leads to

$\forall y\in\mathbb{R},\quad\underset{k\leq
K}{\min}\;\hat{u^{\mathcal{A}^{(k)}}}(y)\geq\frac{y^{2}}{2}=\underset{k\leq
K}{\max}\;\hat{u^{\mathcal{A}^{(k)}}}(y).$

Consequently, we deduce that $\underset{k\leq
K}{\min}\;\hat{u^{\mathcal{A}^{(k)}}}=\underset{k\leq
K}{\max}\;\hat{u^{\mathcal{A}^{(k)}}}$.

##### End of proof.

The last result implies that

$\forall k\leq K,\quad\forall
y\in\mathbb{R},\quad\hat{u^{\mathcal{A}^{(k)}}}(y)=\underset{k\leq
K}{\max}\;\hat{u^{\mathcal{A}^{(k)}}}(y)=\frac{y^{2}}{2}.$

From the latter we compute the bi-conjugates
$\hat{\hat{u^{\mathcal{A}^{(k)}}}}:z\mapsto\frac{(z-z^{*})^{2}}{2}$. Since
$z\mapsto\frac{z-z^{*}}{2}$ is strictly convex and it is the lower convex
envelope of $u^{\mathcal{A}^{(k)}}$, we obtain that

$\forall k\leq K,\quad\forall z\in\mathbb{R},\quad
u^{\mathcal{A}^{(k)}}(z)=\frac{(z-z^{*})^{2}}{2}.$

∎

##  Appendix C Formal justification of the constraints (C) on the main terms
$u_{0}^{A}$ and $u_{0}^{a}$

We drop the index $i$ indicating the habitat and the time dependence $t$ for
this appendix for the sake of simpler notations.

Let us first formally justify that $U_{0}^{A}$ and $U_{0}^{a}$ are positive
almost everywhere and cancelling somewhere. As we are interested in the
maintenance of the polymorphism at the major-effect locus, we consider that no
major-effect allele has yet fixed. Hence, $N^{A}_{\varepsilon}$ and
$N^{a}_{\varepsilon}$ need to remain positive and bounded when $\varepsilon$
vanishes. Using the Hopf-Cole transforms on $n^{A}_{\varepsilon}$ and
$n^{a}_{\varepsilon}$ (5) along with the formal Taylor expansions (6) on
$U^{A}_{\varepsilon}$ and $U^{a}_{\varepsilon}$ leads to

$N_{\varepsilon}^{A}=\displaystyle\int_{\mathbb{R}}n_{\varepsilon}^{A}(z^{\prime})\,dz^{\prime}=\displaystyle\int_{\mathbb{R}}\frac{1}{\sqrt{2\pi}\varepsilon}e^{-\frac{U^{A}_{\varepsilon}(z^{\prime})}{\varepsilon^{2}}}\,dz^{\prime}=\displaystyle\int_{\mathbb{R}}\frac{1}{\sqrt{2\pi}\varepsilon}e^{-\frac{u^{A}_{0}(z^{\prime})}{\varepsilon^{2}}}e^{-u_{1}^{A}+\varepsilon^{2}v^{A}_{\varepsilon}}\,dz^{\prime}.$
(27)

If we assume that the residues $u_{1}^{A}$ and $v_{\varepsilon}^{A}$ stay
bounded when $\varepsilon$ vanishes (as [CGP19] suggests it), then (27)
implies that $u^{A}_{0}$ must be non-negative for $N_{\varepsilon}^{A}$ to
remain bounded when $\varepsilon$ vanishes. For $N_{\varepsilon}^{A}$ not to
vanish asymptotically, $u^{A}_{0}$ must cancel. Moreover, for any interval
$I\subset\mathbb{R}$, $u^{A}_{0}$ cannot cancel on $I$, or we would have:

$N^{A}_{\varepsilon}>\displaystyle\int_{I}\frac{1}{\sqrt{2\pi}\varepsilon}e^{\frac{1}{\varepsilon^{2}}}e^{-u_{1}^{A}+\varepsilon^{2}v^{A}_{\varepsilon}}\,dz^{\prime}\rightarrow+\infty.$

So $u^{A}_{0}$ is positive almost everywhere, and cancelling somewhere. The
same holds for $u^{a}_{0}$.

Now, for determining the constraints $\eqref{constraint_u0}$, let us notice
that if we divide the right-hand side of the first equality of (3) by
$n^{A}_{\varepsilon}(z)$, the reproduction term
$\frac{\mathcal{B^{A}_{\varepsilon}}(n^{A}_{\varepsilon},n^{a}_{\varepsilon})(z)}{n^{A}_{\varepsilon}(z)}$
has to remain positive and bounded for all $z\in\mathbb{R}$ when $\varepsilon$
vanishes for the effect of reproduction to remain well-balanced with
selection, migration and competition. We assume henceforth that (6) is the
correct ansatz (as suggested by [CGP19]). Using the Hopf-Cole transforms on
$n^{A}_{\varepsilon}$ and $n^{a}_{\varepsilon}$ (5) along with the formal
Taylor expansions (6) on $U^{A}_{\varepsilon}$ and $U^{a}_{\varepsilon}$ in
(4) leads to

$\displaystyle\frac{\mathcal{B^{A}_{\varepsilon}}(n^{A}_{\varepsilon},n^{a}_{\varepsilon})(t,z)}{n^{A}_{\varepsilon}(z)}=\frac{\mathcal{B^{A}_{\varepsilon}}(n^{A}_{\varepsilon},n^{a}_{\varepsilon})(z)}{\frac{1}{\sqrt{2\pi}\varepsilon}e^{-\frac{u^{A}_{0}(z)}{\varepsilon^{2}}}e^{-u_{1}^{A}(z)+\mathcal{O}(\varepsilon^{2})}}$
$\displaystyle=\frac{\sqrt{2}}{N_{\varepsilon}^{A}}\times$
$\displaystyle\left[\int_{\mathbb{R}^{2}}\exp\left(\frac{1}{\varepsilon^{2}}\left[u_{0}^{A}(z)-\left(z-\frac{z_{1}+z_{2}}{2}\right)^{2}-u_{0}^{A}(z_{1})-u_{0}^{A}(z_{2})\right]\right)\exp\left(u_{1}^{A}(z)-u_{1}^{A}(z_{1})-u_{1}^{A}(z_{2})+\mathcal{O}(\varepsilon^{2})\right)dz_{1}dz_{2}\right.$
$\displaystyle\left.+\\!\int_{\mathbb{R}^{2}}\exp\left(\frac{1}{\varepsilon^{2}}\left[u_{0}^{A}(z)-\left(z-\frac{z_{1}+z_{2}}{2}\right)^{2}-u_{0}^{A}(z_{1})-u_{0}^{a}(z_{2})\right]\right)\exp\left(u_{1}^{A}(z)-u_{1}^{A}(z_{1})-u_{1}^{a}(z_{2})+\mathcal{O}(\varepsilon^{2})\right)dz_{1}dz_{2}\right]\\!\\!.$

