A set of axioms is formulated characterizing ecologically plausible community dynamics. Using these axioms, it is proved that the transients following an invasion into a sufficiently stable equilibrium community by a mutant phenotype similar to one of the community's finitely many resident phenotypes can always be approximated by means of an appropriately chosen Lotka-Volterra model. To this end, the assumption is made that similar phenotypes in the community form clusters that are well-separated from each other, as is expected to be generally the case when evolution proceeds through small mutational steps. Each phenotypic cluster is represented by a single phenotype, which we call an approximate phenotype and assign the cluster's total population density. We present our results in three steps. First, for a set of approximate phenotypes with arbitrary equilibrium population densities before the invasion, the Lotka-Volterra approximation is proved to apply if the changes of the population densities of these phenotypes are sufficiently small during the transient following the invasion. Second, quantitative conditions for such small changes of population densities are derived as a relationship between within-cluster differences and the leading eigenvalue of the community's Jacobian matrix evaluated at the equilibrium population densities before the invasion. Third, to demonstrate the utility of our results, the 'invasion implies substitution' result for monomorphic populations is extended to arbitrarily polymorphic populations consisting of well-recognizable and -separated clusters.

中文翻译：

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进化特质替代过程的Lotka-Volterra近似。

制定了一系列公理来表征生态上合理的社区动态。使用这些公理，证明了通过类似于适当地选择的Lotka-Volterra模型的突变表型，入侵到足够稳定的平衡族之后的瞬态，总是可以近似地近似于该族的有限多个常驻表型之一。为此，假定群落中相似的表型形成彼此良好分离的簇，如预期通过小突变步骤进行进化的情况一般。每个表型簇由一个表型代表，我们称其为近似表型，并为其分配总种群密度。我们分三个步骤介绍结果。第一，对于入侵前具有任意均衡种群密度的一组近似表型，如果入侵后的瞬态过程中这些表型的种群密度变化足够小，则证明可以使用Lotka-Volterra近似。第二，人口密度如此微小变化的定量条件是通过集群内部差异与入侵前在平衡人口密度下评估的社区雅可比矩阵的领先特征值之间的关系得出的。第三，为了证明我们的结果的实用性，单态种群的“入侵意味着替代”结果扩展到了由公认的和分离的簇组成的任意多态种群。