<figure><img src="image_14.png" /><figcaption>FIGURE 11. (a) Zoom near the origin of the relaxation of the normalized profiles $F_n/||F_n||_{L^1(\mathbb{R}^d)}$ towards the eigenfunction $F_\alpha$ along the time iterations $n = 0, 1, 2$ and $15$ for the step function (7.1) as initial datum and strong selection parameter $\alpha = 0.4$. The vertical dotted line represents the location of the mean of the equilibrium profile $F_\alpha$. (b) Relaxation of the growth rates $||F_n||_{L^1(\mathbb{R}^d)}/||F_{n-1}||_{L^1(\mathbb{R}^d)}$ towards the eigenvalue $\lambda_\alpha$.</figcaption></figure>

the survival probability of individuals. Our main result provides quantitative convergence rates of the renormalized distributions towards a unique stationary profile. Indeed, rates are exponential, which can be interpreted loosely as a spectral gap in this non-linear context.

It is noticeable that the sexual reproduction operator is contractive under the Wasserstein distance (see Lemma 2.1). However, the generic incompatibility of multiplicative selection with transport distances becomes an apparent obstruction to the use of a direct perturbative approach in the regime of weak selection (see Example 2.4). This obstacle was circumvented by G. RAOUL [21], assuming that selection is restricted to a compact support. We follow a different route in a purely non-perturbative setting. To this end, we restrict to the specific choice of a quadratic selection function of the trait, for the sake of simplicity. Our alternative approach relies on an appropriate study of the propagation of information along large binary trees of ancestors, by a suitable reformulation of high-dimensional integrals, inspired by the changes of variables performed in [10] in the regime of small variance. This reveals an ergodicity property where the exact shape of the initial distribution is quickly forgotten across generations.

We remark that the above heuristic arguments indicate that the quadratic Wasserstein metric is not fully appropriate for dealing with the problem at hand. However, we could not identify yet a proper metric that extends the contraction property of the neutral case to the quadratic selection case.

Several perspectives are envisaged. First, there is an apparent price to pay with our method, in which we fully exploit the Gaussian structure induced by quadratic selection in order to perform tractable computations within the high-dimensional integrals. We believe though that the restriction to quadratic selection might be overcome in future works to allow for more general selection functions. Second, as studied in [10], the case of multiple minima on the selection function leads to non-uniqueness of stable equilibria. Then, uncovering the hidden metastability and quantifying the relaxation towards a specific equilibrium is of great interest for its applications in quantitative genetics. Finally, a major problem is to transcend non-overlapping generations and tackle the full time-continuous model as presented in previous literature.

## APPENDIX A. NONDIMENSIONALIZATION AND DERIVATION OF THE TIME-DISCRETE VERSION

In this section, we shall nondimensionalize the time-continuous model (1.6)-(1.3) with Gaussian mixing kernel $G$ and quadratic selection function $m$. By time discretization on the Duhamel formulation, we will derive the time-discrete version (1.1) which has been central in this paper. Indeed, we shall show that we can reduce parameters into only one, namely, parameter $\alpha \in \mathbb{R}_+$ in the quadratic selection function (1.5). Bearing in mind all the biological parameters and dimensions, our evolution problem reads:

$$ \left\{ \begin{array}{ll} \partial_t f = -r_m m(x)f + r_b \mathcal{B}_{\sigma^2}[f], & t \ge 0, x \in \mathbb{R}^d \\ f(0,x) = f_0(x), & x \in \mathbb{R}^d. \end{array} \right. \qquad (A.1) $$
<figure><img src="image_14.png" /><figcaption>FIGURE 11. (a) Zoom near the origin of the relaxation of the normalized profiles $F_n/||F_n||_{L^1(\mathbb{R}^d)}$ towards the eigenfunction $F_\alpha$ along the time iterations $n = 0, 1, 2$ and $15$ for the step function (7.1) as initial datum and strong selection parameter $\alpha = 0.4$. The vertical dotted line represents the location of the mean of the equilibrium profile $F_\alpha$. (b) Relaxation of the growth rates $||F_n||_{L^1(\mathbb{R}^d)}/||F_{n-1}||_{L^1(\mathbb{R}^d)}$ towards the eigenvalue $\lambda_\alpha$.</figcaption></figure>

the survival probability of individuals. Our main result provides quantitative convergence rates of the renormalized distributions towards a unique stationary profile. Indeed, rates are exponential, which can be interpreted loosely as a spectral gap in this non-linear context.

