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\begin{document}
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\begin{frontmatter}
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\dochead{--DIR--}
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\title{Sample article for DIR}
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\author[
addressref={aff1}, % id's of addresses, e.g. {aff1,aff2}
corref={aff1}, % id of corresponding address, if any
noteref={n1}, % id's of article notes, if any
email={jane.e.doe@cambridge.co.uk} % email address
]{\inits{JE}\fnm{Jane E} \snm{Doe}}
\author[
addressref={aff1,aff2},
email={john.RS.Smith@cambridge.co.uk}
]{\inits{JRS}\fnm{John RS} \snm{Smith}}
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\orgname{Department of Zoology, Cambridge}, % university, etc
\street{Waterloo Road}, %
%\postcode{} % post or zip code
\city{London}, % city
\cny{UK} % country
}
\address[id=aff2]{%
\orgname{Marine Ecology Department, Institute of Marine Sciences Kiel},
\street{D\"{u}sternbrooker Weg 20},
\postcode{24105}
\city{Kiel},
\cny{Germany}
}
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%\note{Sample of title note} % note to the article
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% <put your article body there>
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\section*{Content}
Text and results for this section, as per the individual journal's instructions for authors. %\cite{koon,oreg,khar,zvai,xjon,schn,pond,smith,marg,hunn,advi,koha,mouse}
\section*{Section title}
Text for this section \ldots
\subsection*{Sub-heading for section}
Text for this sub-heading \ldots
\subsubsection*{Sub-sub heading for section}
Text for this sub-sub-heading \ldots
\paragraph*{Sub-sub-sub heading for section}
Text for this sub-sub-sub-heading \ldots
In this section we examine the growth rate of the mean of $Z_0$, $Z_1$ and $Z_2$. In
addition, we examine a common modeling assumption and note the
importance of considering the tails of the extinction time $T_x$ in
studies of escape dynamics.
We will first consider the expected resistant population at $vT_x$ for
some $v>0$, (and temporarily assume $\alpha=0$)
%
\[
E \bigl[Z_1(vT_x) \bigr]= E
\biggl[\mu T_x\int_0^{v\wedge
1}Z_0(uT_x)
\exp \bigl(\lambda_1T_x(v-u) \bigr)\,du \biggr].
\]
%
If we assume that sensitive cells follow a deterministic decay
$Z_0(t)=xe^{\lambda_0 t}$ and approximate their extinction time as
$T_x\approx-\frac{1}{\lambda_0}\log x$, then we can heuristically
estimate the expected value as
%
\begin{eqnarray}\label{eqexpmuts}
E\bigl[Z_1(vT_x)\bigr] &=& \frac{\mu}{r}\log x
\int_0^{v\wedge1}x^{1-u}x^{({\lambda_1}/{r})(v-u)}\,du
\nonumber\\
&=& \frac{\mu}{r}x^{1-{\lambda_1}/{\lambda_0}v}\log x\int_0^{v\wedge
1}x^{-u(1+{\lambda_1}/{r})}\,du
\nonumber\\
&=& \frac{\mu}{\lambda_1-\lambda_0}x^{1+{\lambda_1}/{r}v} \biggl(1-\exp \biggl[-(v\wedge1) \biggl(1+
\frac{\lambda_1}{r}\biggr)\log x \biggr] \biggr).
\end{eqnarray}
%
Thus we observe that this expected value is finite for all $v>0$ (also see \cite{koon,khar,zvai,xjon,marg}).
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\begin{backmatter}
\section*{Competing interests}
The authors declare that they have no competing interests.
\section*{Author's contributions}
Text for this section \ldots
\section*{Acknowledgements}
Text for this section \ldots
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\end{document}
