Math Insight

Quiz 11

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Total points: 8
  1. Consider the SIR model for an infectious disease, \begin{align*} S'(t) &= - 0.7SI\\ I'(t) &= 0.7SI - 0.21 I. \end{align*} Since $S(t)$ and $I(t)$ represent the fraction of the population that is susceptible and infected, these state variables are always between 0 and 1.

    Sketch the phase plane of this dynamical system, including nullclines and a direction vector in each region of the phase plane divided by the nullclines. Only consider biologically plausible regions of the phase plane.

    Feedback from applet
    direction of vectors:
    location of vectors:
    nullclines:

    Use the above applet to sketch the phase plane of this dynamical system. Move the thick solid blue lines so that, together, they form the $S$-nullcline. Move the thin dashed green lines so that, together, they form the $I$-nullcline. Move the two black points into different regions of the phase plane that are divided by the nullclines. Rotate each vector around the points so that the vector indicates the rough direction that a solution should move in the phase plane through the point. Each vector can point in one of eight directions. These directions simply indicate whether $S$ and $I$ are increasing, decreasing, or staying constant at that point.

  2. A population of sponges of size $s(t)$ and a population of corals of size $c(t)$ compete for space in a reef, where $t$ is time in years. If either population were alone in the reef, the reef could support a population of 2100 sponges (i.e., the carrying capacity for sponges alone would be 2100) or a population of 2900 corals (i.e., the carrying capacity for corals alone would be 2900). Imagine that each additional sponge effectively decreases the carrying capacity for corals by $\frac{4}{5}$ and vice versa. The result is that the competition of the species can be modeled by \begin{align*} \diff{ s }{t} &= 0.2s\left(1 - \frac{ s + \frac{4}{5}c }{ 2100 } \right)\\ \diff{ c }{t} &= 0.6c\left(1 - \frac{ c + \frac{4}{5}s }{ 2900 } \right). \end{align*} Follow the below steps to analyze the system. Use this phase plane applet to report the results of the analysis.
    Feedback from applet
    Step 1: nullclines:
    Step 2: equilibria:
    Step 2: number of equilibria:
    Step 3: vector directions in regions:
    Step 3: vector locations in regions:
    Step 4: vector directions on nullclines:
    Step 4: vector locations on nullclines:
    Step 5: initial condition:
    Step 5: solution trajectory end point:
    Step 5: solution trajectory follows vector field:
    1. Calculate the nullclines. The $s$-nullcline is made of two pieces. Enter the simpler equation first:

       or

      The $c$-nullcline is made of two pieces. Enter the simpler equation first:

       or

      Use the above phase plane applet to plot the nullcline. Use the thick solid blue lines to plot the $s$-nullcline and the thin dashed green lines to plot the $c$-nullcline. When the slider for step is set to step=1, you can drag the points on the nullclines to move them. Dotted lines are included as a hint to the location of part of the nullclines.

    2. Identify all equilibria. (Include only biologically plausible equilibria, i.e., equilibria where both population sizes are positive.) Give their values and show them in the phase plane.

      Equilibria:

      Separate multiple answers by commas. For example if equilibria were $(s,c)=(1,2)$ and $(s,c)=(3,4)$, enter (1,2),(3,4). You can round your answer to the nearest integer.

      To show the equilibria on above phase plane, drag the step slider to step=2, which will cause a second slider labeled $n_e$ to appear. Use the $n_e$ slider to indicate the number of equilibria. Then drag the red points that appear to the correct locations of the equilibria. When step=2, the coordinates of the red points are displayed to allow you to estimate the equilibria values.

    3. The nullclines divide the phase plane into regions. Sketch a direction vector in each of these regions. You only need to consider the biologically plausible part of the phase plane, i.e., where $s(t) \ge 0 $ and $c(t) \ge 0$.

      To plot the direction vectors, set step=3 in the above phase plane to reveal the vectors. The black vectors can be dragged into the regions divided by the nullclines; drag one into each region. Then, for each vector, drag the point by the arrowhead to change the direction so that it indicates whether $s$ and $c$ increase, decrease, or stay constant at that point.

    4. Sketch a direction vector on each segment of each nullcline. The segments of a nullcline are the parts divided by the other nullcline.

      In the above phase plane, set step=4 to reveal purple arrows on each nullcline branch. The number of arrows that appear on a nullcline branch will match the number of segments of that branch (in the biologically plausible section of the phase plane). Move each vector to a different nullcline segment. Then, for each vector, drag the point by the arrowhead to change the direction so that it indicates whether $s$ and $c$ increase, decrease, or stay constant at that point.

    5. At time $t=0$, imagine that there are $s(0)=750$ sponges and $c(0) = 580$ corals. Sketch a plausible solution to the dynamical system with these initial conditions on the above phase plane. The solution curve should represent $(s(t),c(t))$ from the initial condition $t=0$ through very large values of $t$.

      Set step=5 to reveal the green point for the initial condition and the cyan line segments to use to approximate the solution trajectory (the curve $(s(t),c(t))$ through the phase plane). Drag the green point to the initial condition. Then drag the cyan points so that the cyan line segments form a plausible solution trajectory. If you would like more or fewer segments with which to draw the curve, you can change the number of segments with the nsegs slider.

      (For the solution trajectory to be valid, it must respect the direction arrows you determined in the previous steps. When it crosses a nullcline, it should move exactly in the direction given by the corresponding purple arrow. In a region between nullclines, the solution can change directions as long as it continues to follow the general direction of the corresponding black arrow, so that $s$ and $c$ increase or decrease as required.)

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Math 1241, Fall 2013