It is noticeable that the sexual reproduction operator is contractive under the Wasserstein distance (see Lemma 2.1). However, the generic incompatibility of multiplicative selection with transport distances becomes an apparent obstruction to the use of a direct perturbative approach in the regime of weak selection (see Example 2.4). This obstacle was circumvented by G. RAOUL [21], assuming that selection is restricted to a compact support. We follow a different route in a purely non-perturbative setting. To this end, we restrict to the specific choice of a quadratic selection function of the trait, for the sake of simplicity. Our alternative approach relies on an appropriate study of the propagation of information along large binary trees of ancestors, by a suitable reformulation of high-dimensional integrals, inspired by the changes of variables performed in [10] in the regime of small variance. This reveals an ergodicity property where the exact shape of the initial distribution is quickly forgotten across generations.

We remark that the above heuristic arguments indicate that the quadratic Wasserstein metric is not fully appropriate for dealing with the problem at hand. However, we could not identify yet a proper metric that extends the contraction property of the neutral case to the quadratic selection case.

Several perspectives are envisaged. First, there is an apparent price to pay with our method, in which we fully exploit the Gaussian structure induced by quadratic selection in order to perform tractable computations within the high-dimensional integrals. We believe though that the restriction to quadratic selection might be overcome in future works to allow for more general selection functions. Second, as studied in [10], the case of multiple minima on the selection function leads to non-uniqueness of stable equilibria. Then, uncovering the hidden metastability and quantifying the relaxation towards a specific equilibrium is of great interest for its applications in quantitative genetics. Finally, a major problem is to transcend non-overlapping generations and tackle the full time-continuous model as presented in previous literature.

## APPENDIX A. NONDIMENSIONALIZATION AND DERIVATION OF THE TIME-DISCRETE VERSION

In this section, we shall nondimensionalize the time-continuous model (1.6)-(1.3) with Gaussian mixing kernel $G$ and quadratic selection function $m$. By time discretization on the Duhamel formulation, we will derive the time-discrete version (1.1) which has been central in this paper. Indeed, we shall show that we can reduce parameters into only one, namely, parameter $\alpha \in \mathbb{R}_+$ in the quadratic selection function (1.5). Bearing in mind all the biological parameters and dimensions, our evolution problem reads:

$$ \left\{ \begin{array}{ll} \partial_t f = -r_m m(x)f + r_b \mathcal{B}_{\sigma^2}[f], & t \ge 0, x \in \mathbb{R}^d \\ f(0,x) = f_0(x), & x \in \mathbb{R}^d. \end{array} \right. \qquad (A.1) $$
Here, $r_m, r_b \in \mathbb{R}_+$ represent the mortality and birth rates and have frequency units, whilst the dimensionless corrections of $m$ and $\mathcal{B}$ take the form

$$m(x) = \frac{|x|^2}{\sigma_m^2}, \quad \mathcal{B}_{\sigma^2}[f](t, x) = \int_{\mathbb{R}^{2d}} G_{0,\sigma^2} \left( x - \frac{x_1 + x_2}{2} \right) \frac{f(t, x_1)f(t, x_2)}{\int_{\mathbb{R}^d} f(t, x') dx'} dx_1 dx_2,$$

for each $x \in \mathbb{R}^d$. Here, $G_{0,\sigma^2}$ represents the Gaussian centered at 0 with variance $\sigma^2$, and $\sigma_m$ and $\sigma$ have the same units as the quantitative trait. On the one hand, $\sigma_m$ can be regarded as a characteristic unit quantifying the effective range of selection. Namely, if $|x| \ge \sigma_m$ then $m(x) \ge 1$. On the other hand, $\sigma^2$ is the genetic variance and $2\sigma^2$ represents the variance at linkage equilibrium often denoted by $\sigma_{LE}^2 := 2\sigma^2$ (cf. [9]). We nondimensionalize our system by appropriately scaling our variables as follows

$$\hat{t} = \frac{t}{T}, \quad \hat{x} = \frac{x}{L}, \quad \hat{f}(\hat{t}, \hat{x}) = L^d f(t, x).$$