As $N^{A}_{\varepsilon}$ remains bounded and does not vanish asymptotically,
we need the maximum of the two integrals nor to vanish, nor to diverge to
infinity when $\varepsilon$ vanishes, for all $z\in\mathbb{R}$. Therefore the
maximum of the terms into brackets that are multiplied by
$\frac{1}{\varepsilon^{2}}$ needs to be null for all $z\in\mathbb{R}$:

$\displaystyle\forall
z\in\mathbb{R},\quad\max\Big{[}\underset{z_{1},z_{2}}{\sup}\;u_{0}^{A}(z)$
$\displaystyle-\left(z-\frac{z_{1}+z_{2}}{2}\right)^{2}-u_{0}^{A}(z_{1})-u_{0}^{A}(z_{2}),$
$\displaystyle\underset{z_{1},z_{2}}{\sup}\;u_{0}^{A}(z)-\left(z-\frac{z_{1}+z_{2}}{2}\right)^{2}-u_{0}^{A}(z_{1})-u_{0}^{a}(z_{2})\Big{]}=0,$

which is the first constraint of (C). The same holds for
$\frac{\mathcal{B^{a}_{\varepsilon}}(n^{a}_{\varepsilon},n^{A}_{\varepsilon})(z)}{n^{a}_{\varepsilon}(z)}$,
which gives the second constraint of (C).

##  Appendix D Slow-fast analysis underlying the separation of time scales

Under the change of variable (11), the system (10) is equivalent to the
following:

$\displaystyle\begin{cases}\varepsilon^{2}\frac{d\,N^{a}_{\varepsilon,i}}{dt}=\lambda^{i-1}N^{a}_{\varepsilon,i}-\left[N^{A}_{\varepsilon,i}+N^{a}_{\varepsilon,i}\right]N^{a}_{\varepsilon,i}-g_{i}\left[Z_{\varepsilon}+\eta^{a}-(-1)^{i}\right]^{2}N^{a}_{\varepsilon,i}\\\
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\alpha^{(-1)^{j}}m_{j}\,N^{a}_{\varepsilon,j}-m_{i}\,N^{a}_{\varepsilon,i}+\mathcal{O}(\varepsilon^{2}),\\\
\\\
\varepsilon^{2}\frac{d\,N^{A}_{\varepsilon,i}}{dt}=\lambda^{i-1}\,N^{A}_{\varepsilon,i}-\left[N^{A}_{\varepsilon,i}+N^{a}_{\varepsilon,i}\right]N^{A}_{\varepsilon,i}-g_{i}\left[Z_{\varepsilon}+\eta^{A}-(-1)^{i}\right]^{2}N^{A}_{\varepsilon,i}\\\
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\alpha^{(-1)^{j}}m_{j}\,N^{A}_{\varepsilon,j}-m_{i}\,N^{A}_{\varepsilon,i}+\mathcal{O}(\varepsilon^{2}),\\\
\\\
\varepsilon^{2}\frac{d\,\delta^{a}_{\varepsilon}}{dt}=g_{1}+g_{2}+(g_{1}-g_{2})\,(Z_{\varepsilon}+\eta^{a})+\frac{\delta_{\varepsilon}}{2}\left[\frac{N_{\varepsilon,2}^{A}}{N_{\varepsilon,2}^{a}+N_{\varepsilon,2}^{A}}-\frac{N_{\varepsilon,1}^{A}}{N_{\varepsilon,1}^{a}+N_{\varepsilon,1}^{A}}\right]\\\
\qquad\qquad\qquad\qquad\qquad-\delta^{a}_{\varepsilon}\left[\frac{m_{2}\alpha\,N^{a}_{\varepsilon,2}}{N^{a}_{\varepsilon,1}}+\frac{m_{1}N^{a}_{\varepsilon,1}}{\alpha\,N^{a}_{\varepsilon,2}}\right]+\frac{\delta^{A}_{\varepsilon}-\delta^{a}_{\varepsilon}}{4}\left[\frac{N_{\varepsilon,2}^{A}}{N_{\varepsilon,2}^{a}+N_{\varepsilon,1}^{A}}+\frac{N_{\varepsilon,1}^{A}}{N_{\varepsilon,1}^{a}+N_{\varepsilon,1}^{A}}\right]+\mathcal{O}(\varepsilon^{2}),\\\
\varepsilon^{2}\frac{d\,\delta^{A}_{\varepsilon}}{dt}=g_{1}+g_{2}+(g_{1}-g_{2})\,(Z_{\varepsilon}+\eta^{A})+\frac{\delta_{\varepsilon}}{2}\left[\frac{N_{\varepsilon,1}^{a}}{N_{\varepsilon,1}^{a}+N_{\varepsilon,1}^{A}}-\frac{N_{\varepsilon,2}^{a}}{N_{\varepsilon,2}^{a}+N_{\varepsilon,2}^{A}}\right]\\\
\qquad\qquad\qquad\qquad\qquad-\delta^{A}_{\varepsilon}\left[\frac{m_{2}\alpha
N^{A}_{\varepsilon,2}}{N^{A}_{\varepsilon,1}}+\frac{m_{1}N^{A}_{\varepsilon,1}}{\alpha
N^{A}_{\varepsilon,2}}\right]+\frac{\delta^{a}_{\varepsilon}-\delta^{A}_{\varepsilon}}{4}\left[\frac{N_{\varepsilon,2}^{a}}{N_{\varepsilon,2}^{a}+N_{\varepsilon,2}^{A}}+\frac{N_{\varepsilon,1}^{a}}{N_{\varepsilon,1}^{a}+N_{\varepsilon,1}^{A}}\right]+\mathcal{O}(\varepsilon^{2}),\\\
\varepsilon^{2}\frac{d\,\delta_{\varepsilon}}{dt}=-\frac{\delta_{\varepsilon}}{2}-(g_{1}+g_{2})\frac{\eta^{A}-\eta^{a}}{2}+\left(\frac{\delta_{\varepsilon}^{A}}{2}\left[\frac{\alpha
m_{2}N^{A}_{\varepsilon,2}}{N^{A}_{\varepsilon,1}}-\frac{m_{1}N^{A}_{\varepsilon,1}}{\alpha
N^{A}_{\varepsilon,2}}\right]-\frac{\delta_{\varepsilon}^{a}}{2}\left[\frac{\alpha
m_{2}N^{a}_{\varepsilon,2}}{N^{a}_{\varepsilon,1}}-\frac{m_{1}N^{a}_{\varepsilon,1}}{\alpha
N^{a}_{\varepsilon,2}}\right]\right)\\\
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\;\;+\mathcal{O}(\varepsilon^{2}),\\\
\\\
\frac{dZ_{\varepsilon}}{dt}=(g_{2}-g_{1})-(g_{1}+g_{2})\,\left(Z_{\varepsilon}+\frac{\eta^{A}+\eta^{a}}{2}\right)+\frac{\delta_{\varepsilon}}{2}\left[\frac{N^{A}_{1,\varepsilon}}{N^{A}_{1,\varepsilon}+N^{a}_{1,\varepsilon}}+\frac{N^{A}_{2,\varepsilon}}{N^{A}_{2,\varepsilon}+N^{a}_{2,\varepsilon}}-1\right]\\\
\qquad\qquad\qquad\qquad\qquad\qquad+\left(\frac{\delta_{\varepsilon}^{a}}{2}\left[\frac{\alpha
m_{2}N^{a}_{\varepsilon,2}}{N^{a}_{\varepsilon,1}}-\frac{m_{1}N^{a}_{\varepsilon,1}}{\alpha
N^{a}_{\varepsilon,2}}\right]+\frac{\delta_{\varepsilon}^{A}}{2}\left[\frac{\alpha
m_{2}N^{A}_{\varepsilon,2}}{N^{A}_{\varepsilon,1}}-\frac{m_{1}N^{A}_{\varepsilon,1}}{\alpha
N^{A}_{\varepsilon,2}}\right]\right)\\\
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad+\frac{\delta_{\varepsilon}^{A}-\delta_{\varepsilon}^{a}}{4}\left[\frac{N^{A}_{2,\varepsilon}}{N^{A}_{2,\varepsilon}+N^{a}_{2,\varepsilon}}-\frac{N^{A}_{1,\varepsilon}}{N^{A}_{1,\varepsilon}+N^{a}_{1,\varepsilon}}\right]+\mathcal{O}(\varepsilon^{2}).\end{cases}$
(28)

