Exponential Population Growth: Videos & Practice Problems

The exponential population growth model describes how populations grow exponentially when the per capita growth rate (r) is positive and constant. This growth is represented by the equation nt=n0⁢ert, where n0 is the initial population size. The growth rate can be analyzed using dn/dt=rn for instantaneous rates and ∆n/∆t=r∆tn for average rates over intervals. This model highlights the limitations of resource availability in natural settings.

Exponential Population Growth

The exponential population growth model describes how populations can grow rapidly under ideal conditions, characterized by a constant and positive per capita growth rate, denoted as r. In this model, the growth of a population is directly proportional to its current size, meaning that as the population size n increases, the rate of growth, represented as \(\Delta n / \Delta t\), also increases. This relationship leads to a distinctive J-shaped curve when graphed, with time on the x-axis and population size on the y-axis.

Exponential growth occurs in nature only under specific circumstances and typically over short periods, as resources are limited. These limitations prevent indefinite exponential growth. The model assumes a constant growth rate r, which can vary among species. The maximum growth rate, referred to as r_max, represents the highest possible growth rate under ideal conditions. For example, if r_max = 1.5, the population will grow more rapidly than if r = 0.5, which still allows for exponential growth but at a slower pace.

In summary, the exponential population growth model illustrates how populations can increase rapidly when conditions are favorable, but it is essential to recognize the limitations imposed by environmental factors that ultimately restrict long-term growth. Understanding this model is crucial for applying these concepts in ecological studies and population management.

Exponential Population Growth Example 1

In understanding population growth, it is essential to recognize the conditions that allow for exponential growth. Exponential growth occurs when a population increases rapidly without significant limitations. This scenario typically arises under specific conditions, which include the availability of resources and the absence of limiting factors.

Among the conditions listed, the key concept to grasp is that density-dependent factors play a crucial role in regulating population growth. These factors include the availability of food, space, and competition from other species. As the population density increases, the impact of these factors becomes more pronounced, often leading to a decrease in growth rates as the population approaches its carrying capacity.

In this context, the best answer to the question of which conditions allow for exponential growth is option d: no density-dependent factors. This option effectively encompasses the implications of unlimited food, space, and lack of competition, as all these elements contribute to density-dependent limitations. When density-dependent factors are absent, populations can grow uninhibited, leading to exponential growth.

In summary, for a population to grow exponentially, it must be free from density-dependent factors that would otherwise limit its growth. Therefore, option d is the most appropriate choice in this scenario.

You’re studying 2 different populations of the same bacteria in the lab; both in the same unlimited, ideal conditions. Population-A starts with 2,000 individuals, while Population-B starts with 20,000 individuals. Assuming both populations grow exponentially, which population will have a higher population growth rate & why?

• A. Population-A because it has fewer individuals & therefore has more space to grow.
• B. Population-A because populations in ideal environments have a higher growth rate when N is smaller.
• C. Population-B because it can utilize resources more efficiently.
• D. Population-B because with exponential growth, larger populations have a higher growth rate.

Answer: (D)

Exponential Population Growth Equations

Understanding exponential population growth is essential in ecology and biology. The three key equations that describe this phenomenon illustrate how populations increase over time, forming a characteristic J-shaped curve on a graph where time is plotted on the x-axis and population size on the y-axis.

The first equation, n_t = n_0 e^{rt}, represents the final population size (n_t) at a given time t. Here, n_0 is the initial population size, e is Euler's number (approximately 2.71828), r is the per capita growth rate, and t is the elapsed time. This equation shows that as time progresses, the population grows exponentially, reflecting the intrinsic growth potential of the species.

The second equation, \frac{dn}{dt} = r n, uses calculus notation to describe the instantaneous growth rate of the population at any given moment. Here, \frac{dn}{dt} represents the slope of the tangent line at a specific point on the growth curve, indicating how quickly the population is changing at that instant. The variable n denotes the population size at that moment, while r remains the constant per capita growth rate.

The third equation, \Delta n = r \Delta t \cdot n, describes the average growth rate over a discrete time interval. In this case, \Delta n represents the change in population size over the interval \Delta t. Unlike the previous equations, where r is constant, here r \Delta t reflects the apparent per capita growth rate during that specific time interval, making it variable depending on the duration of \Delta t.

In summary, these equations collectively illustrate the dynamics of population growth, emphasizing the role of the per capita growth rate and the impact of time on population size. Understanding these relationships is crucial for applying these concepts in ecological studies and population management.

Exponential Population Growth Example 2

In this example problem, we explore the exponential population growth model through three distinct parts, applying relevant equations to calculate population size and growth rates.

In the first part, we determine the population size at t = 3 years, starting with an initial population size of n₀ = 100 and a constant per capita growth rate r = 2.303 individuals per year per individual. The equation used for this calculation is:

n_t = n₀ \cdot e^{(r \cdot t)}

Substituting the known values, we have:

n_t = 100 \cdot e^{(2.303 \cdot 3)}

Calculating this yields a final population size of approximately 100,125 individuals after 3 years, confirming the exponential growth due to the positive and constant growth rate.