Here, $T$ and $L$ are characteristic units for time and trait. For simplicity, we set them as follows in terms of the parameters of the system:

$$T := \frac{1}{r_m}, \quad L := \sigma.$$

In other words, we scale time according to the mortality rate $r_m$ and the trait variable according to the genetic variance $\sigma^2$. Dropping hats for simplicity yields the following dimensionless form of the equation

$$\left\{ \begin{array}{ll} \partial_t f = -\frac{\alpha}{2}|x|^2 f + \beta \mathcal{B}[f], & t \ge 0, x \in \mathbb{R}^d, \\ f(0, x) = f_0(x), & x \in \mathbb{R}^d, \end{array} \right. \qquad (A.2)$$

where $\mathcal{B} = \mathcal{B}_1$ involves again unit genetic variance thanks to our scaling assumption. This reduces the amount of free parameters to only two of them $\alpha, \beta \in \mathbb{R}_+$ defined as follows:

$$\alpha := \frac{\sigma_{LE}^2}{\sigma_m^2}, \quad \beta := \frac{r_b}{r_m}.$$

In the sequel, we justify our time-discrete version (1.11). Specifically, let us integrate equation (A.2)$_1$ for $t \in [t_n, t_{n+1}]$ where $\{t_n\}_{n \in \mathbb{N}} \subseteq \mathbb{R}_+$ is any increasing sequence. Then, we recover Duhamel's formula

$$f(t_n, x) = e^{-\frac{\alpha}{2}|x|^2(t_n - t_{n-1})} f(t_{n-1}, x) + \beta \int_{t_{n-1}}^{t_n} e^{-\frac{\alpha}{2}|x|^2(t_n - s)} \mathcal{B}[f(s, \cdot)](x) ds,$$

for any $n \in \mathbb{N}$ and each $x \in \mathbb{R}^d$. Using the rectangle rule for the integral in the right hand side leads to the approximations $F_n(x) \approx f(t_n, x)$ of the above time-continuous problem for $f = f(t, x)$:

$$F_n(x) = e^{-\frac{\alpha}{2}|x|^2} F_{n-1}(x) + \beta e^{-\frac{\alpha}{2}|x|^2} \mathcal{B}[F_{n-1}](x), \qquad (A.3)$$

for any $n \in \mathbb{N}$ and each $x \in \mathbb{R}^d$. Here, we have set unitary time steps $t_n - t_{n-1} = 1$ for simplicity and $t_0 = 0$. We emphasize that the right hand side of (A.3) consists of two different terms. On the one hand, the first term describes the amount of old individuals from the previously generation $n-1$ having resisted the effect of selection in generation $n$. On the other hand, the second term computes the amount of offspring conceived by individuals in the previous generation $n-1$, where an eventual decline due to selection has also been taken into account. Let us emphasize that in (A.3) individuals from generation $n-1$ can merge (after breeding) with their offspring to form next generation $n$. This is typical in most biological populations, which have *overlapping generations* (e.g., some multivoltine flies like Drosophila melanogaster). However, some other systems have *non-overlapping generations* so that the full generation $n-1$ gets extinct after breeding and only their offspring survive at generation $n$ (e.g., some univoltine insects like the Dawson's burrowing bee). We refer to [17] for a discussion on the various types of voltinism and the role of diapause. In that case, by canceling the first term (A.3) reduces to

$$F_n(x) = \beta e^{-\frac{\alpha}{2}|x|^2} \mathcal{B}[F_{n-1}](x), \qquad (A.4)$$

