§7.6 Population growth and the logistic equation

Introduction

The earth's population

Solving the logistic differential equation

Summary

Recall that one model for population growth states that a population grows at a rate proportional to its size. In symbols: $\frac{dP}{dt} = kP$.

We begin with the differential equation

. Sketch a slope field as well as a few typical solutions on the axes provided.

Use the data in the table to estimate the derivative $P'(0)$ using a central difference. Assume that $t=0$ corresponds to the year 2000.

What is the population $P(0)$?

Use your results from (a) and (b) to estimate the constant of proportionality $k$ in the differential equation.

Now that we know the value of $k$, we have the initial value problem

. Find the solution to this initial value problem.

What does your solution predict for the population in the year 2010? Is this close to the actual population given in the table?

When does your solution predict that the population will reach 12 billion?

What does your solution predict for the population in the year 2500?

At what value of $P$ is the rate of change greatest?

Consider the model for the earth's population that we recently created: $\frac{dP}{dt} = P(0.025-0.002P)$. At what value of $P$ is the rate of change greatest? How does that compare to the population in recent years?

According to the model we developed (recall we found that $P = \frac{12.5}{1.0546e^{-0.025t} + 1}$), what will the population be in the year 2100?

According to the model we developed, when will the population reach 9 billion?

Now consider the general solution to the general logistic initial value problem that we found, given by

. Verify algebraically that $P(0) = P_0$ and that $\lim_{t\to\infty} P(t) = N$.

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, $p(0) = 0.1$. - What happens to $p(t)$ after a very long time? - Determine a formula for the function $p(t)$. - At what time is $p$ changing most rapidly? - How long does it take before 80% of the people have heard the rumor?

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. - Assume that the per capita growth rate $\frac{db/dt}{b}$ is a linear function of $b$. Use the measurements to find this function and write a logistic equation to describe $\frac{db}{dt}$. - What is the carrying capacity for the bacteria? - At what population is the number of bacteria increasing most rapidly? - If there are initially 1,000 bacteria, how long will it take to reach 80% of the carrying capacity?

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. - What is the carrying capacity of this population? - 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 equation by adding a term to incorporate the harvesting of fish. - What is the new carrying capacity? - What will the fish population be one year after the harvesting begins? - How long will it take for the population to be within 10% of the carrying capacity?