多态演化序列与演化分支
Polymorphic evolution sequence and evolutionary branching
摘要：
我们感兴趣的研究模型描述的多态种群的进化与突变和选择的特定尺度的生物框架的自适应动力学。假设总体规模较大，突变率较小。我们证明了在这两个尺度的良好组合下，种群过程在突变的长时间尺度上可以用描述种群连续性状平衡的马尔可夫纯跳过程来近似。这一过程，即所谓的性状替代序列的推广，被称为多态进化序列。在此基础上，我们引入了一种突变大小的标度，研究了小突变极限下的多态进化序列。从这项研究在附近的进化奇点，我们得到一个完整的数学理由的启发式标准的现象进化分支。为此，我们精细地分析了三维竞争Lotka-Volterra系统的渐近行为。
英文摘要：
We are interested in the study of models describing the evolution of a polymorphic population with mutation and selection in the specific scales of the biological framework of adaptive dynamics. The population size is assumed to be large and the mutation rate small. We prove that under a good combination of these two scales, the population process is approximated in the long time scale of mutations by a Markov pure jump process describing the successive trait equilibria of the population. This process, which generalizes the so-called trait substitution sequence, is called polymorphic evolution sequence. Then we introduce a scaling of the size of mutations and we study the polymorphic evolution sequence in the limit of small mutations. From this study in the neighborhood of evolutionary singularities, we obtain a full mathematical justification of a heuristic criterion for the phenomenon of evolutionary branching. To this end we finely analyze the asymptotic behavior of 3-dimensional competitive Lotka-Volterra systems.
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