The system (28) can be recasted more compactly into $P_{\varepsilon}$. The
main slow-fast analysis result is Theorem 2.1, which states the convergence of
$P_{\varepsilon}$ towards a limit system $P_{0}$ which separates ecological
and evolutionary time scales. The arguments of the proof of Theorem 2.1 are
similar to the analogous theorems proved in [LL54, Dek22]. The proof requires
some preliminaries results, particularly of stability, to which we dedicate
the rest of this section. The structure of this section is represented in Fig.
7.

$\boldsymbol{(P_{\varepsilon})}$ $\begin{aligned} \begin{cases}\text{Fast
dynamics (allelic subpopulations sizes}\\\ \text{and spatial discrepancies
between mean infinitesimal parts)},\\\ \\\ \text{Slow dynamics (mean
infinitesimal part)}.\end{cases}\end{aligned}$Theorem
2.1$\varepsilon\rightarrow 0$$\boldsymbol{(P_{0})}$ $\begin{aligned}
\begin{cases}\text{Fast ecological equilibria},\\\ \\\ \text{Slow evolutionary
dynamics}.\end{cases}\end{aligned}$ $\begin{aligned} &\text{\ref{S_0} (allelic
subpopulation sizes)},\\\ &\text{\scriptsize\emph{Resolution}
\ref{prop:solut_S0}, \lx@cref{creftype~refnum}{rem:degrees_of_freedom}},\\\
&\text{\scriptsize$\definecolor{cb@saved}{rgb}{0,0.75,0.16}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@cmyk@stroke{0}{0}{0}{1}\pgfsys@color@cmyk@fill{0}{0}{0}{1}\boxed{\color[rgb]{0,0.75,0.16}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.75,0.16}\pgfsys@color@cmyk@stroke{0.92}{0}{0.59}{0.25}\pgfsys@color<EMAIL_ADDRESS>\ref{prop:stability_S0}}.\end{aligned}$$\begin{aligned}
&\text{\ref{linear_syst} (mean trait discrepancies)},\\\
&\text{\scriptsize\emph{Resolution} \ref{prop:linear_syst}},\\\
&\text{\scriptsize$\definecolor{cb@saved}{rgb}{0,0.75,0.16}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@cmyk@stroke{0}{0}{0}{1}\pgfsys@color@cmyk@fill{0}{0}{0}{1}\boxed{\color[rgb]{0,0.75,0.16}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.75,0.16}\pgfsys@color@cmyk@stroke{0.92}{0}{0.59}{0.25}\pgfsys@color<EMAIL_ADDRESS>\ref{prop:stability_linear}}.\end{aligned}$One-locus haploid model4 ($Z=0$)
Figure 7: Layout of the slow-fast analysis in Appendix D. This figure presents
the key elements of the separation of time scales leading from
$P_{\varepsilon}$ to $P_{0}$. The stability of the fast equilibria (studied in
the two subsystems $S_{0}(Z)$ and $S_{L}(Z)$) is the crucial argument
underlying the convergence result stated in Theorem 2.1. The resolution of
$S_{0}(Z)$ leads to a uniqueness result that is unexpected with regard to the
analogous resolution in [Dek22] (see 5).

