Arms Races, Conflict Costs and Evolutionary Dynamics

In conflicts and fights, the winner is often determined by the difference in resource-holding potential, e.g. size, weaponry, strength (RHP). I model the evolution of RHP in a symmetric game with continuous strategies. I show that there is a convergence stable ESS level of RHP if the cost of the trait increases faster than linearly, and that this is the only solution if the cost increases fast enough with RHP. Otherwise with slowly increasing cost, the solution is a cyclically fluctuating level of RHP. It is also shown that if the cost increases linearly with RHP, the only solution to the game is neutrally stable cycles, the amplitude determined by the initial conditions. The cycles come about because in a population drawn into an arms race of RHP, individuals after a while suffer costs so high that mutants with the lowest possible armament level may invade.