19.4 Exercises

Exercise 19.1 (Inspired by Logan and Wolesensky (2009)) Compute the trace and determinant for each of these linear systems. Use the trace-determinant condition to classify the stability of the equilibrium solutions. Verify your stability results are consistent when analyzing stability from calculating the eigenvalues.

  1. \(\displaystyle \frac{dx}{dt} = -x, \; \frac{dy}{dt} = -2y\)
  2. \(\displaystyle \frac{dx}{dt} = 3x+y, \; \frac{dy}{dt} = 2x+4y\)
  3. \(\displaystyle \frac{dx}{dt} = 8x-11y, \; \frac{dy}{dt} = 6x-9y\)
  4. \(\displaystyle \frac{dx}{dt}= 3x-y, \; \frac{dy}{dt}=3y\)
  5. \(\displaystyle \frac{dx}{dt} = -2x-3y, \; \frac{dy}{dt} = 3x-2y\)
 

Exercise 19.2 (Inspired by Logan and Wolesensky (2009)) The following equation can be applied to study cell differentiation:

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

  1. Previously you verified that \((x,y)=(1,1)\) is an equilibrium solution for this system. What is the Jacobian matrix at that equilibrium solution?
  2. What is tr\((J)\) and det\((J)\) at that equilibrium soltion?
  3. Evaluate the stability of the equilibrium solution solution using relationships between the trace and determinant.

 

Exercise 19.3 (Inspired by Logan and Wolesensky (2009)) Consider the following predator-prey model, where the carrying capacity of the predator (\(y\)) depends on the prey population (\(x\)):

\[\begin{equation} \begin{split} x' &= \frac{2}{3} x \cdot \left(1- \frac{x}{4} \right) - \frac{1}{6} xy \\ y' &= 0.5y \cdot \left(1 - \frac{y}{x} \right), \end{split} \end{equation}\]

  1. There are two equilibrium solutions for this differential equation. What are they? Hint: first determine where \(y'=0\) and then substitute your solutions into \(x'=0\).
  2. Use the command phaseplane to visualize this system of equations.
  3. Compute the Jacobian matrix for both equilibrium solutions.
  4. Use the trace-determinant relationships to evaluate the stability of the equilibrium solutions. Is that analysis consistent with your phaseplane?

 

Exercise 19.4 (Inspired by Logan and Wolesensky (2009)) Consider the linear system of equations: \[\begin{equation} \begin{split} \frac{dx}{dt}&=-ax-y \\ \frac{dy}{dt} &= -x-ay \end{split} \end{equation}\]

Apply the relationships between the trace and determinant to classify the stability of the equilibrium solution for different values of \(a\). Be sure to include cases where the system will be a spiral source or sink.

 

Exercise 19.5 All of the following systems have an equilibrium solution at the origin (0,0). Compute the Jacobian of these solutions and apply the trace and determinant conditions to analyze the stability. The stability will be a function of the parameter \(\mu\). In your stability analysis you only need to classify differences between a source, sink, or saddle.

  1. \[\begin{equation} \begin{split} x' &=x+ \mu y \\ y' &= \mu x -y, \end{split} \end{equation}\]
  2. \[\begin{equation} \begin{split} x' &=x+y\\ y' &= \mu x +y, \end{split} \end{equation}\]
  3. \[\begin{equation} \begin{split} x' &= y \\ y' &=x^{2}-x+\mu y, \end{split} \end{equation}\]

 

Exercise 19.6 For the lynx-hare system equilibrium \(\displaystyle \left( \frac{d}{eb}, \frac{r}{b} \right)\), determine conditions on the parameters where the system will be a spiral source and sprial sink.
 

Exercise 19.7 (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:

\[\begin{equation} \begin{split} \frac{dC}{dt}&=P - aC - bHC \\ \frac{dH}{dt} &= ebHC-dC \end{split} \tag{19.5} \end{equation}\]

  1. Equation (19.5) has two equilibrium solutions. What are they?
  2. Evaluate the Jacobian at each of the equilibrium solutions.
  3. Evaluate the stability of each equilibrium solution using relationships between the trace and determinant. Be sure to include cases where the system will be a spiral source or sink.
Exercise 19.8 Apply the quadratic formula to \(\lambda^{2} - \mbox{tr}(A)\lambda + \det(A)=0\) obtain Equation (19.4).

 

Exercise 19.9 Assume that you have two complex conjugate eigenvalues: \(\lambda_{1} = a \pm bi\) and \(\lambda_{2} = a - bi\).

  1. What is an expression for \(\lambda_{1} + \lambda_{2}\)?
  2. What is an expression for \(\lambda_{1} \cdot \lambda_{2}\)?
  3. Use your answers from the previous two results to show that tr\((A)=2a\) and \(\det(A)=a^{2}+b^{2}\).
  4. If you were to have a system that is a spiral sink, what conditions on \(a\) and \(b\) need to hold?
  5. Create a linear two-dimensional system where the equilibrium solution is a spiral sink. Show your system and the corresponding phaseplane.
 

Exercise 19.10 Consider a two-dimensional system where tr\((A)=0\) and det\((A)>0\).

  1. Given those conditions, explain why \(\lambda_{1} + \lambda_{2}=0\) and \(\lambda_{1} \cdot \lambda_{2}>0\).
  2. What does \(\lambda_{1} + \lambda_{2}=0\) tell you about the relationship between \(\lambda_{1}\) and \(\lambda_{2}\)?
  3. What does \(\lambda_{1} \cdot \lambda_{2}>0\) tell you about the relationship between \(\lambda_{1}\) and \(\lambda_{2}\)?
  4. Look back to your previous two responses. First explain why \(\lambda_{1}\) and \(\lambda_{2}\) must be imaginary eigenvalues (in other words, not real values). Then explain why \(\lambda_{1,2}= \pm bi\).
  5. Given these constraints, what would the phaseplane for this system be?
  6. Create a linear two-dimensional system where tr\((A)=0\) and det\((A)>0\). Show your system and the phaseplane.

References

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