In the rest of this section, we first solve the slow manifold algebraic system
$G(\bar{Y},Z)=0$, showing that there can only exist one instantaneous
ecological equilibrium at a given $Z\in]-1,1[$ (30 and D.2). Surprisingly,
this resolution is easier than the analogous one in [Dek22] (see 5). Next, in
Section D.1.2, we show a stability criterion of the slow manifold (D.3 and
D.4), which justifies the separation of time scales approach.

###  D.1 Analysis of the fast equilibria.

The fast equilibria, for $Z\in]-1,1[$, are defined as the solutions $\bar{Y}$
to the algebraic system $G(\bar{Y},Z)=0$, or equivalently seven equations that
we group in two subsystems $S_{0}(Z)$ and $S_{L}(Z)$:

$\displaystyle\begin{cases}\alpha\,m_{2}\,N_{2}^{a}-m_{1}N_{1}^{a}+N_{1}^{a}\,\left[1-(N_{1}^{a}+N_{1}^{A})-g_{1}\,(Z+\eta^{a}+1)^{2}\right]&=0,\\\
\alpha\,m_{2}\,N_{2}^{A}-m_{1}N_{1}^{A}+N_{1}^{A}\,\left[1-(N_{1}^{a}+N_{1}^{A})-g_{1}\,(Z+\eta^{A}+1)^{2}\right]&=0,\\\
\frac{m_{1}}{\alpha}\,N_{1}^{a}-m_{2}\,N_{2}^{a}+N_{2}^{a}\,\left[\lambda-(N_{2}^{a}+N_{2}^{A})-g_{2}\,(Z+\eta^{a}-1)^{2}\right]&=0,\\\
\frac{m_{1}}{\alpha}\,N_{1}^{A}-m_{2}\,N_{2}^{A}+N_{2}^{A}\,\left[\lambda-(N_{2}^{a}+N_{2}^{A})-g_{2}\,(Z+\eta^{A}-1)^{2}\right]&=0.\\\
\end{cases}$ ($S_{0}(Z)$) $J_{S_{L}}\begin{pmatrix}\delta\\\ \delta^{A}\\\
\delta^{a}\end{pmatrix}=\begin{pmatrix}(g_{1}+g_{2})\frac{\eta^{A}-\eta^{a}}{2}\\\
-(g_{1}+g_{2})+(g_{2}-g_{1})(Z+\eta^{A})\\\
-(g_{1}+g_{2})+(g_{2}-g_{1})(Z+\eta^{a})\end{pmatrix},$ ($S_{L}(Z)$)

where:

$\tiny{J_{S_{L}}\\!\\!:=\\!\\!\begin{pmatrix}-\frac{1}{2}&\frac{1}{2}\left[\frac{\alpha\,m_{2}\,N_{2}^{A}}{N_{1}^{A}}-\frac{m_{1}\,N_{1}^{A}}{\alpha\,N_{2}^{A}}\right]&-\frac{1}{2}\left[\frac{\alpha\,m_{2}\,N_{2}^{a}}{N_{1}^{a}}-\frac{m_{1}\,N_{1}^{a}}{\alpha\,N_{2}^{a}}\right]\\\
\frac{\frac{N_{1}^{a}}{N_{1}^{A}+N_{1}^{a}}-\frac{N_{2}^{a}}{N_{2}^{A}+N_{2}^{a}}}{2}&-\left[\frac{\alpha\,m_{2}\,N_{2}^{A}}{N_{1}^{A}}+\frac{m_{1}\,N_{1}^{A}}{\alpha\,N_{2}^{A}}\right]-\frac{\frac{N_{1}^{a}}{N_{1}^{A}+N_{1}^{a}}+\frac{N_{2}^{a}}{N_{2}^{A}+N_{2}^{a}}}{4}&\frac{\frac{N_{1}^{a}}{N_{1}^{A}+N_{1}^{a}}+\frac{N_{2}^{a}}{N_{2}^{A}+N_{2}^{a}}}{4}\\\
\frac{\frac{N_{2}^{A}}{N_{2}^{A}+N_{2}^{a}}-\frac{N_{1}^{A}}{N_{1}^{A}+N_{1}^{a}}}{2}&\frac{\frac{N_{1}^{A}}{N_{1}^{A}+N_{1}^{a}}+\frac{N_{2}^{A}}{N_{2}^{A}+N_{2}^{a}}}{4}&-\left[\frac{\alpha\,m_{2}\,N_{2}^{a}}{N_{1}^{a}}+\frac{m_{1}\,N_{1}^{a}}{\alpha\,N_{2}^{a}}\right]-\frac{\frac{N_{1}^{A}}{N_{1}^{A}+N_{1}^{a}}+\frac{N_{2}^{A}}{N_{2}^{A}+N_{2}^{a}}}{4}\end{pmatrix}}$

####  D.1.1 Resolution.

Following 1, we recall that we assume that no major-effect allele has fixed.
Here, we show that there is at most one instantaneous ecological equilibrium
at each $Z$-level (for
$Z\in]-1-\frac{\eta^{A}+\eta^{a}}{2},1-\frac{\eta^{A}+\eta^{a}}{2}[$), thanks
to 30 and D.2.

