Problem 3 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

Suppose a population is initially of size 1,000,000 and grows at the rate of \(2 \%\) per year. What will be the size of the population after 50 years?

Step-by-Step Solution

Answer

The population will be approximately 2,691,590 after 50 years.

Step 1: Understand the Problem

We are given an initial population size of 1,000,000 and a growth rate of 2% per year. We need to calculate the population size after 50 years.

Step 2: Determine the Formula

To find the future population, use the formula for exponential growth: \[ P(t) = P_0 \times (1 + r)^t \] where \( P(t) \) is the population at time \( t \), \( P_0 \) is the initial population, \( r \) is the growth rate per period, and \( t \) is the number of time periods.

Step 3: Assign Given Values

Identify the given values: - \( P_0 = 1,000,000 \) (initial population)- \( r = 0.02 \) (growth rate as a decimal)- \( t = 50 \) (number of years)

Step 4: Substitute Values into the Formula

Substitute the given values into the exponential growth formula:\[ P(50) = 1,000,000 \times (1 + 0.02)^{50} \]

Step 5: Calculate the Population Size

First, calculate \((1 + 0.02)^{50}\):\[ (1.02)^{50} \approx 2.69159 \] (using a calculator)Now, multiply this result by the initial population:\[ P(50) = 1,000,000 \times 2.69159 \approx 2,691,590 \]

Step 6: Finalize the Answer

The population size after 50 years will be approximately 2,691,590.

Key Concepts

Population Dynamics

Population dynamics refers to the study of how and why population sizes change over time. This can include increases, decreases, and fluctuations in population due to various factors like birth rates, death rates, and migration. In our example, we focus on an increasing population affected solely by a constant growth rate.

Studying population dynamics helps us understand biological processes and environmental influences. It's crucial for resource planning, environmental conservation, and understanding ecological balance.

• Key Influences: Birth and death rates, immigration, and emigration.
• Applications: Wildlife management, human population studies, and urban planning.

By examining population dynamics, we gain insights into how populations interact with their environments, which is essential for sustainable development and conservation efforts.

Growth Rate

The growth rate is a critical component in understanding population dynamics. It indicates how quickly a population is increasing or decreasing. The growth rate is often expressed as a percentage and can be positive (for growth) or negative (for decline).

In the exercise, the growth rate is set at 2% per year, meaning the population grows by 2% of its previous size annually. This consistent growth rate simplifies predicting future population sizes using mathematical models.

• Positive Growth Rate: Population trends upward.
• Negative Growth Rate: Population trends downward.

A clear understanding of growth rates allows us to project future changes and plan accordingly for challenges such as resource allocation and infrastructure development.

Exponential Function

An exponential function is a mathematical representation used to model phenomena where growth accelerates rapidly over time. It is characterized by a constant percentage increase, leading to the doubling of quantities within consistent intervals.

In the given problem, the population grows exponentially. This means each year, the increase in population is proportional to its current size. The exponential growth formula used is:
\[P(t) = P_0 \times (1 + r)^t\]

• Base Component: \(1 + r\), where \(r\) is the growth rate.
• Exponent: \(t\), representing the number of growth periods.

Exponential functions are pivotal for modeling biological and financial growth, capturing scenarios like population increases and compound interest in economics.

Mathematical Modeling

Mathematical modeling involves using mathematics to represent and analyze real-world systems. It provides a framework to understand dynamics, predict future trends, or propose scenarios inside complex systems.

In our scenario, we model the population growth with an exponential function. This helps in predicting the future size based on existing data. By using the exponential growth formula, we create a simplified version of reality that's able to yield useful predictions for future planning.

• Real-World Examples: Climate change models, epidemiological forecasting, and financial market analysis.
• Benefits: Allows scenario analysis, improves decision-making, and informs policy development.

Effectively utilizing mathematical modeling in fields like demography, economics, or physics, equips us with powerful tools to tackle complex issues systematically and efficiently.