# In a fitness landscape, is fitness relative or absolute?

I was asked if the fitness of a fitness landscape is absolute or relative.

Fitness landscapes are usually graphed with a fitness function that uses either phenotypes or genotypes in relation to fitness or one of its component.

More specifically, if the fitness measured is survival, and it's computed based on a quadratic model, does that means that every individual survival probability is an absolute measure of "fitness" (survival)?

Or since it's computed from a model, would it be better to find the relative contribution of a variable on the predictor in the model and say that the "fitness" is relative to a trait if it's computed with multiple traits?

How does the mean fitness of the population influence the model, which will determinate the fitness of all individuals? Is this "mean contribution of the population's fitness" a relative measure of fitness?

My specific questions are:

1. What type (relative vs absolute) of fitness is shown in a fitness landscape?
2. Does the type of fitness change if you change way to model it (For example, I want to know specifically for a quadratic model vs a spline)?
3. Does the type of fitness change if you change way to model it (For example, I want to know specifically for a quadratic model vs a spline)?
4. When using a mark recapture model (you use the survival of each individual to find a fitness function) is the survival estimated by the model relative or absolute? Does the mathematical formulation of the model changes something?

In the image below, I've graphed all the points (black line) and I've removed the 3 lowest values of y for the 3 largest x values (red line). If I was calculating the fitness of the individuals with or without the 3 smallest points for the 3 largest x values, then the fitness of all the other individuals would be affected (except at x~38 and y~38). x = c(1:100,101,102,103)
y =  c(x[1:100]+rnorm(100, sd = 6), 50,60,30)

x = c(1:100,101,102,103)
y =  c(x+rnorm(103, sd = 6))

summary(lm(y~x))
plot(y~x)
abline(lm(y~x),col = "red")
points(38,38,col="blue")


Another example with crossbills: Would you say it's relative or absolute fitness. Here the fitness measure is "performance" of eating certain types of seed.

Regression analyses are central to characterization of the form and strength of natural selection in nature. Two common analyses that are currently used to characterize selection are (1) least squares–based approximation of the individual relative fitness surface for the purpose of obtaining quantitatively useful selection gradients, and (2) spline-based estimation of (absolute) fitness functions to obtain flexible inference of the shape of functions by which fitness and phenotype are related. These two sets of methodologies are often implemented in parallel to provide complementary inferences of the form of natural selection.

Are least squares-based approximation of the individual always computing relative fitness surfaces? For spline, is it always absolute values?

• I put the 2 questions in bold. The rest is more of an explanation or avenue of research. – M. Beausoleil Aug 15 '17 at 18:44
• Could you provide labeling for your computed graph, or just mention what they are? Thanks. – user22020 Aug 16 '17 at 23:43
• This graph is just an example of a linear regression. Both lines are identical, but the red one, I removed the 3 "outliers" at the far right. That way you see how the line shifts and how it affects the expected value on the left and the right of the $(38,38)$ point – M. Beausoleil Aug 17 '17 at 15:17
• Let me ask you this: When considering the crossbills, do you think that, based on the nature of what the data represents, the datapoints are directly related to each other? i.e., is the performance of one type of crossbill immediately dependent upon the other profile types, in most cases? Which is the same as asking -- do you think that crossbills of significantly different bill profiles are directly competing with each other? Or, do you think it's that, since the crossbills beak profiles are different enough to then harvest different seeds, that they "stay out of each others way"? – user22020 Aug 17 '17 at 15:31
• From the website: [...]The potential benefits of specializing on a key conifer species, the diversity of key conifer cones, and the method that crossbills use to extract seeds from these cones, it is perhaps not surprising that crossbills have diversified into several forms, many of which appear to be more or less specialized for foraging on a single or a few key conifers. While the association between bill depth and cone size is pretty striking, there's more evidence inside each bill that some degree of specialization is occurring. – M. Beausoleil Aug 17 '17 at 20:50

By experience, most of the time, authors are talking about relative fitness. To confirm my feeling, I looked at the three randomly chosen theory papers from the first page of results that Google Scholar returned when searching for fitness landscape and they all use fitness as a relative measure. The papers are Derrida and Peliti (1991), Merz and Freisleben (2000) and Woodcock and Higgs (1995).

However, I do not think this is in the definition of a fitness landscape. One could work with absolute fitness and still call his model a fitness landscape. I think that therefore the answer is specific to every model considered. Note that even the "landscape axes" for which a fitness value is assigned can be of different nature. Typically, one can associate phenotypic or genotypic and/or even environmental axes.