######  Proposition D.1.

Suppose that no major-effect allele has fixed. Then, for
$Z\in]-1-\frac{\eta^{A}+\eta^{a}}{2},1-\frac{\eta^{A}+\eta^{a}}{2}[$,
$S_{0}(Z)$ has exactly one solution
$(N_{1}^{a},N_{1}^{A},N_{2}^{a},N_{2}^{A})\in(\mathbb{R}^{*})^{4}$, given by:

$N^{a}_{1}=\frac{Y^{A}\,N_{1}-N_{2}}{Y^{A}-Y^{a}},\quad
N^{a}_{2}=Y^{a}\frac{Y^{A}\,N_{1}-N_{2}}{Y^{A}-Y^{a}},\quad
N^{A}_{1}=\frac{N_{2}-Y^{a}\,N_{1}}{Y^{A}-Y^{a}},\quad
N^{A}_{2}=Y^{A}\frac{N_{2}-Y^{a}\,N_{1}}{Y^{A}-Y^{a}},$

where the quantities $(Y^{A},Y^{a},N_{1},N_{2})$ are defined by (29):

$\displaystyle\begin{cases}Y^{A}=\frac{g_{1}}{\alpha\,m_{2}}\,\left(\eta^{A}+\eta^{a}+2\,(Z+1)\right)\frac{\eta^{A}-\eta^{a}}{2}\left[\sqrt{1+\frac{m_{1}\,m_{2}}{4\,g_{1}\,g_{2}\,\left(\frac{\eta^{A}-\eta^{a}}{2}\right)^{2}\left(1-\left(\frac{\eta^{A}+\eta^{a}}{2}+Z\right)^{2}\right)}}+1\right],\\\
Y^{a}=\frac{g_{1}}{\alpha\,m_{2}}\,\left(\eta^{A}+\eta^{a}+2\,(Z+1)\right)\frac{\eta^{A}-\eta^{a}}{2}\left[\sqrt{1+\frac{m_{1}\,m_{2}}{4\,g_{1}\,g_{2}\,\left(\frac{\eta^{A}-\eta^{a}}{2}\right)^{2}\left(1-\left(\frac{\eta^{A}+\eta^{a}}{2}+Z\right)^{2}\right)}}-1\right],\\\
N_{1}=1-g_{1}\,(Z+1+\eta^{A})^{2}-m_{1}+\alpha\,m_{2}\,Y^{A},\\\
N_{2}=\lambda-
g_{2}\,(Z-1+\eta^{A})^{2}-m_{2}+\frac{m_{1}}{\alpha\,Y^{A}}.\end{cases}$ (29)

This solution $(N_{1}^{a},N_{1}^{A},N_{2}^{a},N_{2}^{A})$ defines viable
numbers of each allele in each sub-populations if and only if:

$[Y^{A}\,N_{1}>N_{2}]\text{ and }[N_{2}>Y^{a}\,N_{1}].$ (30)

###### Proof.

Let us introduce the following change of variables, valid under the assumption
that no major-effect allele has fixed:

$N_{1}:=N_{1}^{A}+N_{1}^{a},\quad N_{2}:=N_{2}^{A}+N_{2}^{a},\quad
Y^{A}:=\frac{N_{2}^{A}}{N_{1}^{A}},\quad Y^{a}:=\frac{N_{2}^{a}}{N_{1}^{a}}.$

Then, under the assumptions made in 1, the system ($S_{0}(Z)$) is equivalent
to:

$\displaystyle\begin{cases}\alpha\,m_{2}\,Y^{a}-m_{1}+\left[1-N_{1}-g_{1}\,(Z+\eta^{a}+1)^{2}\right]&=0,\\\
\alpha\,m_{2}\,Y^{A}-m_{1}+\left[1-N_{1}-g_{1}\,(Z+\eta^{A}+1)^{2}\right]&=0,\\\
\frac{m_{1}}{\alpha}\,\frac{1}{Y^{a}}-m_{2}+\left[\lambda-
N_{2}-g_{2}\,(Z+\eta^{a}-1)^{2}\right]&=0,\\\
\frac{m_{1}}{\alpha}\,\frac{1}{Y^{A}}-m_{2}+\left[\lambda-
N_{2}-g_{2}\,(Z+\eta^{A}-1)^{2}\right]&=0.\\\ \end{cases}$ (31)

This is equivalent to the following system:

$\displaystyle\begin{cases}\alpha\,m_{2}\,(Y^{a}-Y^{A})+\,g_{1}\,\left(\eta^{A}+\eta^{a}+2\,(Z+1)\right)(\eta^{A}-\eta^{a})&=0,\\\
\frac{m_{1}}{\alpha}\,(\frac{1}{Y^{a}}-\frac{1}{Y^{A}})+g_{2}\,\left(\eta^{A}+\eta^{a}+2\,(Z-1)\right)(\eta^{A}-\eta^{a})&=0,\\\
N_{1}-\left(1-g_{1}\,(Z+1+\eta^{A})^{2}-m_{1}+\alpha\,m_{2}\,Y^{A}\right)&=0,\\\
N_{2}-\left(\lambda-
g_{2}\,(Z-1+\eta^{A})^{2}-m_{2}+\frac{m_{1}}{\alpha\,Y^{A}}\right)&=0.\\\
\end{cases}$

As $Z\neq 1-\frac{\eta^{A}+\eta^{a}}{2}$, the closed subsystem on
$(Y^{A},Y^{a})$ is, in turn, equivalent to:

$\displaystyle\begin{cases}Y^{A}-Y^{a}&=A_{1}(Z):=\frac{g_{1}}{\alpha\,m_{2}}\,\left(\eta^{A}+\eta^{a}+2\,(Z+1)\right)(\eta^{A}-\eta^{a}),\\\
-Y^{A}\,Y^{a}&=A_{0}(Z):=\frac{g_{1}\,m_{1}}{\alpha^{2}\,g_{2}\,m_{2}}\frac{\eta^{A}+\eta^{a}+2\,(Z+1)}{\eta^{A}+\eta^{a}+2\,(Z-1)}.\\\
\end{cases}$

$Y^{A}$ and $-Y^{a}$ are the roots of the polynomial:

$P(X)=X^{2}-A_{1}(Z)\,X+A_{0}(Z).$

$P$ has two real roots of opposite signs if and only if:

$\left[A_{0}(Z)<0\right],$

which is equivalent to:

$-1-\frac{\eta^{A}+\eta^{a}}{2}<Z<1-\frac{\eta^{A}+\eta^{a}}{2}.$

Under the last condition on $Z$, $A_{1}(Z)$ is positive, $A_{0}(Z)$ is
negative and we get:

$\displaystyle\begin{cases}Y^{A}=\frac{A_{1}(Z)}{2}\,\left[\sqrt{1-\frac{A_{0}(Z)}{\left(\frac{A_{1}(Z)}{2}\right)^{2}}}+1\right],\\\
Y^{a}=\frac{A_{1}(Z)}{2}\,\left[\sqrt{1-\frac{A_{0}(Z)}{\left(\frac{A_{1}(Z)}{2}\right)^{2}}}-1\right],\end{cases}$
(32)

which is equivalent to (29).

