Problem 92 - College Algebra with Modeling and Visualization

Question

If bacteria are cultured in a medium with limited nutrients, competition ensues and growth slows. According to Verhulst's model, the number of bacteria at 40 -minute intervals is given by $$a_{n}=\left(\frac{2}{1+a_{n-1} / K}\right) a_{n-1}$$ where \(K\) is a constant. (a) Let \(a_{1}=200\) and \(K=10,000\). Graph the sequence for \(n=1,2,3, \ldots, 20\). (b) Describe the growth of these bacteria. (c) Trace the graph of the sequence. Make a conjecture as to why \(K\) is called the saturation constant. Test your conjecture by changing the value of \(K\).

Step-by-Step Solution

Answer

The bacteria initially grow rapidly, then slow and stabilize, reflecting Verhulst's model. The saturation constant \(K\) dictates the carrying capacity.

Step 1: Initialize and Calculate First Term

Given the initial term \(a_1 = 200\) and \(K = 10,000\), we start calculating the sequence using the formula: \[a_n = \left(\frac{2}{1 + \frac{a_{n-1}}{K}}\right) a_{n-1}\]For \(n=2\), substitute \(a_{1}\) into the equation:\[a_2 = \left(\frac{2}{1 + \frac{200}{10,000}}\right) \times 200\]Simplifying, \[a_2 = \left(\frac{2}{1.02}\right) \times 200 \approx 392.16\]

Step 2: Calculate Subsequence Values

Continue calculating for \(a_3, a_4, \ldots, a_{20}\) using the same formula:For \(a_3\), substitute \(a_2\):\[a_3 = \left(\frac{2}{1 + \frac{392.16}{10,000}}\right) \times 392.16\]Simplifying, \[a_3 \approx 768.19\]Repeat this process for each subsequent \(a_n\) up to \(a_{20}\).

Step 3: Graph the Sequence

Use a graphing tool to plot the points \((n, a_n)\) for \(n=1, 2, 3, \ldots, 20\). Observe how the values start to flatline as \(n\) increases, indicating the saturation effect due to the limited nutrient medium.

Step 4: Describe the Growth

The growth demonstrates an initial rapid increase when nutrients are more abundant, followed by a leveling off as the nutrient limits cause saturation. This behavior aligns with Verhulst's logistic growth model, showing how population growth slows as it approaches a carrying capacity.

Step 5: Conjecture and Test the Saturation Constant

The parameter \(K\) in the equation represents the saturation constant because it determines the carrying capacity limit of the population given the fixed resources. Adjust \(K\) to test its effect:- Increasing \(K\) should result in a higher carrying capacity, leading to a higher saturation level.- Decreasing \(K\) reduces the limit, causing quicker saturation.

Key Concepts

Bacteria Growth

When bacteria are cultured in a medium with limited nutrients, their growth doesn't proceed infinitely. **In the initial stages** of bacterial growth, when nutrients are ample, the population tends to increase rapidly. This is known as the **logarithmic phase** where the bacteria multiply at an exponential rate.

However, as nutrients start depleting, this **exponential growth can't be sustained**. **Competition for available resources intensifies** among the bacteria, which results in a steady reduction in the growth rate. Eventually, this growth levels off, illustrating a hallmark pattern of bacteria in closed systems.

Through Verhulst's logistic growth model, we can mathematically describe this behavior. It allows for the prediction of how bacteria populations evolve over time when faced with nutrient constraints.

Saturation Constant

The saturation constant, denoted as \(K\), is a crucial component of the logistic growth model. It essentially reflects the maximum population level, or **carrying capacity**, that the environment can sustain for the bacteria given limited resources.

In the formula provided for the exercise: \[ a_{n} = \left(\frac{2}{1 + \frac{a_{n-1}}{K}}\right) a_{n-1} \] **Changes to \(K\) result in variations** within the population's dynamics. **If \(K\) is larger**, it indicates that the environment can support more bacteria. This results in the population's eventual saturation happening at a higher number. Conversely, **a smaller \(K\) means** fewer available resources, thus the population reaches its saturation point sooner.

The saturation constant therefore dictates **how fast the bacteria population saturates**, making it integral in predicting bacterial growth phases.

Carrying Capacity

Carrying capacity symbolizes the **upper limit of population growth** sustained by a particular environment. It is not just a theoretical number but a real measure of how many individuals the environment can sustainably support.

In bacteria growth studies, the carrying capacity is reached when growth ceases to advance at the same rate. The population tends to stabilize around this number because all available resources are maximized.

**Factors that influence carrying capacity** include:

• **Resource availability**: More nutrients allow a larger population.
• **Environmental conditions**: Changes in temperature, pH levels, etc., might affect the carrying capacity.
• **Space constraints**: Physical space needed for additional bacteria also plays a role.

Understanding carrying capacity helps in predicting when the **growth will plateau** and aids in designing effective cultivation strategies for controlled environments.

Population Dynamics

Population dynamics delve into the changes within a population over time and how they are influenced by various biological and environmental factors. In the context of bacteria growth, these dynamics offer insight into how populations grow, stabilize, or die off.

Verhulst's logistic model showcases that **bacterial populations do not grow unchecked**. Instead, they follow distinct **growth phases**: initial rapid growth, deceleration, and eventual steady state.

Key factors to consider in population dynamics include:

• **Birth and death rates**: How quickly bacteria reproduce versus how quickly they die.
• **Resource limitation**: Availability of nutrients affects growth rates.
• **Intrinsic growth rate**: The natural rate at which reproduction occurs when resources are abundant.

These insights are not only theoretical but have practical applications, such as in optimizing bacterial cultures for research and industrial purposes. By understanding these dynamics, one can predict and manage bacterial population changes effectively.