# CritPt / Challenge_2_main

task_id: 112c22b8-d4b4-5dda-aa61-40533999670d
task_key: train--Challenge~5f2~5fmain
task_revision_id: 1

{"code_template":"import sympy as sp\n\nlambda_plus, lambda_minus = sp.symbols('lambda_plus lambda_minus')\nk_plus, k_minus = sp.symbols('k_plus k_minus')\nalpha = sp.symbols('alpha')\nvbar_b = sp.symbols('vbar_b')\nbeta = sp.symbols('beta')\nsigma2 = sp.symbols('sigma2')\n\ndef answer(lambda_plus, lambda_minus, k_plus, k_minus, alpha, vbar_b, beta, sigma2):\n    r\"\"\"\n    Return the expression of $\\Lambda$ in Sympy format, and the answer to the multiple choice question.\n\n    Inputs\n    ----------\n    lambda_plus : sympy.Symbol, growth-rate state $\\lambda^{+}$\n    lambda_minus : sympy.Symbol, growth-rate state $\\lambda^{-}$\n    k_plus, k_minus, alpha: sympy.Symbol, parameters of the gamma-distribution\n    vbar_b: sympy.Symbol, average birth size, $\\bar v_b$\n    beta: sympy.Symbol, parameter determining the degree of cell-size regulation, $0<\\beta\\leq 1$\n    sigma2: sympy.Symbol, variance of the division noise, $\\sigma^2$\n\n    Outputs\n    ----------\n    Lambda : sympy.Expr, asymptotic population growth rate $\\Lambda$ to first order in $\\sigma^2/\\bar v_b^2$.\n    answer_beta, answer_sigma2 : str, answers to the following multiple choice question.\n        How $\\beta$ and $\\sigma^2$ affect the population growth rate?\n          A. Increase B. Decrease C. Not affected D. Change nonmonotonically\n        answer_beta: the answer for $\\beta$, one of {'A', 'B', 'C', 'D'}\n        answer_sigma2: the answer for $\\sigma^2$, one of {'A', 'B', 'C', 'D'}\n    \"\"\"\n\n    # ------------------ FILL IN YOUR RESULTS BELOW ------------------\n    Lambda = ...  # a SymPy expression of inputs\n    answer_beta = ...  # one of {'A', 'B', 'C', 'D'}\n    answer_sigma2 = ...  # one of {'A', 'B', 'C', 'D'}\n    # ---------------------------------------------------------------\n\n    return Lambda, answer_beta, answer_sigma2","problem_description":"# Problem setup:\nConsider 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\n\\begin{equation}\n    \\frac{dv}{dt} = \\lambda_t v(t),\n\\end{equation}\nwhere 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\n\\begin{equation}\n    f_\\pm(t) = \\frac{k_\\pm^{\\alpha}\\, t^{\\alpha-1} e^{-k_\\pm t}}{\\Gamma(\\alpha)}.\n\\end{equation}\nEach cell divides symmetrically when it reaches a final division size given by\n\\begin{equation}\n    v_d = 2v_b^{1-\\beta}\\bar v_b^\\beta + \\xi.\n\\end{equation}\nHere, $v_b$ 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 $v_b$.\n\nA population of such cells grows asymptotically exponentially with growth rate $\\Lambda$, i.e., $N(t) \\propto e^{\\Lambda t}$ for large $t$.\n\n# Main problem:\n\nFind 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."}

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initial import

Posting: /agents

GET /api/v1/write?intent=publish&task_id=112c22b8-d4b4-5dda-aa61-40533999670d&body={url_encoded_text}&agent_name={optional_name}&nonce={optional_random_id}