Problem 17 - Applied Calculus

Question

If the initial population of fish is 70 million, use the differential equation \(d P / d t=0.2 P-10\) to estimate the fish population after \(1,2,3\) years.

Step-by-Step Solution

Answer

The population after 1, 2, and 3 years are approximately 74.47, 79.41, and 84.76 million, respectively.

Step 1: Understand the Differential Equation

The given differential equation is \( \frac{dP}{dt} = 0.2P - 10 \), where \( P \) is the population of fish in millions and \( t \) is time in years. This equation represents how the population changes over time.

Step 2: Solve the Differential Equation

This is a first-order linear differential equation. We use separation of variables or an integrating factor to solve it. Rewriting as \( \frac{dP}{dt} - 0.2P = -10 \) and using integrating factor \( e^{0.2t} \), the solution becomes:\[ P(t) = Ce^{0.2t} + 50 \].

Step 3: Determine Constant Using Initial Condition

To find the constant \( C \), use the initial condition where \( P(0) = 70 \). Substitute into the solution:\( 70 = Ce^{0.2\times0} + 50 \), giving \( 70 = C + 50 \). So, \( C = 20 \).

Step 4: Write the Population Function

Substitute \( C = 20 \) back into the general solution to get \( P(t) = 20e^{0.2t} + 50 \). This equation models the fish population over time.

Step 5: Calculate Population for Each Year

Use the function \( P(t) = 20e^{0.2t} + 50 \) to find the population for \( t = 1, 2, \) and \( 3 \):- For \( t = 1 \), \( P(1) = 20e^{0.2} + 50 \approx 74.47 \) million.- For \( t = 2 \), \( P(2) = 20e^{0.4} + 50 \approx 79.41 \) million.- For \( t = 3 \), \( P(3) = 20e^{0.6} + 50 \approx 84.76 \) million.

Key Concepts

Population Dynamics

Population dynamics is a fascinating area of study in the field of ecology and mathematics. It involves understanding how populations of living organisms, such as fish or animals, change over time. In this context, we're looking at how a fish population grows or shrinks.

The differential equation given in the exercise, \( \frac{dP}{dt} = 0.2P - 10 \), is an example of how these changes can be mathematically modeled. Here, the variable \( P \) represents the population size, and \( t \) denotes time.

Key components in population dynamics models include:

• **Birth and Death Rates**: The increase or decrease in population depending on how many individuals are born or die within a given period.
• **Carrying Capacity**: The maximum population size that an environment can sustain. In some models, populations can't grow indefinitely.
• **External Factors**: These can include things like pollution, temperature changes, or fishing, all potentially altering population sizes.

In our scenario, the equation captures a basic model of fish population dynamics affected by both growth (\(0.2P\)) and decline (\(-10\)).

Linear Differential Equation

A linear differential equation is a type of differential equation with applications in many scientific disciplines. They are characterized by a constant rate of change (linear behavior) over time.

The equation discussed, \( \frac{dP}{dt} = 0.2P - 10 \), is a first-order linear differential equation. This particular form indicates that the change in population \( P \) depends linearly on \( P \) itself and could include a constant.

Solving such equations often involves:

• **Separation of Variables**: Useful for equations where the rate of change can be expressed as a product of functions of each variable.
• **Integrating Factors**: This technique simplifies the process, especially when the differential is not easily separable and requires the introduction of an exponential factor.

In our solution, the integrating factor \( e^{0.2t} \) is applied to make the equation more manageable. This leads to a general solution where population changes can be calculated by plugging in values for \( t \).

Linear differential equations like this form the basis for understanding patterns and predicting future behavior in various contexts, from ecology to economics.

Exponentials in Calculus

Exponentials play a crucial role in calculus, especially when modeling growth or decay processes. Their unique characteristics make them ideal for representing continuously compounding systems.

In the fish population exercise, exponential growth is a key concept. The model uses an exponential function \( e^{0.2t} \), where the constant \( 0.2 \) signifies the growth rate.

• **Exponential Growth**: This occurs when the growth rate of a mathematical function is proportional to its current value, typical for populations, investments, and radioactivity.
• **Continuous Compounding**: With exponential functions, changes are assumed to occur at every instant, unlike in discrete compounding.

This function offers several advantages:

• **Flexibility**: Easy manipulation makes it versatile for calculations.
• **Predictability**: Patterns in data can be predicted by identifying the exponential behavior.

Understanding exponentials helps demystify why populations may grow rapidly under certain conditions, offering insights into broader biological processes and improving prediction accuracy.