for any $n \in \mathbb{N}$ and each $x \in \mathbb{R}$. We remark that two different parameters $\alpha$ and $\beta$ are still present in (A.4). However, given that $F \mapsto e^{-\frac{\alpha}{2}|x|^2} \mathcal{B}[F]$ is a 1-homogeneous operator we can kill the parameter $\beta$ by rescaling the trait distributions $F_n$. Specifically, define $\hat{F}_n = \beta^{-n} F_n$ and note that (A.4) reduces to our time-discrete problem (1.1) with only one parameter $\alpha$.
Here, $r_m, r_b \in \mathbb{R}_+$ represent the mortality and birth rates and have frequency units, whilst the dimensionless corrections of $m$ and $\mathcal{B}$ take the form

$$m(x) = \frac{|x|^2}{\sigma_m^2}, \quad \mathcal{B}_{\sigma^2}[f](t, x) = \int_{\mathbb{R}^{2d}} G_{0,\sigma^2} \left( x - \frac{x_1 + x_2}{2} \right) \frac{f(t, x_1)f(t, x_2)}{\int_{\mathbb{R}^d} f(t, x') dx'} dx_1 dx_2,$$

for each $x \in \mathbb{R}^d$. Here, $G_{0,\sigma^2}$ represents the Gaussian centered at 0 with variance $\sigma^2$, and $\sigma_m$ and $\sigma$ have the same units as the quantitative trait. On the one hand, $\sigma_m$ can be regarded as a characteristic unit quantifying the effective range of selection. Namely, if $|x| \ge \sigma_m$ then $m(x) \ge 1$. On the other hand, $\sigma^2$ is the genetic variance and $2\sigma^2$ represents the variance at linkage equilibrium often denoted by $\sigma_{LE}^2 := 2\sigma^2$ (cf. [9]). We nondimensionalize our system by appropriately scaling our variables as follows

$$\hat{t} = \frac{t}{T}, \quad \hat{x} = \frac{x}{L}, \quad \hat{f}(\hat{t}, \hat{x}) = L^d f(t, x).$$

Here, $T$ and $L$ are characteristic units for time and trait. For simplicity, we set them as follows in terms of the parameters of the system:

$$T := \frac{1}{r_m}, \quad L := \sigma.$$

In other words, we scale time according to the mortality rate $r_m$ and the trait variable according to the genetic variance $\sigma^2$. Dropping hats for simplicity yields the following dimensionless form of the equation

$$\left\{ \begin{array}{ll} \partial_t f = -\frac{\alpha}{2}|x|^2 f + \beta \mathcal{B}[f], & t \ge 0, x \in \mathbb{R}^d, \\ f(0, x) = f_0(x), & x \in \mathbb{R}^d, \end{array} \right. \qquad (A.2)$$

where $\mathcal{B} = \mathcal{B}_1$ involves again unit genetic variance thanks to our scaling assumption. This reduces the amount of free parameters to only two of them $\alpha, \beta \in \mathbb{R}_+$ defined as follows:

$$\alpha := \frac{\sigma_{LE}^2}{\sigma_m^2}, \quad \beta := \frac{r_b}{r_m}.$$

In the sequel, we justify our time-discrete version (1.11). Specifically, let us integrate equation (A.2)$_1$ for $t \in [t_n, t_{n+1}]$ where $\{t_n\}_{n \in \mathbb{N}} \subseteq \mathbb{R}_+$ is any increasing sequence. Then, we recover Duhamel's formula

$$f(t_n, x) = e^{-\frac{\alpha}{2}|x|^2(t_n - t_{n-1})} f(t_{n-1}, x) + \beta \int_{t_{n-1}}^{t_n} e^{-\frac{\alpha}{2}|x|^2(t_n - s)} \mathcal{B}[f(s, \cdot)](x) ds,$$

for any $n \in \mathbb{N}$ and each $x \in \mathbb{R}^d$. Using the rectangle rule for the integral in the right hand side leads to the approximations $F_n(x) \approx f(t_n, x)$ of the above time-continuous problem for $f = f(t, x)$:

$$F_n(x) = e^{-\frac{\alpha}{2}|x|^2} F_{n-1}(x) + \beta e^{-\frac{\alpha}{2}|x|^2} \mathcal{B}[F_{n-1}](x), \qquad (A.3)$$