Inverting the initial change of variables leads to:

$N^{a}_{1}=\frac{Y^{A}\,N_{1}-N_{2}}{Y^{A}-Y^{a}},\quad
N^{a}_{2}=Y^{a}\frac{Y^{A}\,N_{1}-N_{2}}{Y^{A}-Y^{a}},\quad
N^{A}_{1}=\frac{N_{2}-Y^{a}\,N_{1}}{Y^{A}-Y^{a}},\quad
N^{A}_{2}=Y^{A}\frac{N_{2}-Y^{a}\,N_{1}}{Y^{A}-Y^{a}},$

hence (29). It defines a viable solution with positive entries if and only if
$Y^{A}\,N_{1}>N_{2}$ and $N_{2}>Y^{a}\,N_{1}$. ∎

######  Proposition D.2.

For all allelic sizes of subpopulations
$(N_{1}^{a},N_{1}^{A},N_{2}^{a},N_{2}^{A})\in\left(\mathbb{R}_{+}^{*}\right)^{4}$
and $Z\in\mathbb{R}$, there exists a unique solution
$(\delta,\delta^{A},\delta^{a})$ to the system $S_{L}(Z)$.

###### Proof.

Using the notation $N_{1}:=N_{1}^{A}+N_{1}^{a}$ and
$N_{2}:=N_{2}^{A}+N_{2}^{a}$, we compute thanks to a symbolic computation tool
(Mathematica©):

$\displaystyle\det(J_{S_{L}})=$
$\displaystyle-\frac{1}{4}\left[\frac{m_{1}}{\alpha}\frac{N_{1}}{N_{2}}+\alpha\,m_{2}\frac{N_{2}}{N_{1}}+2\frac{m_{1}^{2}}{\alpha^{2}}\frac{N_{1}^{a}\,N_{1}^{A}}{N_{2}^{a}\,N_{2}^{A}}+2\alpha^{2}m_{2}^{2}\frac{N_{2}^{a}\,N_{2}^{A}}{N_{1}^{a}\,N_{1}^{A}}\right.$
$\displaystyle+\left.2m_{1}m_{2}\left(\frac{{N_{1}^{A}}^{2}N_{2}^{a}}{N_{1}^{a}N_{1}N_{2}}+\frac{{N_{1}^{a}}^{2}N_{2}^{A}}{N_{1}^{A}N_{1}N_{2}}+\frac{{N_{2}^{A}}^{2}N_{1}^{a}}{N_{2}^{a}N_{1}N_{2}}+\frac{{N_{2}^{a}}^{2}N_{1}^{A}}{N_{2}^{A}N_{1}N_{2}}+2\frac{N_{2}^{A}N_{1}^{a}}{N_{1}\,N_{2}}+2\frac{N_{1}^{a}N_{2}^{a}}{N_{1}N_{2}}\right)\right]$
$\displaystyle<0.$

∎

####  D.1.2 Stability.

Convergence toward a limit system locally in time in a slow-fast analysis
relies essentially on a stability criterion of the fast equilibria which
constitute the slow manifold ([LL54, Dek22]). In this subsection, we show that
all fast equilibria found in 30 and D.2 for a level
$Z\in]-1-\frac{\eta^{A}+\eta^{a}}{2},1-\frac{\eta^{A}+\eta^{a}}{2}[$, are
stable. Due to the particular shape of the slow manifold, it is sufficient to
study separately the Jacobian matrix associated to $S_{0}(Z)$ denoted
$J_{S_{0}}$ (D.3) and the Jacobian matrix associated to the linear system
$S_{L}(Z)$, which is exactly $J_{S_{L}}$ (D.4).

######  Proposition D.3.

Let $Z\in]-1-\frac{\eta^{A}+\eta^{a}}{2},1-\frac{\eta^{A}+\eta^{a}}{2}[$ such
that ($S_{0}(Z)$) has a unique solution
$(N_{1}^{a},N_{1}^{A},N_{2}^{a},N_{2}^{A})\in\left(\mathbb{R}_{+}^{*}\right)^{4}$.
Let us define the following matrix:

$J_{S_{0}}=\begin{pmatrix}-\frac{\alpha\,m_{2}\,N_{2}^{a}}{N_{1}^{a}}-N_{1}^{a}&\alpha\,m_{2}&-N_{1}^{a}&0\\\
\frac{m_{1}}{\alpha}&-\frac{m_{1}\,N_{1}^{a}}{\alpha\,N_{2}^{a}}-N_{2}^{a}&0&-N_{2}^{a}\\\
-N_{1}^{A}&0&-\frac{\alpha\,m_{2}\,N_{2}^{A}}{N_{1}^{A}}-N_{1}^{A}&\alpha\,m_{2}\\\
0&-N_{2}^{A}&\frac{m_{1}}{\alpha}&-\frac{m_{1}\,N_{1}^{A}}{\alpha\,N_{2}^{A}}-N_{2}^{A}\\\
\end{pmatrix}.$ (33)

Then:

  1. 1.

$J_{S_{0}}$ is the Jacobian of $S_{0}(Z)$ at
$(N_{1}^{a},N_{1}^{A},N_{2}^{a},N_{2}^{A})$.

  2. 2.

All the eigenvalues of $J_{S_{0}}$ are located in the left open half plane.

###### Proof.

1\. Let $(N_{1}^{a},N_{1}^{A},N_{2}^{a},N_{2}^{A})$ be solution of $S_{0}(Z)$.
One can verify that:

$\displaystyle\frac{\partial\left[\alpha\,m_{2}\,N_{2}^{a}-m_{1}N_{1}^{a}+N_{1}^{a}\,\left[1-(N_{1}^{a}+N_{1}^{A})-g_{1}\,(Z+\eta^{a}+1)^{2}\right]\right]}{\partial
N_{1}^{a}}$
$\displaystyle=\left[1-(N_{1}^{a}+N_{1}^{A})-g_{1}\,(Z+\eta^{a}+1)^{2}-m_{1}\right]-N_{1}^{a}=-\frac{\alpha\,m_{2}\,N_{2}^{a}}{N_{1}^{a}}-N_{1}^{a},$

for $(N_{1}^{a},N_{1}^{A},N_{2}^{a},N_{2}^{A})$ solves $S_{0}(Z)$. The same
holds for the other diagonal entries.

