Asymptotic analysis of a quantitative genetics model with nonlinear integral operator (Q2315131)

The paper deals with the following stationary problem which is a quantitative genetics model \[ \lambda_{\varepsilon}F_{\varepsilon}(z)+m(z)F_{\varepsilon}(z)=\mathcal{B}_{\varepsilon}(F_{\varepsilon})(z),\,\,\,z\in\mathbb{R}^d,\tag{1} \] where \(\mathcal{B}_{\varepsilon}(f)\) is the nonlinear, homogeneous integral operator associated to an infinitesimal model, defined as \[ \mathcal{B}_{\varepsilon}(f)(z):=\frac{1}{\varepsilon^d\pi^{d/2}}\iint_{\mathbb{R}^{2d}}\exp\left[-\frac{1}{\varepsilon^2}\left(z-\frac{z_1+z_2}{2}\right)^2\right]f(z_1)\frac{f(z_2)}{\int_{\mathbb{R}^d}f(z_2')\,dz_2'}\,dz_1dz_2. \] Here \(z\) denotes the multi-dimensional phenotypic trait, \(F_{\varepsilon}(z)\) is the phenotypic distribution of the population, \(m(z)\) is the trait-dependent mortality rate, the mixing operator \(\mathcal{B}_{\varepsilon}\) represents the inheritance of quantitative traits in a population with a sexual mode of reproduction, and the parameter \(\frac{\varepsilon^2}{2}\) describes the deviation between the offspring and the mean parental trait. Under suitable regularity and growth assumptions on the mortality rate, the authors prove the existence and local uniqueness of solution \((\lambda_{\varepsilon},F_{\varepsilon})\in\mathbb{R}\times L^1(\mathbb{R})\) of problem (1), and its asymptotic behaviour as \(\varepsilon\) vanishes. In the proof of the main results they use some perturbative analysis techniques.