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Text for this sub-sub-heading …Text for this sub-sub-sub-heading …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 \(\alpha=0\))

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{aligned} E\bigl{[}Z_{1}(vT_{x})\bigr{]} & = & \frac{\mu}{r}\log x\int_{0}^{v\wedge 1}x^{1-u}x^{({\lambda_{1}}/{r})(v-u)}\,du \\ & = & \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\wedge 1)\biggl{(}1+\frac{\lambda_{1}}{r}\biggr{)}\log x\biggr{]}\biggr{)}.\nonumber \\ \end{aligned}

Thus we observe that this expected value is finite for all \(v>0\) (also see (Koonin 1996, Zvaifler 1999, Jones 1996, Margulis 1970)).

\begin{aligned} E\bigl{[}Z_{1}(vT_{x})\bigr{]}=E\biggl{[}\mu T_{x}\int_{0}^{v\wedge 1}Z_{0}(uT_{x})\exp\bigl{(}\lambda_{1}T_{x}(v-u)\bigr{)}\,du\biggr{]}.\\ \end{aligned}

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