for any $n \in \mathbb{N}$ and each $x \in \mathbb{R}^d$. Here, we have set unitary time steps $t_n - t_{n-1} = 1$ for simplicity and $t_0 = 0$. We emphasize that the right hand side of (A.3) consists of two different terms. On the one hand, the first term describes the amount of old individuals from the previously generation $n-1$ having resisted the effect of selection in generation $n$. On the other hand, the second term computes the amount of offspring conceived by individuals in the previous generation $n-1$, where an eventual decline due to selection has also been taken into account. Let us emphasize that in (A.3) individuals from generation $n-1$ can merge (after breeding) with their offspring to form next generation $n$. This is typical in most biological populations, which have *overlapping generations* (e.g., some multivoltine flies like Drosophila melanogaster). However, some other systems have *non-overlapping generations* so that the full generation $n-1$ gets extinct after breeding and only their offspring survive at generation $n$ (e.g., some univoltine insects like the Dawson's burrowing bee). We refer to [17] for a discussion on the various types of voltinism and the role of diapause. In that case, by canceling the first term (A.3) reduces to

$$F_n(x) = \beta e^{-\frac{\alpha}{2}|x|^2} \mathcal{B}[F_{n-1}](x), \qquad (A.4)$$

for any $n \in \mathbb{N}$ and each $x \in \mathbb{R}$. We remark that two different parameters $\alpha$ and $\beta$ are still present in (A.4). However, given that $F \mapsto e^{-\frac{\alpha}{2}|x|^2} \mathcal{B}[F]$ is a 1-homogeneous operator we can kill the parameter $\beta$ by rescaling the trait distributions $F_n$. Specifically, define $\hat{F}_n = \beta^{-n} F_n$ and note that (A.4) reduces to our time-discrete problem (1.1) with only one parameter $\alpha$.
REFERENCES

1. L. Ambrosio, N. Gigli, and G. Savaré, Gradient flows in metric spaces and in the space of probability measures, Birkhäuser, Basel, 2008.

2. D. Bakry, I. Gentil, and M. Ledoux, Analysis and Geometry of Markov Diffusion Operators, A Series of Comprehensive Studies in Mathematics, vol. 348, Springer, Cham, 2014.

3. G. Barles and S. Mirrahimi, Concentrations and constrained Hamilton-Jacobi equations arising in adaptive dynamics, Recent Developments in Nonlinear Partial Differential Equations (D. Danielli, ed.), Contemp. Math, vol. 439, American Mathematical Society, Providence, RI, 2007, pp. 57–68.

4. G. Barles, S. Mirrahimi, and B. Perthame, Concentration in Lotka-Volterra parabolic or integral equations: a general convergence result, Methods App. Anal. 16 (2009), no. 3, 321–340.

5. N. H. Barton, A. M. Etheridge, and A. Véber, The infinitesimal model: Definition, derivation, and implications, Theoret. Population Biol. 118 (2017), 50–73.

6. H. Berestycki, J. Coville, and H.-H. Vo, Persistence criteria for populations with non-local dispersion, J. Math. Biol. 72 (2016), 1693–1745.

7. F. Bolley, I. Gentil, and A. Guillin, Convergence to equilibrium in Wasserstein distance for Fokker-Planck equations, J. Funct. Anal. 263 (2012), no. 8, 2430–2457.

8. T. Bourgeron, V. Calvez, J. Garnier, and L. Lepoutre, Existence of recombination-selection equilibria for sexual populations, 2017, arXiv:1703.09078.

9. M. G. Bulmer, The mathematical theory of quantitative genetics, Clarendon Press, Oxford, 1980.

10. V. Calvez, J. Garnier, and F. Patout, Asymptotic analysis of a quantitative genetics model with nonlinear integral operator, J. Éc. polytech. Math. 6 (2019), 537–579.

11. V. Calvez and K.-Y. Lam, Uniqueness of the viscosity solution of a constrained Hamilton-Jacobi equation, Calc. Var. 59 (2020), no. 163.

12. V. Calvez, D. Poyato, and F. Santambrogio, Uniform contractivity of the Fisher infinitesimal model with strongly convex selection, 2023, arXiv:2302.12063.

