A few weeks back, I reflected on why mathematical biology can be so hard to learn—much harder, indeed, than the mathematics itself would warrant.
The application of mathematics to biological evolution is rooted, historically, in statistics rather than in dynamics. Consequently, a lot of model-building starts with tools that belong, essentially, to descriptive statistics (e.g., linear regression). This is fine, but then people turn around and discuss those models in language that implies they have constructed a dynamical system. This makes life quite difficult for the student trying to learn the subject by reading papers! The problem is not the algebra, but the assumptions; not the derivations, but the discourse.
Hamilton’s rule asserts that a trait is favored by natural selection if the benefit to others, $B$, multiplied by relatedness, $R$, exceeds the cost to self, $C$. Specifically, Hamilton’s rule states that the change in average trait value in a population is proportional to $BR – C$. This rule is commonly believed to be a natural law making important predictions in biology, and its influence has spread from evolutionary biology to other fields including the social sciences. Whereas many feel that Hamilton’s rule provides valuable intuition, there is disagreement even among experts as to how the quantities $B$, $R$, and $C$ should be defined for a given system. Here, we investigate a widely endorsed formulation of Hamilton’s rule, which is said to be as general as natural selection itself. We show that, in this formulation, Hamilton’s rule does not make predictions and cannot be tested empirically. It turns out that the parameters $B$ and $C$ depend on the change in average trait value and therefore cannot predict that change. In this formulation, which has been called “exact and general” by its proponents, Hamilton’s rule can “predict” only the data that have already been given.