3.5 Exercises

Exercise 3.1 Consider the following type of functional responses: $$\begin{align} \mbox{ Type I: } \frac{dP}{dt} &= 0.1 P \\ \mbox{ Type II: } \frac{dP}{dt} &= \frac{0.1P}{1+.03P} \\ \mbox{ Type III: } \frac{dP}{dt} &= \frac{0.1P^{2}}{1+.05P^{2}} \end{align}$$

For each of the functional responses evaluate $\displaystyle \lim_{P \rightarrow \infty} \frac{dP}{dt}$. 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 $\displaystyle \frac{dP}{dt} = \frac{0.1P}{1+.05P} -P$.

1. 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?
2. 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 $\displaystyle \frac{0.1P}{1+5P} -P$.)
3. Does your algebraic solution match your graphical solutions?

 

Exercise 3.3 A population grows according to the equation $\displaystyle \frac{dP}{dt} = 2P - \frac{4P^{2}}{1+P^{2}}$.

1. 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?
2. 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 $\displaystyle 2P - \frac{4P^{2}}{1+P^{2}}$.)
3. Does your algebraic solution match your graphical solutions?

 

Exercise 3.4 A population grows according to the equation $\displaystyle \frac{dP}{dt} = \frac{aP}{1+abP} - dP$, where $a$, $b$ and $d$ are parameters. Determine the two steady state values of $P$, that is solve $\displaystyle \frac{dP}{dt}=0$ for $P$.

 

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.

1. The rate of change of an animal’s body temperature is proportional to the difference in temperature between the environment.
2. 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 $\mu$.
3. 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.
4. 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.
5. 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.
6. 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 $H$.
7. 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.

Exercise 3.7 You are tasked with the job of investigating the effect of a pesticide on water quality, in terms of its effects on the health of the plants and fish in the ecosystem. Different models can be created that investigate the effect of the pesticide. Different types of reaction schemes for this system are shown in Figure 3.7, where $F$ represents the amount of pesticide in the fish, $W$ the amount of pesticide in the water, and $S$ the amount of pesticide in the soil. The prime (e.g. $F'$, $W'$, and $S'$ represent other bound forms of the respective state). In all seven different models can be derived. For each of the model schematics, apply the Law of Mass Action to write down a system of differential equations.