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\begin{document}

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\begin{flushleft}
{\Large \textbf{Text S2} }
% Insert Author names, affiliations and corresponding author email.
%\\
%Te Wu$^{1,\ast}$, Feng Fu$^{2,3,\dagger}$, Yanling Zhang$^{1,\S}$,
%Long Wang$^{1,\ddagger}$
%\\
%\bf{1} Center for Systems and Control, State Key Laboratory for
%Turbulence and Complex Systems, College of Engineering, Peking
% University, Beijing, China,
%\\
%\bf{2} Program for Evolutionary Dynamics, Harvard University,
%Cambridge, Massachusetts, United States of America
%\\
%\bf{3} Institute of Integrative Biology, ETH Zurich, Zurich 8092,
%Switzerland
%\\$\ast$ wute@pku.edu.cn
%\\$\dagger$ fengfu@fas.harvard.edu
%\\$\S$ yanlzhang@pku.edu.cn
%\\$\ddagger$ longwang@pku.edu.cn
\end{flushleft}

% Please keep the abstract between 250 and 300 words
\section*{The first derivative at the real roots of a polynomial changes signs alternatively
between negative and positive}
Let
$F(x)=f_C(x)-f_D(x)=\frac{r}{N}\sum_{k=0}^{N-1}{N-1\choose
k}x^k(1-x)^{N-1-k}[(k+1)g(k+1)-kg(k)]-1$ and $G(x)=x(1-x)F(x)$. We
here intend to demonstrate that the first-order derivative of $G(x)$
changes signs alternately between negative and positive at the real
roots of $G(x)$.

Obviously, $G(x)$ is a polynomial function of $x$, and thus can be
written as
\begin{equation}G(x)=c\prod_{i=1}^l(x-x_i)\prod_j(x^2+a_jx +
b_j)\end{equation} with $x_i$ being the real root of $G(x)$ and $l$
the number of all $x_i$. The two constraint conditions are
satisfied: ($I$) For any $i_1<i_2$, $x_{i_1}<x_{i_2}$ and ($II$) For
any $j$, $a_j^2-4b_j<0$. It is straightforward to derive the
first-order derivative of $G(x)$ as \begin{equation} G'(x)=(\sum_i
\frac{1}{x-x_i}+\sum_j\frac{2x+a_j}{x^2+a_jx+b_j})G(x)
\end{equation}
We can easily obtain that \begin{eqnarray} G'(x_k)&=&c\prod_{i\neq
k}(x_k-x_i)\prod_j(x_k^2+a_jx_k + b_j)+(\sum_{i\neq
k}\frac{1}{x_k-x_i}+\sum_j
\frac{2x_k+a_j}{x_k^2+a_jx_k+b_j})G(x_k)\nonumber\\&=& c\prod_{i\neq
k}(x_k-x_i)\prod_j(x_k^2+a_jx_k + b_j)
\end{eqnarray}
Similarly we have
\begin{eqnarray} G'(x_{k+1})=c\prod_{i\neq k+1}(x_{k+1}-x_i)\prod_j(x_{k+1}^2+a_jx_{k+1} + b_j)
\end{eqnarray}
Multiplying $G'(x_x)$ and $G'(x_{k+1})$ leads to
\begin{eqnarray}
G'(x_k)G'(x_{k+1})=-c^2(x_k-x_{k+1})^2\prod_{i=1,i\neq k,i\neq
k+1}^{l}(x_k-x_i)(x_{k+1}-x_i)\prod_j(x_k^2+a_jx_k+b_j)(x_{k+1}^2+a_jx_{k+1}
+ b_j)\nonumber\end{eqnarray} Noting that
$x_1<x_2<\cdot\cdot\cdot<x_l$, the above expression indicates
$G'(x_k)G'(x_{k+1})<0$. As a consequent, we claim that the
first-order derivative of $G(x)$ shares opposite sign at any two
contiguous fixed points. In other words, the stability of any two
contiguous fixed points of the replicator equation is opposite.
\end{document}
