16.3 Exercises

Exercise 16.1 Consider the following nonlinear system of equations: \[\begin{equation} \begin{split} \frac{dx}{dt} &= x - .5xy \\ \frac{dy}{dt} &= .5yx-y \end{split} \end{equation}\]

  1. What are the equations for the nullclines for this differential equation?
  2. What are the equilibrium solutions for this differential equation?
  3. Generate a phaseplane that includes all equilibrium solutions.
  4. Based on the phaseplane, evaluate the stability of the equilibrium solution.
 

Exercise 16.2 (Inspired by Logan and Wolesensky (2009)) A population of fish has natural predators. A model that describes this interaction is the following:

\[\begin{equation} \begin{split} \frac{dN}{dt} &= N - .3NP \\ \frac{dP}{dt} &= .5NP - P , \end{split} \end{equation}\]

  1. What are the equations for the nullclines for this differential equation?
  2. What are the equilibrium solutions for this differential equation?
  3. Generate a phaseplane that includes all the equilibrium solutions.
  4. Based on the phaseplane, evaluate the stability of the equilibrium solution.

 

Exercise 16.3 Consider the following system:

\[\begin{equation} \begin{split} \frac{dx}{dt} &= y^{2} \\ \frac{dy}{dt} &= -\frac{2}{3} x, \end{split} \end{equation}\]

  1. What are the nullclines for this system of equations?
  2. What is the equilibrium solution for this system of equations?
  3. Generate a phaseplane that includes the equilibrium solution.
  4. Based on the phaseplane, evaluate the stability of the equilibrium solution.

 

Exercise 16.4 The Van der Pol Equation is a second-order differential equation used to study radio circuits: \(x'' + \mu \cdot (x^{2}-1) x' + x = 0\), where \(\mu\) is a parameter.

  1. Let \(y'=x\). Show that with this change of variables the Van de Pol equation can be written as a system: \[\begin{equation} \begin{split} \frac{dx}{dt} &= y \\ \frac{dy}{dt} &= -x-\mu (x^{2}-1)y \end{split} \end{equation}\]
  2. Verify that the only equilibrium solution is \((0,0)\). Set \(\mu=1\). Generate a phaseplane that includes the equilibrium solution.
  3. Based on the phaseplane, evaluate the stability of the equilibrium solution.
 

Exercise 16.5 Consider the following nonlinear system:

\[\begin{equation} \begin{split} \frac{dx}{dt} &= x-y \\ \frac{dy}{dt} &=-y + \frac{5x^2}{4+x^{2}}, \end{split} \end{equation}\]

  1. What are the two equations for the nullclines?
  2. Using desmos (or some other graphing utility), graph the two nullclines simultaneous. What are the intersection points?
  3. Generate a phaseplane for this system that contains all the equilibrium solutions.
  4. Let’s say instead that \(\displaystyle \frac{dx}{dt} = bx-y\), where \(b\) is a parameter such that \(0 \leq b \leq 2\). How many equilibrium solutions do you have as \(b\) changes?

 

Exercise 16.6 (Inspired by Logan and Wolesensky (2009)) Let \(C\) be the amount of carbon in a forest ecosystem, with \(P\) be the rate of increase due to photosynthesis. Herbivores \(H\) consume carbon on the following predator-prey model:

\[\begin{equation} \begin{split} \frac{dC}{dt}&=- aC - bHC \\ \frac{dH}{dt} &= ebHC-dC \end{split} \end{equation}\]

  1. What are the equations for the nullclines?
  2. Determine the equilibrium solutions.

References

Logan, J. David, and William Wolesensky. 2009. Mathematical Methods in Biology. 1st ed. Hoboken, N.J: Wiley.