Math Insight

Quiz 5

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Total points: 8
  1. A population of cheetahs would be increasing at the rate of $a$ per year, except that due to a scarcity of antelope, there is only enough food to support $W$ cheetahs. Therefore, if $c_t$ is the number of cheetahs in year $t$, the population evolves according to the dynamical system \begin{align*} c_{t+1} - c_{t} = a c_t\left(1 - \frac{ c_t}{ W }\right). \end{align*}
    1. Calculate the equilibria of the dynamical system.

      Equilibria:

      If there is more than one equilibrium, enter them in increasing order, separated by commas. (The number $W$ is positive.)

    2. If $a = 0.9$, determine the stability of the equilibria.

      Specify the stability of each equilibrium in the same order as above. Enter stable if an equilibrium is stable or unstable if it is unstable. Separate answers by commas.

      Equilibria are stable:

      For example, if there are four equilibria and they are, in order, stable, unstable, unstable, and unstable, then you should enter stable, unstable, unstable, unstable in the answer blank.

    3. If $a = 0.9$, $W = 490$, and the initial population is $c_0=770$, what will happen to the population after a long time?

      After a long time, the population size will approach
      cheetahs.

    4. If $a = 3.3$, determine the stability of the equilibria.

      Specify the stability of each equilibrium in the same order as above. Enter stable if an equilibrium is stable or unstable if it is unstable. Separate answers by commas.

      Equilibria are stable:

      For example, if there are four equilibria and they are, in order, stable, unstable, unstable, and unstable, then you should enter stable, unstable, unstable, unstable in the answer blank.

    5. For what range of $a$ is there a stable nonzero equilibrium?

      There is a stable nonzero equilibrium for

      $< a < $
      .

  2. The discrete dynamical system below has an equilibrium $E=5.6$. Determine if the equilibrium point, $E=5.6$, is stable or unstable. Hint: Use the stability theorem for discrete systems. \[ \left\{ \begin{array}{r c l} u_{ n+1} & = & 1.7u_n + \frac{1}{8}u_n^2 \\ u_0 & = & 1\\ \end{array} \right. \] $E = 5.6 $ is
    (write "stable" or "unstable")

  3. Find the linear approximation to the function \[ g( t ) = e^{- t + 7} \left(t - 5\right) \] around $t= 7$.
    $L( t ) =$