Linear Population Growth - Online Tutor, Practice Problems & Exam Prep

The linear population growth model is a simplified framework for understanding population dynamics, characterized by a constant growth rate regardless of population size. This model, represented by the equation n=r⁢t+n₀, is useful for short-term projections and initial growth analysis, despite its limitations in real-world applications. It serves as a foundational concept before exploring more complex models like exponential and logistic growth.

Linear Population Growth

This video, we're going to talk about the very first population growth model in our lesson, which is the linear population growth model. The linear population growth model is a very simplified model, especially in comparison to the other two population growth models that we'll talk about moving forward in our course, which are the exponential and logistic growth models. It's so simplified that it's often an oversimplification for most real-world scenarios. However, this linear population growth model is still very helpful as it can provide a basic framework for understanding population growth in the absence of complicating factors, making it a perfect starting place for beginners. The key to this linear population growth model is that unlike the other two population growth models, the population growth rate in this linear model is constant and always going to be the same regardless of the current population size, which makes it very simple and easy, but not realistic to most situations as you might expect that a larger population size would have more individuals that can contribute to the population growth.

Now, although the linear population growth is often an oversimplification, it's not totally useless as it can be helpful and used for analyzing the initial stages of population growth. It can be helpful for short-term projections of population growth and can be applicable in controlled experimental settings. Notice down below, we're showing you the equation for a line, which you should already be familiar with, which is y=mx+b. We're showing you this because it turns out that the linear population growth equation is exactly the same, except that it substitutes the variables with population growth model variables. And so nt=r(t)+n0, where r is the absolute population growth rate, which is just delta n over delta t, where 'absolute' just implies that it is constant and does not change, and that's going to be multiplied by t, which is the amount of elapsed time, plus the, n₀, which is the initial population size.

On a graph like this one that you can see over here, where we have the time on the x-axis and population size on the y-axis, for the linear population growth, it's no surprise that the data will form a straight line, where you started at some non-zero population size and the population growth rate will be constant, creating a straight line in the data. This here concludes our lesson on the linear population growth model. We'll be able to apply these concepts and problems, and then we'll shift over to the other growth models. So, I'll see you in our next video.

Linear Population Growth Example 1

So here we have a pretty straightforward example problem that says, a proliferating stem cell whose progeny does not reproduce would grow via the linear population growth model. Which of the following best explains why this is the case? And we've got these 4 potential answer options down below. Now option a says, the progeny continue to reproduce leading to exponential growth, but this completely contradicts what the problem initially told us, which is that the progeny does not reproduce. And, also, even if the progeny did continue to reproduce and it did lead to exponential growth, that would not be linear population growth.

And so for that reason, we can eliminate answer option a. Now option b says the progeny only reproduce during certain seasons causing cyclical growth patterns. But cyclical growth patterns are not going to be linear growth patterns, and so for that reason, we can eliminate answer option b. Now, option c says, the progeny exhibit high rates of reproduction, but the growth is offset by a greater rate of cell death. Now, once again, this is contradicting what the problem told us initially, which is that the progeny does not reproduce.

So to say that the progeny exhibit high rates of reproduction is not going to be correct. So we can eliminate option c, and this leaves answer option d as the only option and it is the correct answer, which says that the progeny do not contribute to further population growth resulting in constant addition of cells over time, regardless of the current population size. And, ultimately, this is what makes it grow via the linear population growth model. So we can indicate that option d is the correct answer. That concludes this problem, and I'll see you all in our next video.

In a small, isolated island, a population that initially consists of 50 birds is observed to grow by a constant number of 5 birds per month. What is the population size after 2 years?

• A. 110 birds.
• B. 170 birds.
• C. 250 birds.
• D. 60 birds.

Here’s what students ask on this topic:

What is the linear population growth model and how is it different from other growth models?

The linear population growth model is a simplified framework for understanding population dynamics, characterized by a constant growth rate regardless of population size. This model is represented by the equation $n=rt+n_0$, where $n$ is the final population size, $r$ is the absolute population growth rate, $t$ is the elapsed time, and $n_0$ is the initial population size. Unlike exponential and logistic growth models, which account for varying growth rates depending on population size and other factors, the linear model assumes a constant growth rate, making it less realistic for most real-world scenarios but useful for short-term projections and initial growth analysis.

How is the equation for linear population growth derived?

The equation for linear population growth is derived from the basic equation of a line, $y=mx+b$. In the context of population growth, the variables are substituted as follows: $n$ (final population size) replaces $y$, $r$ (absolute population growth rate) replaces $m$, $t$ (elapsed time) replaces $x$, and $n_0$ (initial population size) replaces $b$. Thus, the equation becomes $n=rt+n_0$. This equation indicates that the population size increases linearly over time at a constant rate.

In what scenarios is the linear population growth model useful?

The linear population growth model is useful in several scenarios despite its limitations. It is particularly helpful for analyzing the initial stages of population growth, where the growth rate can be approximated as constant. Additionally, it is valuable for short-term projections of population growth, providing a simple and straightforward method for estimating future population sizes over a limited time frame. The model is also applicable in controlled experimental settings where external factors influencing population growth are minimized, allowing for a more accurate representation of linear growth. Overall, while the linear model may not be realistic for long-term or complex real-world applications, it serves as a foundational tool for understanding basic population dynamics.

What are the limitations of the linear population growth model?

The linear population growth model has several limitations. Firstly, it assumes a constant growth rate regardless of the population size, which is not realistic for most real-world scenarios where growth rates typically vary with population density and other factors. Secondly, it does not account for environmental constraints, resource limitations, or other factors that can influence population growth. As a result, the model is often an oversimplification and may not accurately predict long-term population dynamics. Despite these limitations, the linear model can still be useful for short-term projections and initial growth analysis, providing a basic framework for understanding population growth before exploring more complex models like exponential and logistic growth.

How does the linear population growth model compare to the exponential and logistic growth models?

The linear population growth model differs significantly from the exponential and logistic growth models. The linear model assumes a constant growth rate, resulting in a straight-line increase in population size over time, represented by the equation $n=rt+n_0$. In contrast, the exponential growth model assumes a growth rate proportional to the current population size, leading to a J-shaped curve where the population size increases rapidly over time. The logistic growth model incorporates environmental constraints and resource limitations, resulting in an S-shaped curve where the population growth rate slows as it approaches the carrying capacity of the environment. While the linear model is simpler and useful for short-term projections, the exponential and logistic models provide more realistic representations of population dynamics over longer periods and in more complex scenarios.