13.2 Logistic growth (and other models)

Spread of infection. In the model for the spread of a disease (starting page  in the text), we used the fact that the total population is constant ($S(t)+I(t)=N=$constant) to eliminate $S(t)$ and analyze a differential equation for $I(t)$ on its own.

Carry out a similar analysis, but eliminate $I(t)$. Then analyze the differential equation you get for $S(t)$ to find its steady states and behaviour, practicing the qualitative analysis discussed in this chapter.

Social media. Sally Sweetstone has invented a new social media App called HeadSpace, which instantly matches compatible mates according to their changing tastes and styles. Users hear about the App from one another by word of mouth and sign up for an account. The account expires randomly, with a half-life of $1$ month. Suppose $y_1(t)$ are the number of individuals who are not subscribers and $y_2(t)$ are the number of are subscribers at time $t$. The following model has been suggested for the evolving subscriber population

A bimolecular reaction. Two molecules of $A$ can react to form a new chemical, $B$. The reaction is reversible

so that $B$ also continually decays back into $2$ molecules of $A$. The differential equation model proposed for this system is

where $a(t), b(t)>0$ are the concentrations of the two chemicals.

Scaling time in the logistic equation. Consider the scaled logistic equation 12.1.3

Recall that $r$ has units of 1/time, so $1/r$ is a quantity with units of time. Now consider scaling the time variable in the displayed equation by defining $t=s/r$. Then $s$ carries no units ($s$ is “dimensionless”).

Substitute this expression for $t$ in the displayed equation and find the differential equation so obtained (for $dy/ds$).

The Law of Mass Action. The Law of Mass Action in Section 12.1 led to the assumption that the rate of a reaction involving two types of molecules (A and B) is proportional to the product of their concentrations, $k \cdot a \cdot b$.

Explain why the sum of the concentrations, $k \cdot (a+b)$, would not make for a sensible assumption about the rate of the reaction.

Glucose solution in a tank. A tank that holds $1$ liter is initially full of plain water. A concentrated solution of glucose, containing $0.25$ gm/cm$^{3}$ is pumped into the tank continuously, at the rate $10$ cm$^{3}$/min and the mixture (which is continuously stirred to keep it uniform) is pumped out at the same rate.

Let $G(t)$ be the amount of glucose in the tank after $t$minutes. Write a differential equation for $G$, and give its initial condition.

How much glucose is in the tank after a long time?

Pollutant in a lake. A lake of constant volume $V$ gallons contains $Q(t)$ pounds of pollutant at time $t$ evenly distributed throughout the lake. Water containing a concentration of $k$ pounds per gallon of pollutant enters the lake at a rate of $r$ gallons per minute, and the well-mixed solution leaves at the same rate.

A sugar solution. Sugar dissolves in water at a rate proportional to the amount of sugar not yet in solution. Let $Q(t)$ be the amount of sugar undissolved at time $t$. The initial amount is $100$ kg and after $4$ hours the amount undissolved is $70$ kg.

Leaking water tank. A cylindrical tank with cross-sectional area $A$ has a small hole through which water drains. The height of the water in the tank $y(t)$ at time $t$ is given by:

where $k, y_{0}$ are constants.