Concentration in selection-mutation models: error estimates and asymptotic expansions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918203284783104 |
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| author | Guinet, Caroline Mirrahimi, Sepideh Roquejoffre, Jean-Michel |
| author_facet | Guinet, Caroline Mirrahimi, Sepideh Roquejoffre, Jean-Michel |
| contents | In this paper, we study an integro-differential equation which describes the evolutionary dynamics of a population structured by a phenotypic trait. This population undergoes asexual reproduction, competition, selection, and mutation. We provide an asymptotic analysis of the model, assuming that the mutations have small effects. A standard approach for the analysis of the qualitative properties of the solutions of such an equation is to apply a logarithmic transformation, which yields a Hamilton-Jacobi equation with constraint. When the reproduction term is a concave function of the trait, it has been established that the solution is classical. We rigorously derive a first-order asymptotic expansion of the solution. This expansion allows us to approximate the moments of the phenotypic density. This result establishes a connection between the approximations of the phenotypic density obtained via the Hamilton-Jacobi approach and relevant biological quantities, which are more suitable from a modeling perspective. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_12141 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Concentration in selection-mutation models: error estimates and asymptotic expansions Guinet, Caroline Mirrahimi, Sepideh Roquejoffre, Jean-Michel Analysis of PDEs 35B25, 35C20, 35F21, 35Q92, 92D15 In this paper, we study an integro-differential equation which describes the evolutionary dynamics of a population structured by a phenotypic trait. This population undergoes asexual reproduction, competition, selection, and mutation. We provide an asymptotic analysis of the model, assuming that the mutations have small effects. A standard approach for the analysis of the qualitative properties of the solutions of such an equation is to apply a logarithmic transformation, which yields a Hamilton-Jacobi equation with constraint. When the reproduction term is a concave function of the trait, it has been established that the solution is classical. We rigorously derive a first-order asymptotic expansion of the solution. This expansion allows us to approximate the moments of the phenotypic density. This result establishes a connection between the approximations of the phenotypic density obtained via the Hamilton-Jacobi approach and relevant biological quantities, which are more suitable from a modeling perspective. |
| title | Concentration in selection-mutation models: error estimates and asymptotic expansions |
| topic | Analysis of PDEs 35B25, 35C20, 35F21, 35Q92, 92D15 |
| url | https://arxiv.org/abs/2511.12141 |