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Elementary discrete dynamical system biology problems

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  1. In a population of pheasants, each year the number of babies is 28% of the population size at the beginning of the year. However, a disease is killing 14% 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 35,000 pheasants, calculate the number of pheasants for the next 4 years. (Don't worry about having fractional pheasants.) 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.

  2. 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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  3. A population of sea lions is decreasing at a rate of about 1% 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 104,000, in how many years will the population reach 13,000 sea lions?

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  4. A population of robins would be increasing at the rate of 66% per year, except that due to a scarcity of worms, the limited food slows down the growth rate as the population gets larger. Therefore, if $0.66_t$ is the number of robins in year $t$, the population evolves according to the dynamical system \begin{align*} 0.66_{t+1} &= g(0.66_t)\\ 0.66_0 &= 940, \end{align*}

    where $g$ is graphed below, along with the diagonal $0.66_{t+1} =0.66_t$.

    1. Starting with the initial population size $0.66_0=940$, cobweb to determine the evolution of the population size $0.66_t$. What happens to the robin population size after one year? After a long time has passed?

    2. Estimate the equilibria of the dynamical system and determine their stability. Label points corresponding to the equilibria on the above graph.

  5. A population of deer is increasing at a rate of about 10% per year. If it continues at this rate, how many years will it take for the population to double? If the current population is 52,000, in how many years will the population reach 208,000 deer?

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  6. 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 $t=0,1,2,3 \ldots$

    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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  7. The mass of a fish is increasing by 44 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 37 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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  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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Math 1241, Fall 2020