2\. Let:

$\chi_{J_{S_{0}}}(X)=X^{4}-\operatorname{tr}\left(J_{S_{0}}\right)X^{3}+b\,X^{2}+c\,X+\det{J_{S_{0}}},$

be the characteristic polynomial of $J_{S_{0}}$. Let us verify the Routh-
Hurwitz criterion: all the eigenvalues of $J_{S_{0}}$ are located in the left
open half plane if and only if:

  * $(i)$

$\det{J_{S_{0}}}>0$,

  * $(ii)$

$-\operatorname{tr}\left(J_{S_{0}}\right)>0$,

  * $(iii)$

$-\operatorname{tr}\left(J_{S_{0}}\right)\,b-c>0$,

  * $(iv)$

$(-\operatorname{tr}\left(J_{S_{0}}\right)\,b-c)\,c-\operatorname{tr}\left(J_{S_{0}}\right)^{2}\,\det{J_{S_{0}}}>0$.

We have:

$\det{J_{S_{0}}}=m_{1}\,m_{2}\left(N_{1}^{a}\,N_{2}^{a}-N_{1}^{A}\,N_{2}^{A}\right)^{2}\left(\frac{1}{N_{1}^{a}\,N_{2}^{A}}+\frac{1}{N_{1}^{A}\,N_{2}^{a}}\right)>0.$

and:

$\displaystyle-\operatorname{tr}\left(J_{S_{0}}\right)$
$\displaystyle=N_{1}+N_{2}+\sqrt{m_{1}\,m_{2}}\left(\frac{\alpha\sqrt{m_{2}}N_{2}^{a}}{\sqrt{m_{1}}N_{1}^{a}}+\frac{\sqrt{m_{1}}N_{1}^{a}}{\alpha\sqrt{m_{2}}N_{2}^{a}}+\frac{\alpha\sqrt{m_{2}}N_{2}^{A}}{\sqrt{m_{1}}N_{1}^{A}}+\frac{\alpha\sqrt{m_{2}}N_{2}^{A}}{\sqrt{m_{1}}N_{1}^{A}}\right)>0.$

With the help of a symbolic computation tool (Mathematica©), we verify that
the left hand side of the two last conditions are sums of positive terms, but
are too long to be displayed here. ∎

The Jacobian matrix of the linear system $S_{L}(Z)$ is exactly $J_{S_{L}}$ and
we also show that $J_{S_{L}}$ satisfies the Routh-Hurwitz criterion:

######  Proposition D.4.

$J_{S_{L}}$ has all its eigenvalues located in the left open half plane.

###### Proof.

Let the following be the characteristic polynomial of $J_{S_{L}}$:

$\chi_{J_{S_{L}}}(X)=X^{3}-\operatorname{tr}(J_{S_{L}})X^{2}-\frac{1}{2}\left(\operatorname{tr}(J_{S_{L}}^{2})-\operatorname{tr}(J_{S_{L}})^{2}\right)X-\det(J_{S_{L}}).$

We show that $J_{S_{L}}$ satisfies the Routh-Hurwitz criterion:

  * $(i)$

$-\det(J_{S_{L}})>0$,

  * $(ii)$

$-\operatorname{tr}(J_{S_{L}})>0$,

  * $(iii)$

$\frac{1}{2}\left(\operatorname{tr}(J_{S_{L}}^{2})-\operatorname{tr}(J_{S_{L}})^{2}\right)\,\operatorname{tr}(J_{S_{L}})+\det(J_{S_{L}})>0.$

We have $-\det(J_{S_{L}})>0$ from the proof of D.2 and:

$-\operatorname{tr}(J_{S_{L}})=1+\sqrt{m_{1}\,m_{2}}\left(\frac{\alpha\sqrt{m_{2}}N_{2}^{a}}{\sqrt{m_{1}}N_{1}^{a}}+\frac{\sqrt{m_{1}}N_{1}^{a}}{\alpha\sqrt{m_{2}}N_{2}^{a}}+\frac{\alpha\sqrt{m_{2}}N_{2}^{A}}{\sqrt{m_{1}}N_{1}^{A}}+\frac{\alpha\sqrt{m_{2}}N_{2}^{A}}{\sqrt{m_{1}}N_{1}^{A}}\right)>1+4\sqrt{m_{1}m_{2}}.$

We verify that the l.h.s. of the last condition is a sum of positive terms. ∎

##  Appendix E Proof of 3.1.

Let us define the quantities:

$\displaystyle\begin{cases}Y^{A,*}=\frac{2\,g\,\eta}{m}\left[\sqrt{1+\frac{m^{2}}{4\,g^{2}\,\eta^{2}}}+1\right],\\\
Y^{a,*}=\frac{2\,g\,\eta}{m}\left[\sqrt{1+\frac{m^{2}}{4\,g^{2}\,\eta^{2}}}-1\right],\\\
N^{*}_{1}=1-g\,\eta^{2}-g-m+2\,g\,\eta\,\sqrt{1+\frac{m^{2}}{4\,g^{2}\,\eta^{2}}},\\\
N^{*}_{2}=N^{*}_{1}.\end{cases}$ (34)

30 states that the latter defines a solution to $S_{0}(Z)$ given that $Z=0$:

$(N^{a,*}_{1},N^{a,*}_{2},N^{A,*}_{1},N^{A,*}_{2})=\left(N^{*}_{1}\frac{Y^{A,*}-1}{Y^{A,*}-Y^{a,*}},\;N^{*}_{1}\frac{1-Y^{a,*}}{Y^{A,*}-Y^{a,*}},\;N^{*}_{1}\frac{1-Y^{a,*}}{Y^{A,*}-Y^{a,*}},\;N^{*}_{1}\frac{Y^{A,*}-1}{Y^{A,*}-Y^{a,*}}\right).$

Since $Y^{A,*}>1$ and $Y^{A,*}\;Y^{a,*}=1$, this solution is viable under the
condition: $N_{1}>0$, hence requiring :

$1+\sqrt{4\,g^{2}\,\eta^{2}+m^{2}}>g\,\eta^{2}+g+m,$

which in turn is equivalent to (13).

