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 \(x_{0}\). For what values of \(b\) will the series converge and diverge? What does the series converge to?

Short answer

The series converges to zero for \(0 < b < 1\) and diverges for \(b > 1\).

Step-by-step solution

Understand the Recursive Formula

The recursive formula given is \(x_{n+1} = b x_n\). This formula explains how the population of houseflies grows from one generation to the next. Here, \(b\) is the constant that denotes the average number of offspring per housefly that survive to the next generation.

Analyze Convergence of the Series

For the series \(x_n\) to converge, the population must stabilize over time, which means \(x_n\) should approach a finite value as \(n\) goes to infinity. This occurs when the multiplier \(b\) is less than 1 (i.e., \(0 < b < 1\)) because the values of \(x_n\) will keep getting smaller and approach zero.

Analyze Divergence of the Series

The series \(x_n\) will diverge if the population does not stabilize, meaning \(x_n\) grows without bound as \(n\) approaches infinity. This happens when \(b > 1\) because with each generation, the population multiplies by more than 1, leading to exponential growth.

Special Case of the Series

If \(b = 1\), the population remains constant across generations because each generation's population equals the previous population. Therefore, \(x_n = x_0\) for all \(n\), neither converging nor diverging.

Conclusion on Convergence

The series converges to zero for \(b < 1\) since each generation's population becomes a smaller fraction of the previous one. For \(b > 1\), the series diverges, meaning the population increases indefinitely.

Key concepts

Population Dynamics

Population dynamics is a fascinating area of study in biological systems. It looks at how and why populations change over time based on birth rates, death rates, and other factors. In our housefly example, understanding these dynamics is key.
We use models to predict changes in population size, which can have significant implications for ecology and environmental science.
Models help scientists track population changes, prevent imbalances, and manage ecosystems sustainably.
In our scenario, the population model examines the growth of houseflies over several generations. This model's insights help us understand the balance needed in natural populations to prevent overpopulation, which could lead to ecological issues.

Recursive Sequences

In mathematics, a recursive sequence is a sequence of numbers where each term after the first is defined as a function of the preceding terms. Essentially, it builds upon itself step-by-step.
The formula given, \(x_{n+1} = b x_n\), is a perfect example of a recursive sequence. It uses the current population \(x_n\) to calculate the next generation's population \(x_{n+1}\).
What is unique about recursive sequences is their reliance on previous terms to predict future outcomes.

• They are useful in modeling real-life problems.
• They help forecast outcomes based on past data.

Recursive sequences were used in our exercise to predict housefly populations, showing how one generation influences the next.

Convergence and Divergence

Convergence and divergence are critical concepts in analyzing the behavior of sequences or series over time.
In our context, it assesses whether the housefly population stabilizes or grows indefinitely.
A sequence converges if it approaches a single value or zero as the number of terms increases. A significant aspect is ensuring the system dynamics balance out over time.

• Convergence is observed when \(0 < b < 1\).
• Divergence occurs when \(b > 1\).
• Special case: when \(b = 1\), we see neither convergence nor divergence, as the population remains constant.

Understanding these behaviors is essential to managing population growth effectively, avoiding scenarios of unmanageable size leading to resource depletion.

Growth Models

Growth models aim to forecast or predict how populations will change over time based on given parameters. They incorporate mathematical functions to describe the growth patterns.
In our exercise, the growth model was defined by the recursive formula \(x_{n+1} = b x_n\). This specific type of model is commonly encountered in discrete-time dynamical systems.
Growth models help in understanding:

• Population control and management strategies.
• Impact of different variables on population growth, like offspring survival rate (the parameter \(b\)).
• Predictive analytics in ecosystems management, influencing decisions and policy-making.

By studying these models, we gain crucial insights into biological systems, making them indispensable tools for ecologists and environmental scientists alike.