The geometric theory of adaptive evolution: trade-off and invasion plots
The geometric theory of adaptive evolution: trade-off and invasion plots
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DOI：
10.1016/j.jtbi.2004.10.017
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发表时间：
2005-04-07
影响因子：
2
通讯作者：
Boots, M
中科院分区：
文献类型：
--
作者：
Bowers, RG;Hoyle, A;Boots, M
本文的目的是采取完全几何路径来确定受权衡影响的生态系统的进化特性。特别是，我们以几何的方式对进化奇点进行分类。为了实现这一目标，我们研究了权衡与入侵图（TIPs），它从三条曲线之间的关系中以图形方式显示进化的结果。在第一个入侵边界（曲线）中，一种菌株作为常驻菌株，另一种菌株作为假定的入侵者，而在第二个边界（曲线）中，菌株的角色发生了反转。用一个应变的参数值作为原点，另一个应变的参数值变化。第三条曲线表示权衡。所有三条曲线都经过tip的原点或尖端。我们证明，在这一点上，入侵边界是相切的。在起始点为进化奇点的奇点上，入侵边界和权衡曲线都是切向的。权衡曲线的曲率决定了它进入奇异TIP的区域。这些区域中的每一个都具有特定的进化特性（EUS、CS、SPR和MI）。因此，我们根据权衡曲线和入侵边界的相对曲率，通过直接几何参数确定这些性质的每个条件。证明了这些条件等价于自适应动力学的标准偏导数条件。我们的结果的意义在于，我们可以通过观察权衡曲线在哪个区域进入奇异TIP来确定奇异策略是吸引点、分支点还是排斥点等。特别地，我们发现当且仅当TIP有一个互不可达的区域时，奇异策略有可能是分支点。我们用一个例子来说明这个理论，并指出前进的方向。(C) 2004 Elsevier Ltd.版权所有。
The purpose of this paper is to take an entirely geometrical path to determine the evolutionary properties of ecological systems subject to trade-offs. In particular we classify evolutionary singularities in a geometrical fashion. To achieve this, we study trade-off and invasion plots (TIPs) which show graphically the outcome of evolution from the relationship between three curves. The first invasion boundary (curve) has one strain as resident and the other strain as putative invader and the second has the roles of the strains reversed. The parameter values for one strain are used as the origin with those of the second strain varying. The third curve represents the trade-off. All three curves pass through the origin or tip of the TIP. We show that at this point the invasion boundaries are tangential. At a singular TIP, in which the origin is an evolutionary singularity, the invasion boundaries and trade-off curve are all tangential. The curvature of the trade-off curve deter-mines the region in which it enters the singular TIP. Each of these regions has particular evolutionary properties (EUS, CS, SPR and MI). Thus we determine by direct geometric argument conditions for each of these properties in terms of the relative curvatures of the trade-off curve and invasion boundaries. We show that these conditions are equivalent to the standard partial derivative conditions of adaptive dynamics. The significance of our results is that we can deter-mine whether the singular strategy is an attractor, branching point, repellor, etc. simply by observing in which region the trade-off curve enters the singular TIP. In particular we find that, if and only if the TIP has a region of mutual invadability, is it possible for the singular strategy to be a branching point. We illustrate the theory with an example and point the way forward. (C) 2004 Elsevier Ltd. All rights reserved.