5.4 Exercises
Exercise 5.1 What are the equilibrium solutions to the following differential equations?
- \(\displaystyle \frac{dS}{dt} = 0.3 \cdot(10-S)\)
- \(\displaystyle \frac{dP}{dt} = P \cdot(P-1)(P-2)\)
Then classify the stability of the equilbrium solutions using the local linearization stability test.
Exercise 5.3 A population grows according to the equation \(\displaystyle \frac{dP}{dt} = \frac{P}{1+2P} - 0.2P\).
- Determine the equilibrium solutions for this differential equation.
- Classify the stability of the equilibrium solutions using the local linearization stability test.
Exercise 5.4 (Inspired by Logan and Wolesensky (2009)) A cell with radius \(r\) assimilates nutrients at a rate proportional to its surface area, but uses nutrients proportional to its volume, according to the following differential equation: \[ \frac{dr}{dt} = 4 \pi r^{2} - \frac{4}{3} \pi r^{3}. \]
- Determine the equilibrium solutions for this differential equation.
- Construct a phase line for this differential equation to classify the stability of the equilibrium solutions.
Exercise 5.5 (Inspired by Thornley and Johnson (1990)) The Chanter equation of growth is the following, where \(W\) is the weight of an object: \[\begin{equation} \frac{dW}{dt} = W(3-W)e^{-Dt}, \end{equation}\]
Use this differential equation to answer the following questions.
- What happens to the rate of growth (\(\displaystyle \frac{dW}{dt}\)) as \(t\) grows large?
- What are the equilibrium solutions to this model? Are they stable or unstable? Notice how the equilbrium solutions are the same as those for the logistic model. Based on your previous work, do why would this model be a more realistic model of growth than the logistic model \(\displaystyle \frac{dW}{dt} = W(3-W)\)?
Exercise 5.6 Red blood cells are formed from stem cells in the bone marrow. The red blood cell density \(r\) satisfies an equation of the form
\[\begin{equation} \frac{dr}{dt} = \frac{br}{1+r^{n}} - c r, \end{equation}\]
where \(n>1\) and \(b>1\) and \(c>0\). Find all the equilibrium solutions \(r_{*}\) to this differential equation. Hint: can you factor an \(r\) from your equation first?
Exercise 5.7 (Inspired by Hugo van den Berg (2011)) Organisms that live in a saline environment biochemically maintain the amount of salt in their blood stream. An equation that represents the level of \(S\) in the blood is the following:
\[ \frac{dS}{dt} = I + p \cdot (W - S) \]
Where the parameter \(I\) represents the active uptake of salt, \(p\) is the permeability of the skin, and \(W\) is the salinity in the water.
- First set \(I=0\). Determine the equilibrium solutions for this differential equation. Express your answer \(S_{*}\) in terms of the parameters \(p\), and \(W\).
- Next consider \(I>0\). Determine the equilibrium solutions for this differential equation. Express your answer \(S_{*}\) in terms of the parameters \(p\), \(W\), and \(I\). Why should your new equilbrium solution be greater than the equilibrium solution from the previous problem?
- Classify the stability of both equilibrium solutions using the local linearization stability test.
Exercise 5.8 (Inspired by Logan and Wolesensky (2009)) The immigration rate of bird species (species per time) from a mainland to an offshore island is \(I_{m} \cdot (1-S/P)\), where \(I_{m}\) is the maximum immigration rate, \(P\) is the size of the source pool of species on the mainland, and \(S\) is the number of species already occupying the island. Further, the extinction rate is \(E \cdot S / P\), where \(E\) is the maximum extinction rate. The growth rate of the number of species on the island is the immigration rate minus the extinction rate, given by the following differential equation:
\[\begin{equation} \frac{dS}{dt} = I_{m} \left(1-\frac{S}{P} \right) - \frac{ES}{P} \end{equation}\]
- Determine the equilibrium solutions \(S_{*}\) for this differential equation. Expression your answer in terms of \(I_{M}\), \(P\), and \(E\).
- Classify the stability of the equilibrium solutions using the local linearization stability test.
Exercise 5.9 A colony of bacteria growing in a nutrient-rich medium deplete the nutrient as they grow. As a result, the nutrient concentration \(x(t)\) is steadily decreasing. The equation describing this decrease is the following: \[ \displaystyle \frac{dx}{dt} = - \mu \frac{x \cdot (\xi- x)}{\kappa + x}, \]
where \(\mu\), \(\kappa\), and \(\xi\) are all parameters greater than zero.
- Determine the equilibrium solutions \(x_{*}\) for this differential equation.
- Construct a phase line for this differential equation and classify the stability of the equilibrium solutions.