Quiz 3

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Total points: 1

The Ricker model for population growth can be written as $$\diff{p}{t} = 0.65 p e^{- 0.00278 p}- 0.01 p$$ where $p$ is the population size (or density) of a population.
1. Calculate the equilibria of the model. Round your answers to the nearest 100.
  
  $p_{eq}=$
  
  (If more than one, separate by commas.)
2. Plot the equilibria on the below phase line. In each segment divided by the equilibria, plot at least one direction vector. From the direction vectors, determine the stability of the equilibria. Represent the stability using a filled circle for a stable equilibrium and an open circle for an unstable equilibrium.
  
  Feedback from applet
  direction vectors: correct directions:
  direction vectors: no stray vectors:
  equilibria: locations:
  equilibria: number of equilibria:
  equilibria: stability:
  solutions: final values:
  solutions: initial conditions:
  solutions: number of solutions:
  
  In the above applet, increase $n_e$ to specify the number of equilibria and move the resulting points to the correct locations. Increase $n_v$ to reveal direction vectors. Move at least one direction vector to each segment of the phase line and point them to specify whether a solution in that segment increases or decreases. Click the equilibria to specify stability by changing them between closed and open circles
3. Represent the solution to differential equation on the above phase line for the following initial conditions.
  1. $p(0)=1000$
  2. $p(0)= 2500$
  Increase $n_s$ to reveal an arrow for each solution. Move the arrow so that it starts at the initial condition and ends where $p(t)$ approaches as $t$ gets large (or to the edge of the shown portion of the phase line if the solution leaves that portion of the phase line).
4. Plot the two solutions versus time using the below applet. Also, represent the equilibria as constant solutions. Draw stable equilibria as solid lines and unstable equilibria as dashed line.
  
  Feedback from applet
  Equilibrium stability:
  Final values:
  Found equilibria:
  Initial conditions:
  Number of curves:
  Valid shapes:
  
  Increase $n_c$ to reveal curves one at a time. (You can only change the last curve displayed.) For each curve, drag the points to determine the general shape of the solution. The applet requires that the initial condition and final value match what you should have determined from the phase line. When you draw a constant equilibrium solution, it will turn red. You can then click it to change between a solid and a dashed line.
Consider the following model of competition between two species with population sizes $p_1(t)$ and $p_2(t)$: \begin{align*} \diff{p_1}{t} & = 1.4 p_1 \left(1-\frac{p_1+0.4p_2}{ 2700 }\right)\\ \diff{p_2}{t} & = 1.3 p_2 \left(1-\frac{p_2+0.4p_1}{ 3500 }\right). \end{align*}
1. Calculate the nullclines.
  The $p_1$-nullcline:
  
  or
  
  The $p_2$-nullcline:
  
  or
2. Graph the nullclines with the following phase plane applet. (Use the thick solid blue lines for the $p_1$-nullcline and the thin dashed green lines for the $p_2$-nullcline.)
  
  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:
3. Graph the equilibria using the above phase plane applet. Set the step counter to 2, increase $n_e$ to reveal points for the equilibria, and move them to the correct locations. Only include biologically plausible equilibria. The (biologically plausible) equilibria are:
  
  (Enter the equilibria in any order, separated by commas, such as (1,3),(2,4),(5,6).)
4. Use the above phase plane applet to sketch direction vectors:
  1. in each region the phase plane separated by the nullclines (include only the biologically plausible quadratic of the phase plane), and
  2. and on each piece of each nullcline, as divided by the opposite nullcline (doing this separately for each line of the nullclines, include only the biologically plausible quadrant of the phase plane).
  Set the step to 3 to reveal vectors for each region of the phase plane. Move one vector to each region and set the corresponding general direction of each vector. Then, set the step counter to 4 to reveal vectors for each segment of the nullclines. Move one vector to each piece of the nullcline and set the corresponding general direction of each vector.
5. Sketch the solution on the phase plane with initial condition $p_1(0)=1750$ and $p_2(0) = 700$. In the applet, change the step counter to 5, move the large green point to the initial condition and move the small cyan point so that the segment between them follows the general direction of the vector in the corresponding region. Increase nsegs to reveal most segments so that you continue to draw the solution trajectory $(p_1(t),p_2(t))$. End the trajectory at the point where the solution must be after a long time.
  
  After a long time, the trajectory $(p_1(t),p_2(t))$ approaches what point?
6. Translate your results to plots of $p_1$ and $p_2$ versus time:
  
  Feedback from applet
  correct shapes:
  final values:
  initial conditions:
  
  Round your answers to the nearest 50 when using this applet.
7. Which species won the competition?

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Quiz 3

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Math 2241, Spring 2023