3.5 Exercises
Exercise 3.1 Consider the following type of functional responses: Type I: dPdt=0.1P Type II: dPdt=0.1P1+.03P Type III: dPdt=0.1P21+.05P2
For each of the functional responses evaluate lim. Since these functional responses represent a rate of change of a population, what are some examples (hypothetical or actual) would each of these responses be appropriate?
Exercise 3.2 A population grows according to the equation \displaystyle \frac{dP}{dt} = \frac{0.1P}{1+.05P} -P.
- On the same axis, plot the equations \displaystyle f(P) = \frac{0.1P}{1+.05P} and g(P)=P. What are the two positive values of P where f(P) and g(P) intersect?
- Next algebraically determine the two steady state values of P, that is solve \displaystyle \frac{dP}{dt}=0 for P. (Hint: factor a P out of the expression \displaystyle \frac{0.1P}{1+5P} -P.)
- Does your algebraic solution match your graphical solutions?
Exercise 3.3 A population grows according to the equation \displaystyle \frac{dP}{dt} = 2P - \frac{4P^{2}}{1+P^{2}}.
- On the same axis, plot the equations \displaystyle f(P) = 2P and \displaystyle g(P)=\frac{4P^{2}}{1+P^{2}}. What are the two positive values of P where f(P) and g(P) intersect?
- Next algebraically determine the two steady state values of P, that is solve \displaystyle \frac{dP}{dt}=0 for P. (Hint: factor a P out of the expression \displaystyle 2P - \frac{4P^{2}}{1+P^{2}}.)
- Does your algebraic solution match your graphical solutions?
Exercise 3.5 A chemical reaction takes two chemicals X and Y to form a substrate Z through the law of mass action. However the substrate can also disassociate. The reaction schematic is the following:
\begin{equation} X + Y \rightleftharpoons Z, \end{equation}
where the proportionality constant k_+ is associated with the formation of the substrate Z and k_- the disassociation (Z decays back to X and Y).
Write down a differential equation that represents the rate of reaction \displaystyle \frac{dZ}{dt}.
Exercise 3.6 For each of the following exercises consider the following contextual situations modeling rates of change. Name variables for each situation and write down a differential equation describing the context. Be sure to identify and briefly describe any parameters you need for your model. For each problem you will need to:
- Name and describe all variables.
- Write down a differential equation.
- Identify and describe any parameters needed.
- Write a brief one-two sentence explanation of why your differential equation models the situation at hand.
- Hand sketch a rough graph of what you think the solution as a function of time - note: your solution needs to be consistent with your explanation and vice versa.
- The rate of change of an animal’s body temperature is proportional to the difference in temperature between the environment.
- A plant grows propritional to its current length L. Assume this proportionality constant is \mu, whose rate also decreases proportional to its current value. You will need to write down a system of two equation with variables L and \mu.
- A patient undergoing chemotherapy receives an injection at rate I. This injection decreases the rate that a tumor accumulates mass. Independent of the injection, the tumor accumulates mass at a rate proportional to the mass of the tumor.
- A cell with radius r assimilates nutrients at a rate proportional to its surface area, but uses nutrients proportional to its volume. Determine an equation that represents the rate of change of the radius.
- A patient undergoing chemotherapy receives an injection at rate I. This injection decreases the rate that a tumor accumulates mass. Independent of the injection, the tumor accumulates mass at a rate proportional to the mass of the tumor.
- The rate that a cancer cell divides (increases in amount) is proportional to the amount of healthy cells in its surrounding environment. You may assume that a healthy cell has a mortality \delta_{H} and a cancer cell has mortality \delta_{C}. Be sure to write down a system of differential equations for the population of cancer cells C and healthy cells H.
- The rate that a virus is spread to the population is proportional to the probability that a person is sick (out of N total sick and healthy individuals).

Figure 3.7: Reaction schemes.