CritPt / Challenge_2_main / Consider a population of genetically identical bacterial cells in balanced…

# Problem setup:
Consider a population of genetically identical bacterial cells in balanced growth. Each cell starts with some initial size v_(b) and grows according to the equation
\begin{equation}
    \frac{dv}{dt} = \lambda_t v(t),
\end{equation}
where the growth rate λ_(t) is a two-state stochastic process that jumps between values λ^(+) and λ^(−) and has the gamma-distributed waiting times with densities
\begin{equation}
    f_\pm(t) = \frac{k_\pm^{\alpha}\, t^{\alpha-1} e^{-k_\pm t}}{\Gamma(\alpha)}.
\end{equation}
Each cell divides symmetrically when it reaches a final division size given by
\begin{equation}
    v_d = 2v_b^{1-\beta}\bar v_b^\beta + \xi.
\end{equation}
Here, v_(b) is the birth size of the cell, (v)¯_(b)>0 is a constant representing average birth size, 0<β≤1 is a parameter determining the degree of cell-size regulation, and the division noise ξ>0 is a narrowly distributed random variable with mean zero and variance σ². You can assume this noise is Gaussian distributed and sufficiently narrow to ignore the probability that v_(d) is ever smaller than v_(b).

A population of such cells grows asymptotically exponentially with growth rate Λ, i.e., N(t)∝e^(Λt) for large t.

# Main problem:

Find the asymptotic population growth rate Λ in terms of the model parameters λ^(+), λ^(−), k_(+), k_(−), α, (v)¯_(b), β, and σ² for small σ²/(v)¯_(b)². Give your answer to first order in σ²/(v)¯_(b)². Explain how β and σ² affect the population growth rate.

# Problem setup:
Consider a population of genetically identical bacterial cells in balanced growth. Each cell starts with some initial size $v_b$ and grows according to the equation
\begin{equation}
    \frac{dv}{dt} = \lambda_t v(t),
\end{equation}
where the growth rate $\lambda_t$ is a two-state stochastic process that jumps between values $\lambda^+$ and $\lambda^-$ and has the gamma-distributed waiting times with densities
\begin{equation}
    f_\pm(t) = \frac{k_\pm^{\alpha}\, t^{\alpha-1} e^{-k_\pm t}}{\Gamma(\alpha)}.
\end{equation}
Each cell divides symmetrically when it reaches a final division size given by
\begin{equation}
    v_d = 2v_b^{1-\beta}\bar v_b^\beta + \xi.
\end{equation}
Here, $v_b$ is the birth size of the cell, $\bar v_b>0$ is a constant representing average birth size, $0<\beta\leq 1$ is a parameter determining the degree of cell-size regulation, and the division noise $\xi>0$ is a narrowly distributed random variable with mean zero and variance $\sigma^2$. You can assume this noise is Gaussian distributed and sufficiently narrow to ignore the probability that $v_d$ is ever smaller than $v_b$.

A population of such cells grows asymptotically exponentially with growth rate $\Lambda$, i.e., $N(t) \propto e^{\Lambda t}$ for large $t$.

# Main problem:

Find the asymptotic population growth rate $\Lambda$ in terms of the model parameters $\lambda^+$, $\lambda^-$, $k_+$, $k_-$, $\alpha$, $\bar v_b$, $\beta$, and $\sigma^2$ for small $\sigma^2/\bar v_b^2$. Give your answer to first order in $\sigma^2/\bar v_b^2$. Explain how $\beta$ and $\sigma^2$ affect the population growth rate.