In the second part, we calculate the average per capita population growth rate r between t = 1 and t = 2 using the coordinates (1, 100) and (2, 1,000). The average growth rate is derived from the equation:

r \Delta t = \frac{\Delta n}{\Delta t} \cdot \frac{1}{n}

Here, Δn is the change in population size, calculated as:

Δn = 1,000 - 100 = 900

And Δt is the change in time:

Δt = 2 - 1 = 1

Using the initial population size n = 100, we find:

r \Delta t = \frac{900}{1} \cdot \frac{1}{100} = 9

This indicates an average growth rate of 9 individuals per year per individual.

In the final part, we calculate the instantaneous population growth rate at t = 1 and t = 2 using the equation:

\frac{d n}{d t} = r \cdot n

For t = 1, substituting r = 2.303 and n = 100 gives:

\frac{d n}{d t} = 2.303 \cdot 100 = 230.3

For t = 2, using n = 1,000 results in:

\frac{d n}{d t} = 2.303 \cdot 1,000 = 2,303

Thus, the instantaneous growth rates are 230.3 individuals per year at t = 1 and 2,303 individuals per year at t = 2.

This comprehensive analysis illustrates the application of the exponential population growth model, highlighting the calculations for both average and instantaneous growth rates based on given parameters.

If a bacterial population starts with 100 cells and has a per capita population growth rate ($r$) of 0.1 cells per hour per cell, what would be the population size after 5 hours?

• A. 165 individuals.
• B. 111 individuals.
• C. 272 individuals.
• D. 150 individuals.

Answer: (A)

Imagine a regional population of Atlantic Bluefin Tuna has an annual intrinsic growth rate ($r$) of 0.45. Their population ($N$) at the beginning of year 1 is 1,500. Calculate the regional population of Atlantic Bluefin Tuna at the end of year 1.

• A. 2,175 Tuna.
• B. 4,077 Tuna.
• C. 1,835 Tuna.
• D. 2,352 Tuna.

Answer: (D)

Using the same data as the previous question, calculate the regional population of Atlantic Bluefin Tuna at the end of year 2 & at the end of year 3.

• A. 3,689 Tuna; 5,786 Tuna.
• B. 3,670 Tuna; 5,505 Tuna.
• C. 3,204 Tuna; 4,056 Tuna.
• D. 4,085 Tuna; 5,963 Tuna.

Answer: (A)

A chameleon population ($N$) in a shrubland in Cyprus has an intrinsic growth rate ($r_{\max}$) of 0.38. You estimate the total population to be 550. Assuming this population has experienced uninhibited exponential growth, what was the chameleon population 3 years ago?

• A. 376 chameleons.
• B. 257 chameleons.
• C. 176 chameleons.
• D. 202 chameleons.

Answer: (C)

Here’s what students ask on this topic:

What is the exponential population growth model?

The exponential population growth model describes how populations grow exponentially when the per capita growth rate (r) is positive and constant. This model assumes that resources are unlimited, leading to uninhibited growth. The population size (N) at any time (t) can be calculated using the equation:

$N_t=N_0{e}^{rt}$

where $N_0$ is the initial population size, $e$ is Euler's number (approximately 2.71828), $r$ is the per capita growth rate, and $t$ is the elapsed time. This model is idealized and does not account for resource limitations, which can inhibit long-term exponential growth in natural settings.

How does the per capita growth rate (r) affect exponential population growth?

The per capita growth rate (r) is crucial in determining the rate at which a population grows exponentially. When $r$ is positive and constant, the population size increases exponentially over time. A higher value of $r$ results in a faster growth rate, leading to a steeper exponential curve. Conversely, a lower value of $r$ results in slower growth. The relationship between population size (N) and time (t) is given by the equation:

$N_t=N_0{e}^{rt}$

where $N_0$ is the initial population size. In natural settings, resource limitations often prevent populations from maintaining a constant $r$ over long periods.

What are the key equations used in the exponential population growth model?

The exponential population growth model uses three key equations:

1. $N_t=N_0{e}^{rt}$: This equation calculates the population size at any time (t).

2. $\frac{dN}{dt}=rN$: This equation represents the instantaneous population growth rate, describing the slope of the tangent line at any point on the exponential curve.

3. $\frac{\Delta N}{\Delta t}=r_{\Delta }N$: This equation calculates the average population growth rate over a discrete time interval, where $r_{\Delta }$ is the apparent per capita growth rate in that interval.

These equations help analyze and predict population growth under the assumption of constant per capita growth rate.

Why is exponential population growth unsustainable in natural settings?

Exponential population growth is unsustainable in natural settings due to the limitation of resources. While the exponential growth model assumes unlimited resources, in reality, resources such as food, water, and space are finite. As a population grows, the demand for these resources increases, leading to competition and resource depletion. This limitation prevents populations from maintaining a constant per capita growth rate (r) over long periods. Consequently, growth rates slow down, and populations may reach a carrying capacity, where the environment can no longer support further growth. This results in a more realistic logistic growth model, which accounts for resource limitations and environmental resistance.

How can the exponential population growth model be applied in real-world scenarios?

The exponential population growth model can be applied in real-world scenarios to understand and predict population dynamics under ideal conditions. For example, it can be used to model the initial growth phase of a population introduced to a new environment with abundant resources. It is also useful in microbiology to predict the growth of bacterial cultures in a controlled laboratory setting. Additionally, the model can help in conservation biology to estimate the potential growth of endangered species populations when provided with optimal conditions. However, it is important to recognize the limitations of the model and consider factors such as resource availability and environmental resistance for long-term predictions.