\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In this problem, we study a death process, which has the unique reaction $A \\to \\emptyset$ with rate constant $\\delta$.\n", "\n", "\n", "**a)** Write down an ODE describing the deterministic kinetics.\n", "\n", "**b)** Write down the Chemical Master Equation for the stochastic kinetics. Qualitatively compare the two.\n", "\n", "**c)** Find the expected value of the extinction time $T_0$ when starting from an initial population of $n$.\n", "\n", "**d)** Find the variance of the extinction time $T_0$.\n", "\n", "**e)** Consider two independent identical death processes, one for species $A$ starting at an initial population of $n_A$ and one for species $B$ starting at an initial population of $n_B$. Denoting by $T_A$ the extinction time of $A$ and by $T_B$ that of $B$, find (an estimation of) the probability that $T_A \\leq T_B$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Problem 2: Birth and Death Process\n", "\n", "

" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In this problem, we add a birth reaction to the death process from the previous problem, i.e., the reaction $A \\to A + A$ with rate constant $\\gamma$.\n", "\n", "\n", "**a)** Write down an ODE describing the deterministic kinetics.\n", "\n", "**b)** Write down the Chemical Master Equation for the stochastic kinetics. Qualitatively compare the two.\n", "\n", "**c)** Find the probability of extinction when starting from an initial population of $n$.\n", "\n", "**d)** For the case of almost-sure extinction, find the expected extinction time. Evaluate this formula for $n=1,2,3$." ] } ], "metadata": { "anaconda-cloud": {}, "kernelspec": { "display_name": "Python 3 (ipykernel)", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.12.4" } }, "nbformat": 4, "nbformat_minor": 4 }