An evolutionarily stable strategy (ESS) is an evolutionary strategy that, if adapted by a population, cannot be invaded by any deviating (mutant) strategy. The concept of ESS has been extensively studied and widely applied in ecology and evolutionary biology [M. Smith, On Evolution (1972)] but typically on the assumption that the system is ecologically stable. With reference to a Rosenzweig–MacArthur predator–prey model [M. Rosenzweig, R. MacArthur, Am. Nat. 97, 209–223 (1963)], we derive the mathematical conditions for the existence of an ESS when the ecological dynamics have asymptotically stable limit points as well as limit cycles. By extending the framework of Reed and Stenseth [J. Reed, N. C. Stenseth, J. Theoret. Biol. 108, 491–508 (1984)], we find that ESSs occur at values of the evolutionary strategies that are local optima of certain functions of the model parameters. These functions are identified and shown to have a similar form for both stable and fluctuating populations. We illustrate these results with a concrete example.

进化稳定策略（ESS）是一种进化策略，如果被群体适应，则不会被任何偏离（突变）策略入侵。 ESS的概念在生态学和进化生物学中得到了广泛的研究和广泛的应用[M. Smith， 《论进化》 （1972）]，但通常基于系统生态稳定的假设。参考 Rosenzweig-MacArthur 捕食者-被捕食者模型 [M.罗森茨威格，R.麦克阿瑟， Am。纳特。 97, 209–223 (1963)]，我们推导了当生态动力学具有渐近稳定极限点和极限环时 ESS 存在的数学条件。通过扩展 Reed 和 Stenseth 的框架 [J.里德，北卡罗来纳州斯坦塞斯， J.理论。生物。 108, 491–508 (1984)]，我们发现 ESS 发生在进化策略的值处，这些进化策略是模型参数的某些函数的局部最优值。这些函数已被识别并显示出对于稳定群体和波动群体都具有相似的形式。我们用一个具体的例子来说明这些结果。