Hamilton's Rule and Altruistic Behavior

Hamilton's rule is a key concept in evolutionary biology, specifically within the field of sociobiology. It helps explain how altruistic behaviors, which seem to reduce an individual's own fitness while benefiting others, can evolve through natural selection.

The rule states that an altruistic act will be favored by natural selection if the benefit to the recipient, weighted by the coefficient of relatedness, outweighs the cost to the donor.

Understanding Hamilton's Rule

Hamilton's rule is expressed by the inequality:

\(rB > C\)

Let's break down the components:

• r: This is the coefficient of relatedness between the donor of the altruistic act and the recipient. It represents the probability that they share the same gene by recent common descent. For example, siblings have an 'r' of 0.5, parent-offspring also have 0.5, while half-siblings have 0.25, and cousins have 0.125.
• B: This is the benefit received by the recipient of the altruistic behavior. It's measured in terms of increased reproductive success or fitness.
• C: This is the cost incurred by the donor of the altruistic behavior. It's measured in terms of decreased reproductive success or fitness.

The rule \(rB > C\) essentially means that altruism is favored when the benefit to relatives, adjusted for how closely related they are, is greater than the cost to the individual performing the action. This is often referred to as kin selection.

Altruistic Condition Analysis

The question asks which relationship results in altruistic behavior according to Hamilton's rule. We are given the inequality \(rB > C\) and several options. We need to see which option is equivalent to this inequality.

Let's look at the given options:

1. \(rB = C\): This means the benefit to the recipient (weighted by relatedness) is exactly equal to the cost to the donor. This condition does not strictly favor the evolution of altruism; it might be neutral. For altruism to be favored, the benefit must exceed the cost.
2. \(rC - B = 0\): This can be rewritten as \(rC = B\). Dividing both sides by C (assuming C > 0, which is true for a cost), we get \(r = B/C\). This is not the condition for altruism to be favored (\(r > C/B\)).
3. \(r > C/B\): To compare this with \(rB > C\), we can multiply both sides of the inequality by B. Since B is a benefit (increase in fitness), B is positive. Multiplying an inequality by a positive number does not change the direction of the inequality sign. So, multiplying \(r > C/B\) by B gives us \(r \times B > (C/B) \times B\), which simplifies to \(rB > C\). This is exactly Hamilton's rule.
4. \(rC - B > 0\): This can be rewritten as \(rC > B\). Dividing both sides by C (assuming C > 0), we get \(r > B/C\). This is not the condition for altruism to be favored (\(r > C/B\)).

Comparing the options to Hamilton's rule \(rB > C\), we find that option 3, \(r > C/B\), is mathematically equivalent to the rule, assuming B > 0.

Therefore, the relationship that will result in an altruistic behavior, according to Hamilton's rule, is \(r > C/B\).