13. J. Coville, F. Li, and X. Wang, On eigenvalue problems arising from nonlocal diffusion models, Discrete Contin. Dyn. Syst. Ser. A 37 (2017), no. 2, 879–903.

14. O. Diekmann, P.-E. Jabin, S. Mischler, and B. Perthame, The dynamics of adaptation: an ilumination example and a Hamilton-Jacobi approach, Theoret. Population Biol. 67 (2005), no. 4, 25–271.

15. R. A. Fisher, The correlation between relatives on the supposition of mendelian inheritance, Trans. Roy. Soc. Edinburgh 52 (1918), 399–433.

16. R. Mahadevan, A note on a non-linear Krein-Rutman theorem, Nonlinear Anal. Theory Methods Appl. 67 (2007), no. 11, 3084–3090.

17. S. Masaki, Geographic variation of diapause in insects, Bull. Fac. Agric. Hirosaki Univ. 7 (1961), 66–98.

18. S. Mirrahimi and G. Raoul, Dynamics of sexual populations structured by a space variable and a phenotypical trait, Theoret. Population Biol. 84 (2013), 87–103.

19. F. Patout, The cauchy problem for the infinitesimal model in the regime of small variance, Anal. PDE 16 (2023), no. 6, 1289–1350.

20. G. Raoul, Macroscopic limit from a structured population model to the Kirkpatrick-Barton model, 2017, arXiv:1706.04094.

21. Exponential convergence to a steady-state for a population genetics model with sexual reproduction and selection, 2021, arXiv:2104.06089.

22. K. Rosen, Discrete Mathematics and Its Applications, McGraw-Hill Science, New York, 2011.

23. M. Talagrand, Transportation cost for Gaussian and other measures, Geom. Funct. Anal. 6 (1996), 587–600.

24. C. Villani, Optimal transport: old and new, Grundlehren der mathematischen Wissenschaften, vol. 338, Springer, Berlin, Heidelberg, 2009.

25. S. G. Williamson, Combinatorics for computer science, Computer Science Press, Rockville, Maryland, 1985.

INSTITUT CAMILLE JORDAN (ICJ), UMR 5208 CNRS & UNIVERSITÉ CLAUDE BERNARD LYON 1, 69100 VILLEURBANNE, FRANCE

Email address: vincent.calvez@math.cnrs.fr

UNIV LYON, INRIA, UNIVERSITÉ CLAUDE BERNARD LYON 1, CNRS UMR5208, INSTITUT CAMILLE JORDAN, F-69603 VILLEURBANNE, FRANCE

Email address: thomas.lepoutre@inria.fr

DEPARTAMENTO DE MATEMÁTICA APLICADA AND RESEARCH UNIT “MODELING NATURE” (MNAT), FACULTAD DE CIENCIAS, UNIVERSIDAD DE GRANADA, 18071 GRANADA, SPAIN

Email address: davidpoyato@ugr.es
REFERENCES

1. L. Ambrosio, N. Gigli, and G. Savaré, Gradient flows in metric spaces and in the space of probability measures, Birkhäuser, Basel, 2008.

2. D. Bakry, I. Gentil, and M. Ledoux, Analysis and Geometry of Markov Diffusion Operators, A Series of Comprehensive Studies in Mathematics, vol. 348, Springer, Cham, 2014.

3. G. Barles and S. Mirrahimi, Concentrations and constrained Hamilton-Jacobi equations arising in adaptive dynamics, Recent Developments in Nonlinear Partial Differential Equations (D. Danielli, ed.), Contemp. Math, vol. 439, American Mathematical Society, Providence, RI, 2007, pp. 57–68.

4. G. Barles, S. Mirrahimi, and B. Perthame, Concentration in Lotka-Volterra parabolic or integral equations: a general convergence result, Methods App. Anal. 16 (2009), no. 3, 321–340.

5. N. H. Barton, A. M. Etheridge, and A. Véber, The infinitesimal model: Definition, derivation, and implications, Theoret. Population Biol. 118 (2017), 50–73.

