Department of Applied Mathematics, Xi'an Jiaotong University, Xi'an 710049, PR China.
Math Biosci. 2011 Jun;231(2):210-24. doi: 10.1016/j.mbs.2011.03.007. Epub 2011 Mar 21.
On the ecological timescale, two predator species with linear functional responses can stably coexist on two competing prey species. In this paper, with the methods of adaptive dynamics and critical function analysis, we investigate under what conditions such a coexistence is also evolutionarily stable, and whether the two predator species may evolve from a single ancestor via evolutionary branching. We assume that predator strategies differ in capture rates and a predator with a high capture rate for one prey has a low capture rate for the other and vice versa. First, by using the method of critical function analysis, we identify the general properties of trade-off functions that allow for evolutionary branching in the predator strategy. It is found that if the trade-off curve is weakly convex in the vicinity of the singular strategy and the interspecific prey competition is not strong, then this singular strategy is an evolutionary branching point, near which the resident and mutant predator populations can coexist and diverge in their strategies. Second, we find that after branching has occurred in the predator phenotype, if the trade-off curve is globally convex, the predator population will eventually branch into two extreme specialists, each completely specializing on a particular prey species. However, in the case of smoothed step function-like trade-off, an interior dimorphic singular coalition becomes possible, the predator population will eventually evolve into two generalist species, each feeding on both of the two prey species. The algebraical analysis reveals that an evolutionarily stable dimorphism will always be attractive and that no further branching is possible under this model.
在生态时间尺度上，具有线性功能反应的两种捕食者物种可以在两种竞争的猎物物种上稳定共存。在本文中，我们采用自适应动力学和关键函数分析的方法，研究了在什么条件下这种共存也是进化稳定的，以及两种捕食者物种是否可能通过进化分支从单一祖先进化而来。我们假设捕食者的策略在捕获率上有所不同，一种捕食者对一种猎物的高捕获率对另一种猎物的捕获率较低，反之亦然。首先，通过使用关键函数分析的方法，我们确定了允许捕食者策略发生进化分支的权衡函数的一般性质。结果发现，如果权衡曲线在奇异策略附近是弱凸的，并且种间猎物竞争不是很强，那么这个奇异策略就是一个进化分支点，在这个点附近，原生和突变的捕食者种群可以共存，并在其策略上发生分歧。其次，我们发现，在捕食者表型发生分支后，如果权衡曲线是全局凸的，那么捕食者种群最终将分支为两种极端的专食者，每种专食者都完全专食于一种特定的猎物物种。然而，在具有平滑阶跃函数样权衡的情况下，可能会出现内部二态奇异联盟，捕食者种群最终将进化为两种广食者物种，每种都以两种猎物物种为食。代数分析表明，进化稳定的二态性总是具有吸引力，在这种模型下，不可能进一步分支。
