20.4 Exercises
Exercise 20.1 Apply local linearization to classify stability of the following differential equations:
- \(\displaystyle \frac{dx}{dt} = x-x^{2}\)
- \(\displaystyle \frac{dx}{dt} = -x^{2}\)
- \(\displaystyle \frac{dx}{dt} = -x-x^{2}\)
Exercise 20.4 (Inspired by Logan and Wolesensky (2009)) Through constructing a bifurcation diagram, determine the behavior of solutions near the origin for the following system:
\[\begin{equation} \frac{\vec{x}}{dt} = \begin{pmatrix} 3 & b \\ b & 1 \end{pmatrix} \vec{x}. \end{equation}\]Exercise 20.5 (Inspired by Logan and Wolesensky (2009)) Consider the linear system of equations:
\[\begin{equation} \begin{split} \frac{dx}{dt}&=-ax-y \\ \frac{dy}{dt} &= -x-ay \end{split} \end{equation}\]
Construct a bifurcation diagram for this system of equations.
Exercise 20.6 Consider the following highly nonlinear system:
\[\begin{equation} \begin{split} \frac{dx}{dt} =-y-x(x^2+y^2-1) \\ \frac{dy}{dt}=x-y(x^2+y^2-1) \end{split} \end{equation}\]
We are going to transform the system by defining new variables \(x=r \cos \theta\) and \(y=r \sin \theta\). Observe that \(r^2=x^2+y^2\).
- Consider the equation \(r^2=x^2+y^2\), where \(r\), \(x\), and \(y\) are all functions of time. Apply implicit differentiation to determine a differential equation for \(\displaystyle \frac{d(r^{2})}{dt}\), expressed in terms of \(x\), \(y\), \(\displaystyle \frac{dx}{dt}\) and \(\displaystyle \frac{dy}{dt}\).
- Multiply \(\displaystyle \frac{dx}{dt}\) by \(2x\) and \(\displaystyle \frac{dy}{dt}\) by \(2y\) on both sides of the equation. Then add the two equations together. You should get an expression for \(\displaystyle \frac{d(r^{2})}{dt}\) in terms of \(x\) and \(y\).
- Re-write the equation for the right hand side of \(\displaystyle \frac{d(r^{2})}{dt}\) in terms of \(r^{2}\).
- Use your equation that you found to verify that \[\begin{equation} \frac{dX}{dt} = -X(X-1), \mbox{ where } X=r^{2} \end{equation}\]
- Verify that \(X=1\) is a stable node and \(X=0\) is unstable.
- As discussed in this section this system has a stable limit cycle. What quick and easy modification to our system could you do to the system to ensure that this is a unstable limit cycle? Justify your work.
Exercise 20.7 Construct a bifurcation diagram for \(\displaystyle \frac{dX}{dt} = - X(X-\mu)\), \(\mu\) is a parameter. Explain how you can apply that result to understanding the bifurcation diagram of the system:
\[\begin{equation} \begin{split} \frac{dx}{dt} =-y- x(x^2+y^2-\mu) \\ \frac{dy}{dt}=x- y(x^2+y^2-\mu) \end{split} \end{equation}\]
This system is an example of a .
Exercise 20.8 (Inspired by Logan and Wolesensky (2009)) Consider following predator-prey model:
\[\begin{equation} \begin{split} \frac{dx}{dt}&=\frac{2}{3}x\left(1 - \frac{x}{4} \right) - \frac{xy}{1+x} \\ \frac{dy}{dt} &= ry\left(1 - \frac{y}{x} \right) \end{split} \end{equation}\]
- Explain the various terms in this model and their biological meaning.
- Determine the equilibrium solutions.
- Evaluate the Jacobian at each of the equilibrium solutions.
- Construct a bifurcation diagram (with the parameter \(r\)) for each of the equilibrium solutions.
Exercise 20.9 (Inspired by Logan and Wolesensky (2009)) The immune response to HIV has been described with differential equations. In the early stages (before the body is swamped by the HIV virions) the dynamics of the virus can be described by the following system of equations, where \(v\) is the virus load and \(x\) the immune response:
\[\begin{equation} \begin{split} \frac{dv}{dt}&=rv - pxv \\ \frac{dx}{dt} &= cv-bx \end{split} \end{equation}\]
- Explain the various terms in this model and their biological meaning.
- Determine the equilibrium solutions.
- Evaluate the Jacobian at each of the equilibrium solutions.
- Construct a bifurcation diagram for each of the equilibrium solutions.