The war of attrition in game theory is a model of a stand-off situation between two opponents where the winner is determined by its persistence. We model a stand-off between a predator and a prey when the prey is hiding and the predator is waiting for the prey to come out from its refuge, or when the two are locked in a situation of mutual threat of injury or even death. The stand-off is resolved when the predator gives up or when the prey tries to escape. Instead of using the asymmetric war of attrition, we embed the stand-off as an integral part of the predator–prey model of Rosenzweig and MacArthur derived from first principles. We apply this model to study the coevolution of the giving-up rates of the prey and the predator, using the adaptive dynamics approach. We find that the long term evolutionary process leads to three qualitatively different scenarios: the predator gives up immediately, while the prey never gives up; the predator never gives up, while the prey adopts any giving-up rate greater than or equal to a given positive threshold value; the predator goes extinct. We observe that some results are the same as for the asymmetric war of attrition, but others are quite different.

博弈论中的消耗战是两个对手之间对峙局面的模型，其中胜利者取决于其坚持。当猎物躲藏而捕食者正在等待猎物从其避难所出来时，或者当两者被锁定在相互伤害甚至死亡的威胁中时，我们模拟了捕食者和猎物之间的对峙。当捕食者放弃或猎物试图逃跑时，僵局就会解决。我们没有使用不对称的消耗战，而是将对峙嵌入为 Rosenzweig 和 MacArthur 的捕食者-猎物模型的一个组成部分，该模型源自第一性原理。我们应用这个模型来研究猎物和捕食者放弃率的共同进化，使用自适应动力学方法。我们发现长期的进化过程会导致三种性质不同的情景：捕食者立即放弃，而猎物永不放弃；捕食者永不放弃，而猎物采用任何大于或等于给定正阈值的放弃率；捕食者灭绝。我们观察到，有些结果与非对称消耗战的结果相同，但有些结果却大不相同。