We study the long-time behaviour of phenotype-structured models describing the evolutionary dynamics of asexual species whose phenotypic fitness landscape is characterised by multiple peaks. First we consider the case where phenotypic variations do not occur, and then we include the effect of heritable phenotypic changes. In the former case the model is formulated in terms of an integrodifferential equation for the phenotype distribution of the individuals of the species, whereas in the latter case the evolution of the phenotype distribution is governed by a non-local parabolic equation whereby a linear diffusion operator captures the presence of phenotypic variations. We prove that the long-time limit of the solution to the integrodifferential equation is unique and given by a measure consisting of a weighted sum of Dirac masses centred at the peaks of the phenotypic fitness landscape. We also derive an explicit formula to compute the weights in front of the Dirac masses. Moreover, we demonstrate that the long-time solution of the non-local parabolic equation exhibits a qualitatively similar behaviour in the asymptotic regime where the diffusion coefficient modelling the rate of phenotypic variations tends to zero. However, we show that the limit measure of the non-local parabolic equation may consist of less Dirac masses, and we provide a sufficient criterion to identify the positions of their centres. Finally, we provide a detailed characterisation of the speed of convergence of the integral of the solution (i.e. the population size) to its long-time limit for both models. Taken together, our results support a more in-depth theoretical understanding of the conditions leading to the emergence of stable phenotypic polymorphism in asexual species.

中文翻译：

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存在多个适应度峰值的选择-突变模型的渐近分析

我们研究了表型结构模型的长期行为，这些模型描述了无性物种的进化动力学，其表型适应性景观以多个峰值为特征。首先我们考虑不发生表型变异的情况，然后我们包括可遗传表型变化的影响。在前一种情况下，该模型根据物种个体表型分布的积分微分方程来制定，而在后一种情况下，表型分布的演变由非局部抛物线方程控制，其中线性扩散算子捕捉表型变异的存在。我们证明了积分微分方程解的长时间限制是唯一的，并且由以表型适应度景观峰值为中心的狄拉克质量的加权总和组成的度量给出。我们还推导出了一个明确的公式来计算狄拉克质量前面的权重。此外，我们证明了非局部抛物线方程的长期解在渐近状态下表现出定性相似的行为，其中模拟表型变化率的扩散系数趋于零。然而，我们表明非局部抛物线方程的极限度量可能由较少的狄拉克质量组成，并且我们提供了一个足够的标准来识别它们的中心位置。最后，我们提供了两个模型的解决方案积分（即人口规模）收敛到其长期限制的速度的详细特征。总之，我们的结果支持对导致无性物种中稳定表型多态性出现的条件的更深入的理论理解。