Introduction to Population Growth Models - Online Tutor, Practice Problems & Exam Prep

Population growth models are mathematical frameworks used to predict changes in population size over time. Key models include linear, exponential, and logistic growth, each with specific assumptions such as a closed population and a homogeneous environment. The population growth rate (Δn/Δt) and per capita growth rate (r) are crucial metrics, with r indicating individual contributions to growth. Understanding these concepts helps in analyzing population dynamics and making informed ecological decisions.

Introduction to Population Growth Models

This video, we're going to begin our introduction to population growth models. Now, a population growth model is really just a mathematical tool or framework that's used to describe and predict how a population size will change over time, allowing us to keep tabs on populations within communities and to make informed decisions about how we either should or shouldn't take action to either fulfill or prevent these predictions depending on the situation. Moving forward in our course, we're going to cover three population growth models in their own separate videos, which are the linear population growth model, the exponential population growth model, and last but not least, the logistic population growth model. And what you should notice here is that each of these models has their own specific theme color that we're going to maintain moving forward in our course. Before we discuss each of these models separately, we need to overview a few things that are common to all three models.

So we'll do that first, and I'll see you all there.

Assumptions of Population Growth Models

This video, we're going to discuss assumptions that are made for all three of the population growth models that we introduced in our last lesson video, which are the linear, exponential, and logistic population growth models. Population growth analyses or models are often simplified by making several different assumptions, including the assumptions that you can see down below in this image. All 3 of the models that we're going to discuss moving forward in our course assume a closed population, meaning that they assume no immigration or emigration is occurring into or out of the population, or if they are occurring, that they completely offset each other so that only birth rates and death rates impact the population size. The next assumption is the assumption of a homogeneous environment, which assumes a completely uniform environment where resources are evenly distributed and all individuals in the population are able to equally access these resources. The next set of assumptions ignore the effects of several different factors, including the effects of age structure, sex ratio, and external factors affecting n or population size.

Age structure is really just a breakdown of the age groups within a population, categorizing individuals into groups such as infants, children, adolescents, adults, and elderly, for example. Age structure certainly can have a significant impact on population size and population growth. But for the sake of simplicity, these models ignore the effects of age structure. They also ignore the effects of sex ratio, which refers to the ratio or proportion of males and females. It is possible for males to significantly outnumber the females or vice versa, for females to significantly outnumber the males.

And that sex ratio certainly can have a significant impact on population size and growth. But, again, these effects are going to be completely ignored in all three of these models for the sake of simplification. Lastly, these models will also ignore the effects of external factors affecting population size, including ignoring density-independent factors such as tornadoes and hurricanes, and things of that nature. Down below, we have a very important note, which is that the assumptions offer both advantages and disadvantages to these population growth models. What you'll notice is that the advantage is certainly going to be simplification, as if these assumptions were not made, the equations in all of these models would be significantly more complex, and the disadvantages are that by making these assumptions, the models will be less accurate.

However, they're still applicable and they can still be used to get us the general idea, and so they're not totally useless when we make these assumptions. This is just something to keep in mind that these models allow us to make predictions, but those predictions are not always going to be perfect in part because of many of these assumptions that are made. So that concludes this video, and I'll see you all in our next one.

Population Growth Rate vs. Per Capita Population Growth Rate

This video, we're going to differentiate between population growth rate and per capita population growth rate, which is important for us to be able to do before we get into the population growth models later in our course. Now recall from our previous lesson videos that population growth rate is simply the overall change in population size expressed as delta n over the elapsed time expressed as delta t. Now, the per capita population growth rate on the other hand is sometimes referred to as the intrinsic rate of increase, and it is expressed with the variable r and it is the average population growth rate per individual, and really that is what per capita means is per individual. Now the per capita population growth rate represents each individual's average contribution to the overall population growth. So let's take a look at this problem down below so that we can better differentiate between these 2.

And the problem says, if a squirrel population has 1,000 individuals at the start of the year and 1,025 individuals at the end of the year, what is the population growth rate or ΔnΔt, and what is the per capita population growth rate or r? So, we'll calculate the population growth rate on the left and the per capita population growth rate on the right. So once again, for population growth rate, it's simply ΔnΔt, which is change in population size over change in time. So all we need to do here is take the difference in population at the end of the year and at the start of the year. So 1,025 minus 1,000 will give us the change in population size and for change in time we just need to realize we're looking at the start of the year versus the end of the year, so 1 year has passed by.

