Question

The wolf population \(P\) in a certain state has been growing at a rate proportional to the cube root of the population size. The population was estimated at 1000 in 1980 and at 1700 in 1990 . (a) Write the differential equation for \(P\) at time \(t\) with the two corresponding conditions. (b) Solve the differential equation. (c) When will the wolf population reach \(4000 ?\)

Short answer

The wolf population will reach 4000 around the year 2029.

Step-by-step solution

Understand Growth Rate

The problem states that the wolf population grows at a rate proportional to the cube root of the population size. This translates to a differential equation of the form \( \frac{dP}{dt} = k \cdot P^{1/3} \), where \( k \) is a proportionality constant, \( P \) is the population, and \( t \) is time.

Set Initial Conditions

The population was 1000 in 1980 and 1700 in 1990, which gives us initial conditions: \( P(1980) = 1000 \) and \( P(1990) = 1700 \). It is convenient to set 1980 as \( t = 0 \), so \( P(0) = 1000 \) and \( P(10) = 1700 \).

Solve the Differential Equation

To solve \( \frac{dP}{dt} = k \cdot P^{1/3} \), separate variables: \[ \frac{dP}{P^{1/3}} = k \, dt \]Integrate both sides:\[ \int P^{-1/3} dP = \int k \, dt \]The left side integrates to \( 3P^{2/3}/2 \) and the right side to \( kt + C \):\[ \frac{3}{2} P^{2/3} = kt + C \]

Solve for the Constant C

Use the initial condition \( P(0) = 1000 \):\[ \frac{3}{2} (1000)^{2/3} = C \]Calculate \( 1000^{2/3} = 100 \), then:\[ \frac{3}{2} \times 100 = 150 \]Thus, \( C = 150 \).

Solve for the Constant k

Use the condition \( P(10) = 1700 \):\[ \frac{3}{2} (1700)^{2/3} = 10k + 150 \]Calculate \( 1700^{2/3} \) (which is approximately 123.3), then:\[ \frac{3}{2} \times 123.3 = 10k + 150 \]Solve for \( k \): \[ 184.95 = 10k + 150 \] \[ 10k = 34.95 \] \[ k = 3.495 \]

Find When Population Reaches 4000

Set \( P = 4000 \) in the formula \( \frac{3}{2} P^{2/3} = kt + 150 \) and solve for \( t \):\[ \frac{3}{2} (4000)^{2/3} = 3.495t + 150 \]Calculate \( 4000^{2/3} = 215.44 \), then:\[ \frac{3}{2} \times 215.44 = 3.495t + 150 \]\[ 323.16 = 3.495t + 150 \]\[ 3.495t = 173.16 \] \[ t = \frac{173.16}{3.495} \approx 49.55 \]

Calculate the Year

Since \( t = 0 \) represents the year 1980, adding 49.55 to 1980 gives the year when the population reaches 4000. This is approximately in the year 2029.

Key concepts

Population Growth Model

In understanding a population growth model, we look at how populations change over time. Population dynamics are influenced by birth rates, death rates, and migration. However, in a mathematical model, we often simplify this to just focus on a particular factor.
In the exercise, we see a wolf population that changes in a specific way: its growth rate is proportional to the cube root of the population size. This kind of modeling helps us predict future population sizes and understand animal or plant population patterns. Mathematically, we represent this as a differential equation. A differential equation is an equation that involves an unknown function and its derivatives, which, in this case, describes how fast the population grows.
The form of the differential equation given in the exercise is:

• \( \frac{dP}{dt} = k \cdot P^{1/3} \)

Here, \( \frac{dP}{dt} \) represents the rate of change of the population concerning time. The cube root, represented by \( P^{1/3} \), indicates how the population influences its growth rate. Understanding this helps us clarify how population characteristics can influence growth patterns over time.

Proportional Growth Rate

Proportional growth rate is a concept where the rate of growth of a population is directly related to its current size. In our wolf population problem, the growth is linked to the cube root of the population size \( P \), which introduces a slight complexity compared to linear growth models where growth is proportional to the size \( P \) itself.
The constant \( k \) in the equation \( \frac{dP}{dt} = k \cdot P^{1/3} \) is called the proportionality constant. It determines how much the cube root of the population size influences the rate of growth. Understanding proportional growth rates involves several important points:

• Growth accelerates as the population size increases, but unlike direct proportionality, the growth here is less aggressive due to the cube root.
• The given differential equation suggests that small increases in the population result in smaller changes in the rate of growth, making it easier to manage the population theoretically.
• In practical terms, different populations have different values of \( k \), which can cause similar initial populations to grow at varying speeds.

Knowing the behavior of the proportional growth rate allows us to better predict when and how the population might reach a certain size under given conditions.

Initial Conditions in Calculus

Initial conditions in calculus are crucial in differential equations. They provide the specific values that allow for the unique solution of differential equations that describe real-world phenomena.
In our problem, the initial conditions are \( P(0) = 1000 \) for the year 1980 and \( P(10) = 1700 \) for the year 1990. These conditions tell us the population size at a specific starting point and a later point. They allow us to solve for the constants in our differential equation. Understanding initial conditions involves appreciating a few fundamental ideas:

• They help find the constant values such as \( k \) and \( C \) in the integration step of solving the differential equation.
• Knowing the exact population at specific points enables us to predict future growth accurately or solve for unknown time periods.
• Initial conditions translate a general solution of a differential equation (with many possible solutions) into a specific one that models the particular problem scenario.

This is why setting and using initial conditions is essential when solving differential equations to model entities like populations.