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Elementary discrete dynamical system biology problems

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  1. Consider the dynamical system \begin{align*} z_{n+1}=f(z_n) \quad \text{for $n=0,1,2,3,\ldots$ ,} \end{align*} where $f$ is defined by \begin{align*} f(\bigstar)=\frac{3\bigstar^2}{2+\bigstar^2}. \end{align*} The graph of $z_{n+1}=f(z_n)$ is shown below along with the diagaonal $z_{n+1}=z_n$.

    Discrete dynamical system example function 3
    1. Write down an equation for the equilibria and show that the system has the equilibria 0, 1, and 2.
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    2. Use cobwebbing to determine what happens to the system if the initial condition $z_0$ is slightly less than 1 and if the initial condition $z_0$ is slightly greater than 1.
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    3. Imagine that $z_n$ represents the firing rate in time step $n$ of a neuron in a monkey's brain. At time step zero, either a banana or a stick is shown briefly to the monkey. If the banana was shown, the neuron starts firing at a rate $z_0$ of about 1.2. If a stick was shown, the neuron starts firing at a rate $z_0$ of about 0.8.

      Imagine that you don't know if the monkey was shown a stick or a banana at time step zero. However, after one of these was briefly shown to the monkey, the firing rate of the neuron has evolved for many time steps, and now the neuron is firing at a rate $z_n$ of about 2. Was the monkey shown a banana or a stick? Justify your answer using the results from the first two parts.

  2. The mass of a fish is increasing by 195 grams per year.

    1. Set up a dynamical system model that describes the evolution of the fish's mass. Be sure to define your notation, including the meaning of any variable for time.
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    2. If the mass of a fish in year 0 is 53 grams, calculate the mass in the next 4 years. Be sure to show how you used the formula in part (a) and to write your results using the notation you defined.
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    3. Find all equilibria of your dynamical system.
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  3. A population of tree frogs is decreasing at a rate of about 9% per year. If it continues at this rate, how many years will it take for the population to decline by one-half? If the current population is 72,000, in how many years will the population reach 18,000 tree frogs?

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  4. Bacteria are growing in a beaker so that the population size increases by 14.87% every minute.
    1. If $b_t$ is the bacteria population size in minute $t$, set up a dynamics system model that describes the evolution of the population size.
      $b_{t+1} - b_t =$
      , for $t=0,1,2,3 \ldots$
      (Here we are explicitly reminding ourselves that the equation is shorthand for many equations, one for $t=0$, one for $t=1$, etc.)
    2. How long does it take the population size to double?

      $T_{\text{double}} = $

      Your answer will look a lot prettier if you round to four significant digits. (In this case, this means round to the nearest thousandth, as there should be one digit to the left of the decimal.) The second blank is for a unit.

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    3. If the population continues to grow at this rate, by what factor does the population size increase in one hour?
      In two hours?
      In four hours?
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    4. An experiment is begun at midnight with just a few bacteria so that the fraction of the beaker that the bacteria occupy is approximately $0.00000005959 = 5.959 \times 10^{-8}$. With this initial condition, the bacteria completely fill the beaker after two hours, at 2 AM. At what time was the beaker half full? The beaker was half full at
      :
      .

      Write your answer in the form: hh:mm AM/PM. Round your answer to the nearest minute.

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    5. Imagine the researchers realized before 2AM that the bacteria were about to overflow the beaker. They found three more empty beakers of the same size as the original beaker so that they had a total of four beakers to hold the bacteria. At what time did the bacteria fill all four beakers?
      :
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  5. In the first days of life, the cells in a human embryo divide into two cells approximately every day.
    1. Assuming the number of cells doubles every day, write down a dynamical system for the evolution of the number of cells. Be sure to define your notation, including the meaning of any variable for time.
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    2. After fertilization, the embryo consists of a single cell. Solve the dynamical system to obtain an expression for the number of cells as a function of the number of days since fertilization.
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    3. If a pregnancy lasts 40 weeks and the cell division continued at the same rate, how many cells would the baby have upon being born? How does this number compare to the number of atoms in the observable universe, which is estimated to be about $10^{80}$? What can you conclude about the rate of cell division during the course of the pregnancy? (I.e., does it seem reasonable that cell division would continue at the same rate, or does this model give strong evidence against that hypothesis.)
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  6. A population of orcs is increasing at a rate of about 8% per year. If it continues at this rate, how many years will it take for the population to double? If the current population is 70,000, in how many years will the population reach 280,000 orcs?

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  7. In a population of wolves, each year the number of babies is 23% of the population size at the beginning of the year. However, a disease is killing 26% of the population each year.
    1. Set up a dynamical system model that describes the evolution of the population size. Be sure to define your notation, including the meaning of any variable for time.
    2. If in one year, there are 2,000 wolves, calculate the number of wolves for the next 4 years. (Don't worry about having fractional wolves.) Be sure to show how you used the formula in part (a) and to write your results using the notation you defined.
    3. Find all equilibria of your dynamical system.

  8. A drug is administered to healthy volunteers and its concentration in the blood is monitored at ten minute intervals. The resulting concentrations are shown in the below table. The change in drug concentration after each interval is calculated (see table) and the drug change is plotted versus drug concentration (see graph). Two possible fits of lines to the data and their slopes are shown.

    Time (min)Drug concentration ($\mu$g/ml)Drug change
    067-25
    1042-14
    2028-13
    3015-8
    407
    Drug concentration change versus concentration line example
    1. Given that a fixed fraction of the drug is removed from the blood in each ten minute interval, set up a dynamical system model that describes the evolution of the drug concentration. Be sure to define your notation, including the meaning of any variable for time. Your model should include an unknown parameter, which you should clearly define. This model should not yet be based on the above data.
    2. Based on the data, determine the unknown parameter from your model. You should use your model to determine which of the above lines will give you the correct value of the parameter. Write down your resulting dynamical system model that describes the data. Be sure to include an initial condition.
    3. Use your dynamical system to compute its prediction for the drug concentration at times 10, 20, 30, and 40 minutes after administration of the drug.

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Elementary dynamical systems