D.2 next states that $S_{L}(Z)$ has a unique solution
$(\delta^{*},\delta^{A,*},\delta^{a,*})$ for such allelic population sizes
$(N^{a,*}_{1},N^{a,*}_{2},N^{A,*}_{1},N^{A,*}_{2})$. One can compute that:

$\delta^{A,*}=\delta^{a,*}=\frac{g\left(1+\eta+Y^{A,*}(1-\eta)\right)}{m(1+Y^{A,*})},\quad\delta^{*}=-\frac{2g\left(1+\eta-{Y^{A,*}}^{2}(1-\eta)\right)}{Y^{A,*}}.$

Finally, one can verify that
$(N^{a,*}_{1},N^{a,*}_{2},N^{A,*}_{1},N^{A,*}_{2},\delta^{*},\delta^{A,*},\delta^{a,*})$
along with setting $Z^{*}=0$ is a solution of the last equation of (12).

##  Appendix F Supplementary IBS with fixed subpopulations sizes

In this appendix, we show in Fig. 8 the analogous results as those presented
in Fig. 5, but with a slightly different procedure for the IBS, which adjusts
the birth rates to compensate exactly for the deaths by selection at each
generation, thus keeping the subpopulations sizes fixed. As mentioned in the
main text, the loss of the polymorphism at the major-effect locus still occurs
with weak selection, but not with strong selection.

(a) Major-effect locus with polygenic background: weak selection ($g=0.1$).

(b) Major-effect locus with polygenic background: intermediate selection
($g=0.5$).

(c) Major-effect locus with polygenic background: strong selection ($g=1$).

Figure 8: Same as the left panel of Fig. 5, but with fixed subpopulations
sizes.

##  Appendix G Supplementary IBS with asymmetrical initial conditions or
different parameters for the genetic architecture ($\boldsymbol{L}=50$ loci
and $\boldsymbol{\sigma_{LE}}$ = 0.2)

In this appendix, we first show that the phenomenon of loss of polymorphism in
the presence of a quantitative background with weak or strong selection is
robust to asymmetrical initial population sizes, and even occurs much faster
(Fig. 9). We emphasize on the excellent agreement of the deterministic
iterations with the individual-based simulations

(a) Major-effect locus with polygenic background: weak selection ($g=0.1$).

(b) Control case without polygenic background: weak selection ($g=0.1$).

(c) major-effect locus with polygenic background: intermediate selection
($g=0.5$).

(d) Control case without polygenic background: intermediate selection
($g=0.5$).

(e) Major-effect locus with polygenic background: strong selection ($g=1$).

(f) Control case without polygenic background : strong selection ($g=1$).

Figure 9: Same as Fig. 5, but with asymmetrical initial subpopulation sizes.

Furthermore, we also show that our findings hold when considering a smaller
number of loci involved in the quantitative background ($\boldsymbol{L}=50$
instead of 200), with increased relative effect
($\boldsymbol{\sigma_{LE}}=0.2$ instead of 0.1), so that the trait range
$[-\boldsymbol{\eta}-\boldsymbol{\sigma_{LE}}\sqrt{\boldsymbol{L}},\boldsymbol{\eta}+\boldsymbol{\sigma_{LE}}\sqrt{\boldsymbol{L}}]\approx[-\boldsymbol{\eta}-1.4,\boldsymbol{\eta}+1.4]$
extends beyond the local optima (-1 and 1) even in the absence of major
effects. We display the results of the IBS with symmetrical initial
subpopulation sizes in Fig. 10 and with asymmetrical initial subpopulation
sizes in Fig. 11. Note that the right panel of each figure does not change
from Fig. 5 and Fig. 9, because the control case does not depend on the number
of loci, but we choose to display it anyway for consistency of comparison. One
can notice that the time to fixation at the major-effect locus in the presence
of a quantitative background under weak (Fig. 10(e), Fig. 11(a)) and strong
selection (Fig. 10(e), Fig. 11(e)) is reduced compared to when the
quantitative background comes from a larger number of loci (Fig. 5, Fig. 9).
Moreover, the sensitivity of the numerical resolutions of (1) with regard to
symmetrical initial states is more pronounced here.

(a) Major-effect locus with polygenic background: weak selection ($g=0.1$).

(b) Control case without polygenic background: weak selection ($g=0.1$).

(c) Major-effect locus with polygenic background: intermediate selection
($g=0.5$).

(d) Control case without polygenic background: intermediate selection
($g=0.5$).

(e) Major-effect locus with polygenic background: strong selection ($g=1$)

(f) Control case without polygenic background : strong selection ($g=1$).

Figure 10: Same as Fig. 5, but with a quantitative background of
$\boldsymbol{L}=50$ loci and $\boldsymbol{\sigma_{LE}}=0.2$.

(a) Major-effect locus with polygenic background: weak selection ($g=0.1$).

(b) Control case without polygenic background: weak selection ($g=0.1$).

(c) Major-effect locus with polygenic background: intermediate selection
($g=0.5$).

(d) Control case without polygenic background: intermediate selection
($g=0.5$).

(e) Major-effect locus with polygenic background: strong selection ($g=1$).

(f) Control case without polygenic background : strong selection ($g=1$).

Figure 11: Same as Fig. 5, but with a quantitative background of
$\boldsymbol{L}=50$ loci and $\boldsymbol{\sigma_{LE}}=0.2$ and asymmetrical
initial subpopulations sizes.

## Acknowledgements

The authors thank John Wakeley and two anonymous reviewers for very
constructive and detailed comments that, in their opinion, greatly improved
the manuscript. L.D also thanks Sepideh Mirrahimi for valuable discussions and
comments, and Florence Débarre for insightful conversations. L.D has received
partial funding from the ANR project DEEV ANR-20-CE40-0011-01 and a Mitacs
Globalink Research Award. S.P.O. has received funding from the NSERC Discovery
Grant: RGPIN-2016-03711. This project has received funding from the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation programm (grant agreement No 865711).

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