Question

The model of exponential population growth predicts that the per capita population growth rate \(r.\) a. does not change as a population gets larger. b. gets larger as a population gets larger. c. gets smaller as a population gets larger. d. is always at its maximum level \(\left(r_{\max }\right)\). e. fluctuates on a regular cycle.

Short answer

The correct answer is a. The per capita population growth rate does not change as a population gets larger.

Step-by-step solution

Understanding the Model of Exponential Population Growth

The model of exponential population growth assumes that the growth rate of a population is constant over time. This means that the per capita growth rate (r) does not vary with the size of the population.

Analyzing the Options

Given this assumption of the exponential growth model, we can evaluate the provided options. We're looking for an option that suggests the per capita growth rate remains constant regardless of population size.

Selecting the Correct Answer

Since the exponential growth model stipulates that the growth rate remains constant, the correct answer must reflect this characteristic. Option a states that the per capita population growth rate does not change as a population gets larger, which aligns with the fundamental assumption of the model.

Key concepts

Per Capita Growth Rate

The concept of the per capita growth rate is fundamental to understanding population dynamics. It refers to the rate at which a population increases in size per individual of the population over a given time period. Essentially, it measures the contribution of each member of the population to the overall population growth. In mathematical terms, if we denote the per capita growth rate by \( r \), and the current population size by \( N \), the change in population size (\( \Delta N \)) over a time period (\( \Delta t \)) can be written as \( \Delta N = r \times N \times \Delta t \.\)

In the context of exponential growth, \( r \) is considered to remain constant; it does not increase or decrease as the population size (\( N \) increases. It is a simplification that is useful for certain applications, such as modeling unchecked biological populations, where resources are not limited. This kind of growth is often depicted as a J-shaped curve when plotted on a graph, with the population growing more rapidly as time progresses.

One interesting way to relate \( r \) to real-world situations is to consider human populations. If we have a city with a per capita growth rate of 0.01, this means that for every 100 individuals, one additional person is added to the population each year. Understanding this rate is crucial for policy planning and resource management. It's also essential in ecology for predicting species' population sizes and assessing potential risks of extinction or overpopulation.

Population Growth Model

A population growth model is a mathematical formula used to predict the future size of a population based on current conditions and growth patterns. The exponential population growth model is one such formula, and it assumes a situation where resources are unlimited, leading to a constant per capita growth rate. Under this model, the population size (\( N \) at any time \( t \) can be described by the equation \( N(t) = N_0e^{rt} \.\), where \( N_0 \) is the initial population size, \( e \) is the base of the natural logarithm, and \( r \) is the constant per capita growth rate.

This exponential model is useful mainly in theoretical scenarios or in the early stages of a population's expansion when resources are plentiful and organisms can reproduce without the constraints of food, space, or competition. However, in reality, many populations will eventually face limitations and undergo logistical growth, which can be modeled by a more complex equation that includes a carrying capacity.

Real-World Implications

The exponential growth model has implications in environmental impact studies, sustainable development, biology, and economics. For example, predicting the growth of invasive species allows ecologists to create management strategies that minimize ecological damage. In economics, understanding population growth helps in the projection of housing needs, job markets, and healthcare services.

Constant Growth Rate

A constant growth rate means that the increase or decrease in population size occurs at a steady rate over time. This is a key characteristic of the exponential growth model in population dynamics. It implies that the proportion of new individuals added to the population is the same in each time period and is crucially independent of the population's size.

A constant growth rate can be both powerful and misleading. It is powerful because it allows for relatively simple calculations and predictions. On the other hand, it can be misleading because no real population can grow indefinitely at a constant rate due to environmental constraints and carrying capacity limitations. The use of a constant growth rate is best suited to short-term projections and must be adjusted to factor in real-world conditions for long-term predictions.

To put it in perspective, a constant growth rate implies that if a population of 100 fish increases by 10 fish per year, it will continue to increase by that amount regardless of whether the population has reached 200, 500, or 1000 fish. In natural ecosystems, such growth is unsustainable, leading to population crashes or the depletion of resources.