And so if we perform this calculation, what we get is 25 over 1, which is simply equal to 25. And so that is the population growth rate. So we could say that the population growth rate in this time period was 25 squirrels per year. Simple as that. Now, for per capita population growth rate, which again is expressed as r, it is the average population growth rate per individual.

So all we need to do is take the population growth rate ΔnΔt that we just calculated over here which is 25 and divided by the initial population size or n0 which is 1,000. And so what we get when we do that is we get r is equal to 251,000 and that's equal to 0.025. And so what we can say is that the per capita population growth rate in this time period was 0.025 squirrels per year, per squirrel, per individual. And so what you'll notice is that there is another way to calculate the per capita population growth rate and that is b minus d, where b is equal to the per capita birth rate and d is equal to the per capita death rate. Now this problem did not give us any per capita birth or death rates, but moving forward, we will see some problems that provide this information and we'll need to know this to solve for the per capita population growth rate.

So this here concludes our video and we'll be able to get some practice moving forward, so I'll see you all in our next one.

Introduction to Population Growth Models Example 1

So here we have an example problem that says, a population of 200 fish in an isolated pond experiences a per capita birth rate or b of 0.3 and a per capita death rate d of 0.1 in a month. Calculate the population growth rate and the intrinsic rate of increase or r. And so recall from our last lesson video that the intrinsic rate of increase is another way to say the per capita population growth rate. Also, recall that this value of r can be calculated using 2 equations, and those two equations can be seen right here. And in this problem, we're given the values of b and d, which again are the per capita birth and death rates.

So all we need to do is take the difference between the two to get the value of r. And so when we do that, what we get is r=0.3-0.1=0.2, and the units of this are going to be fish per month per fish. So now that we've calculated the value of r as 0.2, we can go ahead and plug that into this equation over here. We also have the value of n₀, which is the initial population size given to us as 200 fish. So if we plug those two values in, we can calculate delta n over delta t, which is our population growth rate.

And so we can go ahead and do that now. So what we end up getting is r=deltandeltatdivided byn0, which again is 200 fish. And so if you multiply both sides of the equation by 200, what you'll get is 200×0.2, comes out to 40, and that is going to be equal to the, value of delta n over delta t. And the units of this 40 is going to be fish per month, so forty fish. I'll rewrite it over here.

Forty fish per month. That is going to be what delta n over delta t is. So now that we've calculated delta n over delta t as 40 fish per month, we've calculated the population growth rate, and, of course, we've calculated our 0.2 fish per month per fish, and that is the intrinsic rate of increase. So now that concludes this problem, and I'll see you all on our next video.

An island has a population of 10,000 rabbits at the beginning of the year. During the year, there were 400 births and 150 deaths. Calculate the population growth rate and the per capita population growth rate.

• A. ΔN/Δt = 250 rabbits/year; r = 0.25 rabbits/year/rabbit
• B. ΔN/Δt = 250 rabbits/year; r = 2.5 rabbits/year/rabbit
• C. ΔN/Δt = 250 rabbits/year; r = 0.025 rabbits/year/rabbit
• D. ΔN/Δt = 550 rabbits/year; r = 0.0025 rabbits/year/rabbit

Population Growth Model Variable

This video, we're going to talk more about the population growth model variable r. And so, it turns out that this variable r actually appears in all 3 of the population growth models that we'll talk more about moving forward in our course, which again are the linear, exponential, and logistic population growth models. However, unfortunately, the definition of r is not consistent across these three models because in only the linear population growth model, r is not defined on a per capita basis. Instead, r is just defined as the population growth rate, which we know from our previous lesson videos is simply just Δn/Δt. Now, on the other hand, the other two models, the exponential and logistic population growth models, are defined exactly as we defined it in our previous lesson videos, which is the per capita population growth rate, which we already know from previous lesson videos is defined as the population growth rate divided by the initial population size.

