Question

The following problems consider a simple population model of the housefly, which can be exhibited by the recursive formula \(x_{n+1}=b x_{n}\), where \(x_{n}\) is the population of houseflies at generation \(n\), and \(b\) is the average number of offspring per housefly who survive to the next generation. Assume a starting population \(x_{0}\). Find \(\lim _{n \rightarrow \infty} x_{n}\) if \(b>1, b<1\), and \(b=1\).

Short answer

If \(b > 1\), \(\lim_{n \to \infty} x_{n} = \infty\); if \(b < 1\), \(\lim_{n \to \infty} x_{n} = 0\); if \(b = 1\), \(\lim_{n \to \infty} x_{n} = x_0\).

Step-by-step solution

Understand the Recurrence Relation

The given recurrence formula is \(x_{n+1} = b x_{n}\). This means that each generation is multiplied by a factor \(b\), which represents the average number of offspring that survive to the next generation.

Analyze the Long-Term Behavior for \(b > 1\)

When \(b > 1\), the population is multiplied by a factor greater than 1 in each generation. Consequently, \(x_{n}\) will increase exponentially as \(n\) approaches infinity, leading to \(\lim_{n \to \infty} x_{n} = \infty\).

Analyze the Long-Term Behavior for \(b < 1\)

When \(b < 1\), the population is multiplied by a factor less than 1 in each generation. As a result, \(x_{n}\) will decrease and approach zero as \(n\) approaches infinity, which implies \(\lim_{n \to \infty} x_{n} = 0\).

Analyze the Long-Term Behavior for \(b = 1\)

When \(b = 1\), the population remains constant at each generation since \(x_{n+1} = x_{n}\). Therefore, \(\lim_{n \to \infty} x_{n} = x_{0}\), the initial population.

Key concepts

Recurrence Relations

A recurrence relation is a special equation that defines sequences based on preceding elements. In simpler terms, it gives us a formula to generate a series of numbers step by step. In our context, the housefly population model is expressed by the recurrence relation \(x_{n+1} = b \times x_{n}\). Here, \(x_{n}\) is the population at the \(n\)-th generation, and \(b\) is the factor by which each generation of houseflies is multiplied. The beauty of recurrence relations lies in this repetitive pattern.

• If you know an element in the sequence, you can use the relation repeatedly to calculate all future elements.
• They are extensively used in different areas such as computer science, biology, and economics for modeling growth, populations, and trends.

When tackling a problem that uses recurrence relations, you must first understand how each term depends on the ones before it. That’s why breaking down the sequence step by step often helps insights emerge.

Exponential Growth

Exponential growth is a specific way a quantity increases over time, becoming ever larger at an accelerating pace. In our population model, when the average number of offspring \(b\) is greater than 1, it leads to exponential growth. This means every new generation has more houseflies than the last.
When you multiply a population by a number greater than 1 repeatedly, the effect escalates. Imagine starting with a small number of flies, say 10 flies, with \(b=1.2\):

• The second generation will have \(10 \times 1.2 = 12\) flies.
• The third generation will have \(12 \times 1.2 = 14.4\).
• And this growth goes on accelerating further.

As generations pass, populations can become extremely large, showing the power of exponential growth. It’s essential to understand it can quickly lead to unsustainable numbers in real-life situations.

Limit of a Sequence

The concept of a limit in mathematics helps us understand the behavior of sequences as they approach infinity. Imagine you're observing the population of houseflies generation after generation. The question is, what happens in the long run?
This is where limits come into play. We use limits to predict the population's future:

• If \(b > 1\), the population expands endlessly, thus \(\lim_{n \to \infty} x_{n} = \infty\).
• If \(b < 1\), populations gradually reduce to zero, so \(\lim_{n \to \infty} x_{n} = 0\).
• If \(b = 1\), there’s no change over generations and \(\lim_{n \to \infty} x_{n} = x_{0}\), the initial figure.

In practical terms, knowing the limit of the sequence helps us anticipate long-term behavior of the system we're studying, allowing for more informed decision-making and strategic planning.