How does the interplay between selection, mutation and horizontal gene transfer modify the phenotypic distribution of a bacterial or cell population? While horizontal gene transfer, which corresponds to the exchange of genetic material between individuals, has a major role in the adaptation of many organisms, its impact on the phenotypic density of populations is not yet fully understood.

We study an elliptic integro-differential equation describing the evolutionary equilibrium of the phenotypic density of an asexual population. In a regime of small mutational variance, we characterize the solution which results from the balance between competition for a resource, mutation and horizontal gene transfer. We show that in a certain range of parameters polymorphic equilibria exist, which means that the phenotypic density may concentrate around several dominant traits. Such polymorphic equilibria result from an antagonist interplay between horizontal gene transfer and selection, while similar models which neglect the transfer lead only to monomorphic equilibria.

选择、突变和水平基因转移之间的相互作用如何改变细菌或细胞群体的表型分布？虽然水平基因转移（相当于个体之间遗传物质的交换）在许多生物体的适应中发挥着重要作用，但其对种群表型密度的影响尚未完全了解。

我们研究了一个描述无性种群表型密度进化平衡的椭圆积分微分方程。在小突变方差的情况下，我们描述了资源竞争、突变和水平基因转移之间的平衡所产生的解决方案。我们表明，在一定的参数范围内存在多态平衡，这意味着表型密度可能集中在几个显性性状周围。这种多态性平衡是由水平基因转移和选择之间的拮抗相互作用引起的，而忽略转移的类似模型仅导致单态平衡。