Cushing Jim M
Department of Mathematics, Interdisciplinary Program in Applied Mathematics, 617 N Santa Rita, Tucson, Arizona, 85721, United States.
Math Biosci Eng. 2015 Aug;12(4):643-60. doi: 10.3934/mbe.2015.12.643.
An evolutionary game theoretic model for a population subject to predation and a strong Allee threshold of extinction is analyzed using, among other methods, Poincaré-Bendixson theory. The model is a nonlinear, plane autonomous system whose state variables are population density and the mean of a phenotypic trait, which is subject to Darwinian evolution, that determines the population's inherent (low density) growth rate (fitness). A trade-off is assumed in that an increase in the inherent growth rate results in a proportional increase in the predator's attack rate. The main results are that orbits equilibrate (there are no cycles or cycle chains of saddles), that the extinction set (or Allee basin) shrinks when evolution occurs, and that the meant trait component of survival equilibria occur at maxima of the inherent growth rate (as a function of the trait).
运用庞加莱 - 本迪克松理论等方法，分析了一个适用于受捕食影响且存在强阿利灭绝阈值种群的进化博弈论模型。该模型是一个非线性平面自治系统，其状态变量为种群密度和一个表型性状的均值，该表型性状受达尔文进化影响，决定种群的固有（低密度）增长率（适合度）。假设存在一种权衡，即固有增长率的增加会导致捕食者攻击率成比例增加。主要结果是轨道达到平衡（不存在鞍点的周期或周期链），进化发生时灭绝集（或阿利盆地）会缩小，并且生存平衡点的平均性状分量出现在固有增长率（作为性状的函数）的最大值处。
