Theory for the evolution of diffusible external goods

Organisms from prokaryotes to plants and animals make costly investments in diffusible beneficial external products. While the costs of producing such products are born only by the producer, the benefits may be distributed more widely. How are external goods-producing populations stabilized against invasion by nonproducing variants that receive the benefits without paying the cost? This question parallels the classic question of altruism, but because external goods production need not be altruistic per se, a broader range of conditions may lead to the maintenance of these traits. We start from the physics of diffusion to develop an expression for the conditions that favor the production of diffusible external goods. Important variables in determining the evolutionary outcome include the diffusion coefficient of the good, the distance between individuals, and the uptake rate of the external good. These variables join the coefficient of relatedness and the cost/benefit ratio in an expanded form of Hamilton's rule that includes both selfish and altruistic paths to the evolution of external goods strategies. This expanded framework can be applied to any external goods trait, and is a useful heuristic even when it is difficult to quantify the fitness consequences of producing the good.