Population growth and the logistic equation

Using some recent population data for planet Earth, we want to model the growth of the population over time.

Our first model will be based on the following assumption:

The rate of change of the population is proportional to the population.

On the face of it, this seems pretty reasonable. When there is a relatively small number of people, there will be fewer births and deaths so the rate of change wi…

The logistic equation may be used to model how a rumor spreads through a group of people. Suppose that $p(t)$ is the fraction of people that have heard the rumor on day $t$. The equation describes how $p$ changes. Suppose initially that one-tenth of the people have heard the rumor; that is, $p(0) = 0.1$.

1. What happens to $p(t)$ after a very long time?

2. Determine a formula for the function $p(t)$.

3. At what time is $p$ changing most ra…

Suppose that $b(t)$ measures the number of bacteria living in a colony in a Petri dish, where $b$ is measured in thousands and $t$ is measured in days. One day, you measure that there are 6,000 bacteria and the per capita growth rate is 3. A few days later, you measure that there are 9,000 bacteria and the per capita growth rate is 2.

1. Assume that the per capita growth rate $\frac{db/dt}{b}$ is a linear function of $b$. Use t…

Suppose that the population of a species of fish is controlled by the logistic equation , where $P$ is measured in thousands of fish and $t$ is measured in years.

1. What is the carrying capacity of this population?

2. Suppose that a long time has passed and that the fish population is stable at the carrying capacity. At this time, humans begin harvesting 20% of the fish every year. Modify the differential eq…