16.3 Exercises
Exercise 16.1 Consider the following nonlinear system of equations: dxdt=x−.5xydydt=.5yx−y
- What are the equations for the nullclines for this differential equation?
- What are the equilibrium solutions for this differential equation?
- Generate a phaseplane that includes all equilibrium solutions.
- 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:
dNdt=N−.3NPdPdt=.5NP−P,
- What are the equations for the nullclines for this differential equation?
- What are the equilibrium solutions for this differential equation?
- Generate a phaseplane that includes all the equilibrium solutions.
- Based on the phaseplane, evaluate the stability of the equilibrium solution.
Exercise 16.3 Consider the following system:
dxdt=y2dydt=−23x,
- What are the nullclines for this system of equations?
- What is the equilibrium solution for this system of equations?
- Generate a phaseplane that includes the equilibrium solution.
- 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″, where \mu is a parameter.
- 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}
- Verify that the only equilibrium solution is (0,0). Set \mu=1. Generate a phaseplane that includes the equilibrium solution.
- 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}
- What are the two equations for the nullclines?
- Using desmos (or some other graphing utility), graph the two nullclines simultaneous. What are the intersection points?
- Generate a phaseplane for this system that contains all the equilibrium solutions.
- 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}
- What are the equations for the nullclines?
- Determine the equilibrium solutions.