Arms races in evolution—An ESS to the opponent-independent costs game

The opponent-independent cost game is an animal contest in which the bigger and more heavily-armed opponent wins a disputed resource without significant fighting costs. A strategy is a choice of investment level in armament. Increasing armament is assumed to have fitness costs that are unrelated to contests; i.e. the cost of an individual's investment in arms is independent of the strategy played by an opponent. Previous work with this model showed that no ESS exists if a strategy prescribes an arms level exactly. This is equivalent to the notion that there is no environmental variation in the arms level attained by a given strategy. If environmental variation is introduced, a pure ESS can generally exist. A strategy is assumed to prescribe an exact investment cost, but this is translated into a probability distribution of arms levels attained, rather than an exact arms level. Increasing investment increases the mean of the arms level distribution. The ESS investment level depends both on how environmental factors distribute arms levels, and on the shape of the cost function (i.e. on the way that costs increase with investment); in some instances there is no ESS. Two types of model are investigated; in one fitness is additive (benefit-cost), in the other it is multiplicative (benefit × survivorship). The multiplicative model is likely to apply to the case where contests are between males for access to females. Here the ESS investment level (an ESS degree of risk that a male sustains as a result of armament) increases as fewer individuals guard the available resources. Thus sexual size dimorphism (male/female size) and relative male armament should increase as harem size increases. The ESS investment level will also be highest if most individuals are small and poorly armed, as would often be the case where size increases throughout life. The model can be applied to coevolutionary arms races between two classes of opponent, such as prey and predator, or parent and offspring. Here the ESS is likely to be a pair of ESS arms levels, one for each class of opponent.