• In the three articles, only one was talking about relative fitness. I don't understand what do you mean by 'they all use fitness as a relative measure'. Is it relevant to determine the type of fitness if it was estimated from a model? – M. Beausoleil Aug 17 '17 at 20:56
• I considered their definition of the letter $w$, not whether they commented much on it. But it is not impossible that I'd made mistakes. Do you think two of them use absolute fitness? If yes, please tell me which one(s) and I can have a second look at it. – Remi.b Aug 17 '17 at 21:02

What type (relative vs absolute) of fitness is shown in a fitness landscape?

A fitness landscape is just a graph with a regression line/curve. The reason why this graph is referred to as "fitness" is because it's specifically illustrating how well "fit" an individual is to survive & reproduce within its population; "landscape" however, has no special or unique meaning, and can freely be interchanged with "graph".

With this in mind, a fitness landscape can depict both types of models -- relative and absolute. I believe your confusion/uncertainty may stem from the fact that these two population measurements describe two different things, and, that the use of fitness landscapes isn't unique to just one of them.

(I would really like to address the differences between relative and absolute fitness models at this time, however, I feel that it would get to be off-topic, as that would require somewhat of a lengthy explanation/comparison. I strongly suggest posting a new question that asks this, and I will happily respond. That is, if you even want clarification between the two.)

Does the type of fitness change if you change way to model it (For example, I want to know specifically for a quadratic model vs a spline)?

Depending on the type of fitness model you're portraying (absolute vs. relative), the amount of variability in your data will (most likely) drastically differ; i.e., how sharp of a curve your data may produce within a given interval of your datapoints. That being said, different fitness models do require different regression models in order to most accurately characterize the data.

With absolute fitness, you'll be able to (much) more accurately describe your data using a non-linear equation that can handle a considerable amount of flexibility. This often in the form of a cubic equation (contains a term with $x^{3}$). Higher degree equations can also be used (contains a term with $x^{4}$, or $x^{5}$, ..), however, your regression model may become "too sensative" to variability, and as a result, will cause your (regression) model to be more inaccurate than it should be. Also, because of the sensativity & flexibility of these regression models, smoothening (spline) functions are applied to the model, so that the data isn't quite as chaotic; imagine the difference between a stock market graph (e.g., weierstrass function), with that of a more traditional, "smooth" graph (e.g., sigmoid or logarithmic function).

And then, with relative fitness, you'll want to use either a linear, or quadratic, regression model. Due to the nature of what relative fitness models describe, there isn't nearly as much of a change in your data, given an interval. Because of this, a simple linear (contains $x$ term, and no higher powered term) or quadratic (contains an $x^{2}$ term and nothing higher) regression model will sufficiently represent your datapoints.

(If you'd like more of an explanation as to the nature of absolute & relative fitness models, and why they may or may not bring with them a higher amount of variability in their data, that would be addressed when discussing the differences between the two models, or, you can also open a question that specifically addresses this.)

Are least squares-based approximation of the individual always computing relative fitness surfaces? For spline, is it always absolute values?

Yes. Given what was just discussed, a (near) linear regression model are (typically always) used for relative fitness models, and higher-ordered regression models are (typically always) used for absolute fitness models.

• If there are any points where I am being unclear, or you'd like for me to be more elaborate in explaining, please let me know and I'll gladly make edits. I may make more edits later anyway, but have to leave work right now and get dinner :) – user22020 Aug 15 '17 at 22:27
• Looking at your graph, and given my explanation of which regression model should be used, that fitness landscape is most likely depicting a relative fitness model, and is using a linear regression equation to characterize the data. And to contrast, the crossbill graph is probably an absolute fitness model, and is using a higher powered regression model with smoothing. – user22020 Aug 15 '17 at 22:35
• Cool! Thanks, but I don't understand why you are able to say "it's relative vs absolute". Based on the definition of Absolute vs relative fitness, I feel that it doesn't make sense to call one or the other. The surface of a fitness landscape is actually the "expected fitness". You can estimate the fitness value of an individual by taking the actual parameters values of your model and plugging the phenotypes. In this case, the fitness is relative or absolute? How can you justify that it's one or the other? – M. Beausoleil Aug 15 '17 at 22:56
• @M.Beausoleil "The surface of a fitness landscape is actually the "expected fitness". You can estimate the fitness value of an individual by taking the actual parameters values of your model and plugging the phenotypes." -- Correct; the fact that you're using a regression model is what allows for you to do this. Regression models generalize datasets to where you can consider situations that the data doesn't already contain. – user22020 Aug 16 '17 at 14:01
• Evolutionary biologists are generally most concerned with relative fitness models, which describes the survival & reproduction success of an individual (genotype) with respect to the highest performing genotype. It is an immediately contextual measurement, and requires selection to have occurred. Absolute fitness, however, is neither of these things, and so, is not quite as relavent to the questions that most evolutionary biologists are concerned with (though [absolute fitness] is still important, as it can be used to calculate relative fitness). – user22020 Aug 16 '17 at 14:07