Problem 58 - Calculus with Concepts in Calculus

Question

In 1934 the biologist G. F. Gause placed 20 paramecia in 5 cubic centimeters of a saline solution with a constant amount of food and measured their growth on a daily basis. He found that the population \(f(t)\) at any time \(t\) (in days) was approximately $$ f(t)=\frac{4490}{1+e^{5.4094-1.0235 t}} \text { for } t \geq 0 $$ Determine the carrying capacity of the medium.

Step-by-Step Solution

Answer

The carrying capacity is 4490 paramecia.

Step 1: Identify the Logistic Growth Model

Recognize that the given formula for population growth, \( f(t) = \frac{4490}{1+e^{5.4094-1.0235 t}} \), is a logistic growth model. The general form of a logistic growth function is \( f(t) = \frac{L}{1 + e^{k(b-t)}} \), where \( L \) is the carrying capacity.

Step 2: Determine the Carrying Capacity

In the logistic growth model, the carrying capacity \( L \) represents the maximum population size that the environment can sustain. From the given equation, we identify \( L = 4490 \).

Step 3: Conclusion

The carrying capacity is derived directly from the coefficient in the numerator of the logistic growth function, which is 4490 in this case. Thus, the carrying capacity of the medium is 4490 paramecia.

Key Concepts

Carrying Capacity

In the context of population biology, the carrying capacity refers to the maximum number of individuals in a species that an environment can support sustainably. It’s like the full capacity of a bus, where no more passengers can board. Carrying capacity is determined by environmental limits, like space, food, and other resources.
For example, a forest might have a carrying capacity for 200 deer due to available food and space. If the population grows beyond this number, resources become scarce, leading to competition, which can lower the population back down. This balancing act helps maintain ecosystem stability.
Within the logistic growth model, carrying capacity appears as a constant "L" in the function. It signifies where the population level stabilizes over time. In the exercise provided, this value is 4490 paramecia, indicating that the medium can support a maximum of 4490 paramecia. As the population approaches this number, the growth rate slows down and eventually levels off.

Population Growth

Population growth describes how the number of individuals in a population changes over time. Growth can be linear, exponential, or logistic. The logistic growth model, which is relevant here, is particularly interesting as it accounts for resource limitations, making it more realistic than simple exponential growth.
This model starts with a phase of rapid growth when resources are abundant. However, as the population nears the carrying capacity, resources start to become limited. This limitation slows the growth rate, creating an S-shaped curve typical of logistic growth.
In our scenario with the paramecia, the population was initially small and experienced significant growth. As time progresses and the quantity of available food and space nears depletion, growth slows, preventing the population from surpassing the carrying capacity of 4490 paramecia. This showcases real-world population dynamics, where natural capacity limits control growth.

Differential Equations

Differential equations play a crucial role in expressing population growth mathematically. They provide a way to describe how a population changes over time, considering influences like birth and death rates, as well as carrying capacity.
In the logistic growth model specifically, the differential equation takes the form \[\frac{dP}{dt} = rP\left(1 - \frac{P}{L}\right), \]where:

• \( \frac{dP}{dt} \) is the rate of change of the population over time
• \( r \) is the rate of maximum growth
• \( P \) is the current population size
• \( L \) is the carrying capacity

This equation predicts how fast a population increases when resources are plentiful and how it decelerates as it approaches its environmental limits. By solving such equations, scientists and ecologists can make predictions and manage resources accordingly.
Understanding differential equations provides invaluable insights into the mechanisms driving population dynamics and can be applied beyond biology, into economics, physics, and other fields where change over time needs to be modeled and predicted.