6. H. Berestycki, J. Coville, and H.-H. Vo, Persistence criteria for populations with non-local dispersion, J. Math. Biol. 72 (2016), 1693–1745.

7. F. Bolley, I. Gentil, and A. Guillin, Convergence to equilibrium in Wasserstein distance for Fokker-Planck equations, J. Funct. Anal. 263 (2012), no. 8, 2430–2457.

8. T. Bourgeron, V. Calvez, J. Garnier, and L. Lepoutre, Existence of recombination-selection equilibria for sexual populations, 2017, arXiv:1703.09078.

9. M. G. Bulmer, The mathematical theory of quantitative genetics, Clarendon Press, Oxford, 1980.

10. V. Calvez, J. Garnier, and F. Patout, Asymptotic analysis of a quantitative genetics model with nonlinear integral operator, J. Éc. polytech. Math. 6 (2019), 537–579.

11. V. Calvez and K.-Y. Lam, Uniqueness of the viscosity solution of a constrained Hamilton-Jacobi equation, Calc. Var. 59 (2020), no. 163.

12. V. Calvez, D. Poyato, and F. Santambrogio, Uniform contractivity of the Fisher infinitesimal model with strongly convex selection, 2023, arXiv:2302.12063.

13. J. Coville, F. Li, and X. Wang, On eigenvalue problems arising from nonlocal diffusion models, Discrete Contin. Dyn. Syst. Ser. A 37 (2017), no. 2, 879–903.

14. O. Diekmann, P.-E. Jabin, S. Mischler, and B. Perthame, The dynamics of adaptation: an ilumination example and a Hamilton-Jacobi approach, Theoret. Population Biol. 67 (2005), no. 4, 25–271.

15. R. A. Fisher, The correlation between relatives on the supposition of mendelian inheritance, Trans. Roy. Soc. Edinburgh 52 (1918), 399–433.

16. R. Mahadevan, A note on a non-linear Krein-Rutman theorem, Nonlinear Anal. Theory Methods Appl. 67 (2007), no. 11, 3084–3090.

17. S. Masaki, Geographic variation of diapause in insects, Bull. Fac. Agric. Hirosaki Univ. 7 (1961), 66–98.

18. S. Mirrahimi and G. Raoul, Dynamics of sexual populations structured by a space variable and a phenotypical trait, Theoret. Population Biol. 84 (2013), 87–103.

19. F. Patout, The cauchy problem for the infinitesimal model in the regime of small variance, Anal. PDE 16 (2023), no. 6, 1289–1350.

20. G. Raoul, Macroscopic limit from a structured population model to the Kirkpatrick-Barton model, 2017, arXiv:1706.04094.

21. Exponential convergence to a steady-state for a population genetics model with sexual reproduction and selection, 2021, arXiv:2104.06089.

22. K. Rosen, Discrete Mathematics and Its Applications, McGraw-Hill Science, New York, 2011.

23. M. Talagrand, Transportation cost for Gaussian and other measures, Geom. Funct. Anal. 6 (1996), 587–600.

24. C. Villani, Optimal transport: old and new, Grundlehren der mathematischen Wissenschaften, vol. 338, Springer, Berlin, Heidelberg, 2009.

25. S. G. Williamson, Combinatorics for computer science, Computer Science Press, Rockville, Maryland, 1985.

INSTITUT CAMILLE JORDAN (ICJ), UMR 5208 CNRS & UNIVERSITÉ CLAUDE BERNARD LYON 1, 69100 VILLEURBANNE, FRANCE

Email address: vincent.calvez@math.cnrs.fr

UNIV LYON, INRIA, UNIVERSITÉ CLAUDE BERNARD LYON 1, CNRS UMR5208, INSTITUT CAMILLE JORDAN, F-69603 VILLEURBANNE, FRANCE

Email address: thomas.lepoutre@inria.fr

DEPARTAMENTO DE MATEMÁTICA APLICADA AND RESEARCH UNIT “MODELING NATURE” (MNAT), FACULTAD DE CIENCIAS, UNIVERSIDAD DE GRANADA, 18071 GRANADA, SPAIN

Email address: davidpoyato@ugr.es