Problem 9 - Active Calculus

Question

The Gompertz equation is a model that is used to describe the growth of certain populations. Suppose that \(P(t)\) is the population of some organism and that $$ \frac{d P}{d t}=-P \ln \left(\frac{P}{3}\right)=-P(\ln P-\ln 3) $$ a. Sketch a slope field for \(P(t)\) over the range \(0 \leq P \leq 6\). b. Identify any equilibrium solutions and determine whether they are stable or unstable. c. Find the population \(P(t)\) assuming that \(P(0)=1\) and sketch its graph. What happens to \(P(t)\) after a very long time? d. Find the population \(P(t)\) assuming that \(P(0)=6\) and sketch its graph. What happens to \(P(t)\) after a very long time? e. Verify that the long-term behavior of your solutions agrees with what you predicted by looking at the slope field.

Step-by-Step Solution

Answer

Equilibrium solutions are at \( P = 3\) (stable) and \( P = 0\). For \( P(0)=1 \, \( P(t) \) approaches 3; for \( P(0)=6 \), it also approaches 3.

Step 1 - Sketch a Slope Field

To sketch the slope field for the differential equation \( \frac{dP}{dt} = -P(\ln P - \ln 3) \), calculate the slopes at various points \( P \) between 0 and 6. Plot these slopes on the \( t \)-\( P \) plane.

Step 2 - Identify Equilibrium Solutions

Find equilibrium solutions by setting \( \frac{dP}{dt} = 0\). Solve \( 0 = -P(\ln P - \ln 3) \) to find the population values where the slope is zero. Determine the stability by testing points around these equilibrium solutions.

Step 3 - Population for P(0) = 1

Solve the differential equation with the initial condition \( P(0) = 1 \). Use separation of variables and integration to find an expression for \( P(t) \). Sketch this solution and analyze the behavior as \( t \) approaches infinity.

Step 4 - Population for P(0) = 6

Solve the differential equation with the initial condition \( P(0) = 6 \). Use a similar method as in Step 3 to find \( P(t) \). Sketch this solution and analyze the long-term behavior as \( t \) increases.

Step 5 - Verify Long-Term Behavior

Compare the long-term behavior of the solutions from Steps 3 and 4 with the slope field drawn in Step 1. Confirm if the predictions about stability and equilibrium solutions match the slope field.

Key Concepts

Population Growth Model

The Gompertz equation is a special model used to describe the growth of certain populations. It is particularly useful for populations where growth slows down as the population becomes large. In the Gompertz model, the population growth rate decreases exponentially as the population size increases. This means the larger the population gets, the slower it grows. This type of model can help us understand how populations settle over time and reach a stable size (equilibrium).

Differential Equation

A differential equation involves derivatives, which represent how a quantity changes over time. The Gompertz differential equation we are looking at is \( \frac{dP}{dt} = -P(\ln P - \ln 3) \).
It tells us how the population \(P\) changes as time \(t\) goes on.
When solving differential equations, we can find out how a specific situation evolves by integrating the equation. This means we reverse the process of differentiation to find the original function.

Slope Field

A slope field is a graphical representation that helps us visualize the solutions of a differential equation without actually solving it. For the Gompertz equation, we would plot small lines (slopes) at various points \( P \) over the range from 0 to 6 on the \( t-P \) plane.
Each of these lines represents the slope of the solution curve at that point.
By looking at the slope field, we can