Wikipedia:Reference desk/Archives/Mathematics/2014 August 18

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August 18

Determining relative advantages of traits

So I'm comparing the fitness of organisms. I can take an organism with a collection of traits (say, a, b, and c) and compare it to an organism with another collection of traits (x and y) and see which has greater fitness. I can never compare individual traits. If I've got a sample of a bunch of matchups (not necessarily every possible combination of a given number of traits), how do I go about finding which individual traits, which combinations, etc. best promote organism fitness? --2404:2000:2000:5:0:0:0:C2 (talk) 04:23, 18 August 2014 (UTC)

Okay here's a tentative idea I had: represent each matchup as an inequality (e.g. a+b+c-x-y>0) (note there are only finitely many traits), and I get a system of inequalities. The more samples, the narrower my solution space. But this works under the assumption that each trait contributes a fixed value to fitness, and there's no added value from particular combinations. But it might work reasonably well anyway I think? --2404:2000:2000:5:0:0:0:C2 (talk) 04:56, 18 August 2014 (UTC)

You need to figure out what fitness function you want to optimize, and how the traits contribute to it. For example, what's the better deal? A 2L bottle of soda for $5, or a 500mL bottle for $2? The 2L has a better per-unit cost ($2.5/L versus $4/L) but if you're only going to drink 400mL, with the rest going flat and being dumped out, the 500mL is a better deal, as total cost is better ($2 versus $5). Add in the fact that the 500mL bottle is refrigerated and the 2L isn't (how much is cold soda worth to you?), or perhaps that the 2L comes in your favorite flavor, but the 500mL doesn't, and you have a complex optimization problem that can't just be reduced to a simple "what's the price per liter?". There isn't any single answer as long as you're unclear on how the traits relate to each other, fitnesswise. - What you need to do if you really want to compare fitness is convert each independent trait to a single numerical value that's consistent within and between each other trait. That is, give each trait a certain number of "points", and for each organism sum the number of "fitness points" it has. So you may have 10 points for being cold, but 12 points for coming in your favorite flavor. You can get more complex if your want - trait a and trait b each get so many points individually, but when they occur together, they get a certain number of points for bonus (or penalty). The fitness function need not be a linear combination of the individual traits. - If you can't convert the values to a single consistent "point" scale, you may want to look into Pareto optimization. That is, you can't necessarily rank all the organisms in single file, but you can say "this set of organisms, when taken as a group, have a better fitness than any of these other organisms". Even then, though, you'll want to reduce your dimensionality as much as possible by combining those traits that can be combined into a reduced number of metrics, and if you want to rank organisms on the Pareto front, you'll need to determine a fitness function. -- 160.129.138.186 (talk) 18:39, 18 August 2014 (UTC)

It depends on what you want to do, and there isn't one general answer. I'll assume you are interested in concepts from real biology. One perspective is that traits are only adaptive traits if they lead to a persistent population, i.e. the species/strain doesn't go extinct. Then "fitness" only makes sense in terms of the processes that affect population biology, things like reproduction, predation, resource competition, and so on. This is basically the stuff in Fitness_(biology).

In theoretical ecology, we often use the the long term low density growth rate, as described here [1]. The basic idea is that populations that can reliably grow back from low density while competing with others will not go extinct. If all types can do this, we can infer species coexistence. This LTLDGR can be measured empirically if good data is available, or calculated from a model. Another perspective on fitness is to take the expected value of the number of "grandchildren". Again, this can be computed for a population if data good, but will otherwise need a model. Really, there are just a lot of ways that "fitness" can be quantified, and what is best/most useful depends on the details of your problem. As for your suggestion of just adding together numbers, that is not something that makes much sense. Not only does that disallow interactions between traits, but it also assumes that all traits are somehow on comparable scales... Anyway, if you'd like to know more about any of the stuff I wrote up just ping me, I'll be happy to give more detail/refs. SemanticMantis (talk) 22:50, 18 August 2014 (UTC)

As regards "what I want to do": after taking a sample, I'd like to be able to predict which of two collections of traits has greater fitness. --2404:2000:2000:5:0:0:0:C2 (talk) 23:55, 18 August 2014 (UTC)

The "good" and useful answers for you will be very different, depending on your application. E.g. a highschool project on mice is very different from a research paper on theoretical ecology, which is very different from a video game that uses evolutionary ideas. Unfortunately your response could be true for any of those scenarios, but I'll still try to explain.

If you want to "take a sample" and see which type has "greater fitness", you have to have either (a lot of) data from the real world or a model of some sort. There are a few definitions for measurements at Fitness_(biology)#Measures_of_fitness, and I mentioned the LTDGR above. There are a few others, but they all revolve around the dynamics of the population over time. So, you have to quantify how much each genotype or phenotype contributes to the next generation. The entire notion of "fitness" in biology is about survival and reproduction. No trait is "better" or "worse", except in that context. There is no biologically meaningful way of measuring fitness of traits in a vacuum. You need to account for, at minimum, survival and reproduction over time. This is somewhat a problem of terminology, as this has very little to do with some other uses of the word "fit". I understand this is the math desk, but this is an area of evolution/ecology that actually has a lot of mathematical tools and techniques. If this is just for some small project or video game or something, you can make up whatever you like, or follow some of the other suggestions. If you want your answer to make sense in terms of the modern understanding of fitness in biology, you have to use one of the established mathematical frameworks. I apologize if I sound harsh, but I want to make it clear that this is a well-studied area of research, and making something up ad hoc might not be very useful. SemanticMantis (talk) 15:47, 19 August 2014 (UTC)

If you assume all the traits are independent, then you can do the following:

1) Compare all the cases with trait "a" to all those which lack trait "a", even though each of the two groups will contain a mixture of every other trait.

2) Repeat for trait "b", "c", "x", "y", and any other traits.

3) For each trait, you will then have either a positive or negative differential, or no significant difference in fitness, based on whether that trait is present or absent.

4) You could then conclude that the combination of all the positive traits, and absence of all the negative traits, is the most fit organism.

However, again note that this assumed that all the traits were independent. You may very well find this not to be the case. For example, cats with white fur and blue eyes might be desirable, but such cats also tend to be deaf. So, if this turns out to be the case, things get more complex. StuRat (talk) 00:26, 19 August 2014 (UTC)