Mathematical Model for Population Growth Examples

Understanding Population Growth Models

Creating a mathematical model for population growth involves using mathematical equations to predict how populations change over time based on various factors. These models can vary in complexity, from simple linear growth to more complex exponential or logistic growth models. In this article, we present three diverse, practical examples to illustrate how to create mathematical models for population growth.

Example 1: Linear Population Growth Model

Context

This model assumes a constant rate of growth over time. It can be useful for small populations in stable environments where resources are abundant.

In this example, let’s consider a small town that has a population of 1,000 people. The town experiences a steady increase of 50 people each year due to births and immigration.

The formula for a linear growth model is:

\[ P(t) = P_0 + rt \]

Where:

• P(t) = population at time t
• P_0 = initial population
• r = rate of growth
• t = time (in years)

Using our example values:

• Initial population (P_0) = 1,000
• Rate of growth (r) = 50 people/year
• Time (t) = 5 years

Plugging these values into the equation:

\[ P(5) = 1000 + 50(5) \]

Calculating:

\[ P(5) = 1000 + 250 = 1250 \]

Notes

• This model is simplistic and does not account for factors like resource limitations, which could slow growth in reality.
• Variations could include changing the growth rate or introducing a cap on the population size.

Example 2: Exponential Population Growth Model

Context

Exponential growth occurs when the growth rate of a population is proportional to its current size, often seen in ideal conditions with unlimited resources.

Let’s consider a bacteria culture that starts with 100 bacteria and doubles every hour. The exponential growth formula is:

\[ P(t) = P_0 imes e^{rt} \]

Where:

• P(t) = population at time t
• P_0 = initial population
• e = Euler’s number (approximately 2.718)
• r = growth rate (in this case, we’ll use the doubling time)
• t = time (in hours)

Assuming a doubling time of 1 hour, we can express the growth rate (r) as:

• r = ln(2) (because the population doubles)

Using our example values:

• Initial population (P_0) = 100
• Time (t) = 3 hours

Plugging these values into the equation:

\[ P(3) = 100 imes e^{(ln(2) imes 3)} \]

Calculating:

\[ P(3) = 100 imes 2^3 = 100 imes 8 = 800 \]

Notes

• This model can quickly lead to very large populations, which might not be sustainable in real life.
• A variation could include limiting factors like resource depletion to transition into a logistic growth model.

Example 3: Logistic Population Growth Model

Context

The logistic growth model accounts for environmental limits on population growth. It is often more realistic for populations in a limited environment.

Consider a wildlife reserve that can support a maximum of 5,000 deer. The current deer population is 1,000, and the growth rate is 0.1 per year. The logistic growth model is given by:

\[ P(t) =
rac{K}{1 +
rac{K - P_0}{P_0} e^{-rt}} \]

Where:

• P(t) = population at time t
• K = carrying capacity (maximum population)
• P_0 = initial population
• r = growth rate
• t = time (in years)

Using our example values:

• Carrying capacity (K) = 5,000
• Initial population (P_0) = 1,000
• Growth rate (r) = 0.1
• Time (t) = 10 years

Plugging these values into the equation:

\[ P(10) =
rac{5000}{1 +
rac{5000 - 1000}{1000} e^{-0.1 imes 10}} \]

Calculating:

First, evaluate the exponent:

\[ e^{-1} ext{ (approximately 0.3679)} \]

Then calculate:

\[ P(10) =
rac{5000}{1 + 4 imes 0.3679} =
rac{5000}{1 + 1.4716} =
rac{5000}{2.4716} ext{ (approximately 2029)} \]

Notes

• This model reflects real-world scenarios better than exponential models since it incorporates carrying capacity.
• Variations could include changing the carrying capacity based on seasonal factors or introducing predation dynamics.

By exploring these examples of creating a mathematical model for population growth, students can gain a deeper understanding of how populations change and the factors influencing those changes.