Now, it turns out that there is a maximum value of this per capita population growth rate that we call the r value. And so the r value is just the maximum value of the per capita population growth rate for a particular species in ideal or perfect conditions that supports the growth of that species. Now it is important to emphasize that the r value is a species-specific value as it's determined by the biology of the species. Now, despite the definition of r not being consistent across these three models, one of the big takeaways of this video is that in all three of the population growth models, the value of r will dictate how and how quickly the population size changes over time. And so, when the value of r is equal to 0, the population size is going to remain constant and will not change.

Now when r is less than 0, meaning that r has negative values, the population size is going to decrease. And the more negative the value of r is, the faster the population size will decrease. Now, on the other hand, when r is greater than 0, or in other words, when r has positive values, the population size increases. And, again, the greater the value of r is, the more quickly the population size will increase. But, again, it's important to emphasize that every species will have a maximum value r that we call the r.

So, this here concludes our brief lesson, and we'll be able to apply these concepts and learn more as we move forward. So I'll see you all in our next video.

Introduction to Population Growth Models Example 2

So here we have an example problem that wants us to appropriately match each of the 3 species we have down below with their corresponding rmax values on this scale, and these rmax values are given to us as per capita population growth rates per day under ideal conditions. Recall that the rmax values are species specific as they are determined by the biology of the species. The first species on our list is E. coli, which is a type of bacteria, and each individual bacterium has an incredible capacity to divide and reproduce. We might expect it to have the highest rmax value on the scale, and indeed it does.

We can go ahead and put E. coli over here on the far right and then cross it off our list. This means that each individual E. coli bacterium, on average, contributes 60 individuals to the population growth per day under ideal conditions. The next rmax value we are given is for the Australian spider beetle, which is a value of 0.057. We then need to assign these 2 rmax values here to either the human or the brown rat. Naturally, a population of brown rats has a greater capacity to grow faster than a population of humans, for several different reasons including having shorter generation times and reaching sexual maturity at earlier ages.

It turns out that brown rats have the edge on the rmax value, so we put the brown rat here for a value of 0.015. This leaves humans with the other rmax value of 0.0003. This might seem quite low, but what it is saying is that each individual human, on average, contributes 0.003 humans to the population growth per day under ideal conditions. That is a value for humans, and you can see how this is actually quite low on the spectrum of rmax values when we consider the wide variety of species that exist on this planet. This concludes our problem here, and I will see you in our next video.

Introduction to Population Growth Models Example 3

So here we have an example problem that says, a population of 289 wolves has 27 births and 9 deaths from the year 2023 to the year 2024, a period of 1 year. Calculate the per capita birth rate, the per capita death rate, the per capita population growth rate, and the overall population growth rate. And so we're going to perform each of these calculations one-by-one, starting with the per capita birth rate, which is simply the number of births per individual. So, we can take the 27 births that are given to us and divide it by the 289 individual wolves. The calculation gives us an answer of about 0.0934, and the units are going to be births per wolf.

Next, we can calculate the per capita death rate, which is just going to be the number of deaths per individual. We're given that there are 9 deaths, and again, 289 individual wolves. So, \( \frac{9}{289} \) gives us an answer of about 0.0311, and this is going to be in units of deaths per wolf.

So now we can go ahead and calculate the per capita population growth rate, which, recall, is symbolized with the variable \( r \), and can be calculated by taking the difference between the per capita birth rate and the per capita death rate, \( b - d \). And so, \( 0.0934 - 0.0311 \), gives us an answer of 0.0623, and the units of this are going to be wolves per year per wolf. What this means is that each individual wolf contributes 0.0623 wolves to the population per year.

Last but not least, we can calculate the population growth rate, which is simply \( \Delta n \) over \( \Delta t \) or the change in population size over the change in time. We can calculate this in several different ways, but what we're going to do is simply take the difference in the number of births and the number of deaths, which will give us the change of population size, so 27 births minus 9 deaths. Since \( \Delta t \) is just the elapsed time, which is 1 year, so that's going to be 1. Hence, \( 27 - 9 = 18 \), and over 1, it's just going to come out to 18, and the units of this number are going to be wolves per year. And so what we're seeing is that in this time period, there is an overall growth rate of the population going from 289 wolves to 289 plus 18 wolves; 18 additional wolves grew in that year.

And so, this here concludes this example problem, and I'll see you all in our next video.

In which population growth models is the population growth dependent on the current population size?

• A. Linear model.
• B. Exponential model.
• C. Logistic model.
• D. Exponential & logistic models.