  • $\begingroup$ In principle I definitely agree with you. Maybe we should just clarify the tag wikis, and try to retag questions that use the wrong tags. However, in practice I fear that many not working in the field will fail to see the distinction, and the tags will continue to be muddled. To many, mathematical means theoretical. Wikipedia also mix up the two labels (see wikipedia: Mathematical and theoretical biology). $\endgroup$
    – fileunderwater
    Jan 21, 2015 at 9:04
  • $\begingroup$ I guess the main issue is that theoretical biology can both refer to an research approach (theoretical development of biology) or to the tools used to analyse a problem (as in mathematical/theoretical vs empirical or mathematical/theoretical tools for statistical analysis). $\endgroup$
    – fileunderwater
    Jan 21, 2015 at 9:10
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    $\begingroup$ After considering your answer, I have now edited the tag wikis (see updated question above). $\endgroup$
    – fileunderwater
    Jan 21, 2015 at 9:45
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    $\begingroup$ For what it is worth, I think your new definitions are excellent. $\endgroup$
    – Corvus
    Jan 21, 2015 at 17:06
  • $\begingroup$ A given theoretical idea may be compatible with multiple, but distinct, mathematical models. And the empirical success of a given mathematical model may give weight to our thinking on multiple theoretical ideas. $\endgroup$
    – Galen
    Feb 9, 2022 at 20:17

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