Exponential Growth

Summary

In this educational video, Mr. Andersen explains exponential growth, particularly in population dynamics, using rabbits as an example. He illustrates how the growth rate (r) affects population changes over time, demonstrating the concept with a spreadsheet and algebraic equations. The video also touches on the limitations of exponential growth due to resource constraints, hinting at logistic growth for future discussion.

Takeaways

• 📈 Exponential growth is a concept where populations can increase rapidly, often seen in biological contexts like rabbit populations.
• 🔢 The population size is denoted by 'N' and can change over time due to factors like births and deaths.
• 🐰 The growth rate 'r' is a key factor in determining the change in population size, calculated as the net increase (births minus deaths) divided by the original population size.
• 📚 Understanding the math behind exponential growth can be tricky, but it's essential for grasping how populations can explode in size.
• 📊 The growth rate 'r' is per capita, meaning it's the change in population size relative to the original population size.
• 🌱 In a stable ecosystem, the growth rate 'r' tends to remain constant over time, leading to consistent population growth.
• 🔄 The population increase is calculated by multiplying the current population size by the growth rate 'r', which compounds with each generation.
• 📉 A negative growth rate indicates a population decline, approaching zero as the population shrinks with each generation.
• 📊 Exponential growth can be modeled and visualized using spreadsheets, which can help in understanding and predicting population sizes at different times.
• 📝 Algebraic solutions, such as the formula N(t) = N * (1 + r)^t, provide a mathematical way to calculate population size at any given time 't'.
• 🚫 Exponential growth in real-world scenarios is unsustainable due to limitations in resources, food, and space, which eventually leads to logistic growth.

Q & A

What is exponential growth and why is it significant in the context of populations?

Exponential growth refers to the rapid increase in a population size, often seen in populations such as bacteria or rabbits, where the growth rate is proportional to the current population. It's significant because it can lead to a population explosion if unchecked, as seen in the example of rabbits in the script.

What does the variable 'N' represent in the context of the script?

In the script, 'N' represents the population size at a given time, which can change as the population grows or shrinks due to births and deaths.

What is the role of the growth rate 'r' in determining the change in population?

The growth rate 'r' is a factor that indicates how much the population is changing over time. It is calculated as the difference between births and deaths divided by the original population size, and it determines the increase in population in each generation.

How are births and deaths represented in the calculation of the growth rate 'r'?

Births and deaths are represented as the number of new rabbits (births) and the number of dead rabbits (deaths). The growth rate 'r' is calculated by subtracting the number of deaths from the number of births and then dividing by the original population size 'N'.

Why does the population increase by a larger number in subsequent generations even if the growth rate 'r' remains constant?

The population increase is larger in subsequent generations because the growth rate 'r' is applied to an increasingly larger population size. This means that even a constant growth rate results in a larger absolute increase as the population grows.

What is the significance of the spreadsheet in understanding exponential growth?

The spreadsheet is used to model and visualize the process of exponential growth over time. It allows for easy calculation and iteration of population sizes based on the growth rate, making it simpler to predict future population sizes and observe the exponential growth pattern.

How does changing the growth rate 'r' in the spreadsheet affect the population growth curve?

Changing the growth rate 'r' in the spreadsheet alters the steepness of the exponential growth curve. A higher growth rate results in a steeper curve, indicating a faster increase in population, while a lower growth rate results in a more gradual increase.

What happens when the growth rate 'r' is set to 0 in the spreadsheet?

When the growth rate 'r' is set to 0, the population does not increase at all. The population size remains constant throughout the time period modeled in the spreadsheet.

What is the algebraic solution for calculating the population size at a given time 't' in exponential growth?

The algebraic solution is given by the formula N(t) = N * (1 + r)^t, where N is the initial population size, r is the growth rate, and t is the time. This formula allows for the calculation of the population size at any given time 't' based on the initial conditions and growth rate.

How does the script illustrate the concept of exponential growth with the example of E. coli bacteria?

The script uses the example of E. coli bacteria, which can reproduce every 20 minutes, to illustrate the concept of exponential growth. With a growth rate 'r' of 1 (100% increase per generation), the population doubles with each generation, demonstrating how quickly a population can grow under exponential growth conditions.

What limitations are implied in the script regarding the sustainability of exponential growth?

The script implies that exponential growth is not sustainable in the long term due to limitations such as the availability of food, resources, and space. As the population grows, these factors will eventually cause the growth rate 'r' to change, leading to a shift from exponential to logistic growth.

Outlines

00:00

🐰 Exponential Growth in Rabbit Populations

In this section, Mr. Andersen introduces the concept of exponential growth, focusing on how populations, such as rabbits, can experience rapid increases in numbers. He uses a simple example of a rabbit population starting with 10 rabbits and explains how growth rates (r) affect population changes over time. The growth rate is calculated as the difference between births and deaths divided by the initial population size (N). Andersen demonstrates how the population increases exponentially, not linearly, as the growth rate is applied to an increasingly larger population size in each generation. He also highlights the importance of understanding that the growth rate can remain constant in a stable ecosystem, leading to a significant increase in population over time.

05:03

📈 Understanding Exponential Growth with Spreadsheets

This paragraph delves into how exponential growth can be visualized and calculated using spreadsheets like Excel. Andersen shows how to set up a spreadsheet to model the rabbit population growth over time, emphasizing the exponential curve or J-shaped curve that results from continuous growth. He demonstrates how changing the growth rate in the spreadsheet affects the curve's steepness and the final population size. Andersen also explores what happens when the growth rate is set to zero or negative, illustrating the stability and decline in population, respectively. The use of spreadsheets allows for easy manipulation of variables and quick answers to questions about population size at different times.

10:03

🧪 Exponential Growth in Bacteria and Logistic Growth

In the final paragraph, Andersen shifts the focus from rabbits to bacteria, using E. coli as an example to illustrate how exponential growth can lead to an enormous increase in population in a very short time. He explains that with a growth rate of 1 (100% increase per generation), the population doubles rapidly. Andersen also addresses the limitations of exponential growth, noting that it cannot continue indefinitely due to constraints like resource availability. He introduces the concept of logistic growth, which will be explored in a future video, as a more realistic model for population growth that takes into account carrying capacity and environmental limitations.

Keywords

💡Exponential Growth

Exponential growth refers to a pattern of growth where the rate of increase of a quantity is proportional to the current value of that quantity. In the video, Mr. Andersen uses the example of a rabbit population to illustrate this concept. The growth rate is constant, but the actual increase in population size becomes larger as the population grows, leading to an exponential curve. This concept is central to understanding how populations can rapidly increase under certain conditions.

💡Population Size (N)

Population size, denoted as 'N' in the video, is the total number of individuals in a population at a given time. It is a fundamental variable in the discussion of exponential growth. Mr. Andersen starts with an initial population size of 10 rabbits and uses this as the basis for calculating changes in population over time. The concept is crucial for understanding how the growth rate affects the total number of individuals in a population.

💡Growth Rate (r)

The growth rate, symbolized as 'r', is a measure of how much a population changes over time, typically expressed as a proportion of the current population size. In the video, Mr. Andersen calculates the growth rate by subtracting the number of deaths from the number of births and dividing by the initial population size. This rate is key to determining the speed at which the population increases and is a central component in the mathematical models used to predict population changes.

💡Births and Deaths

Births and deaths are the primary factors affecting the change in population size in the context of exponential growth. In the video, Mr. Andersen demonstrates how the number of new rabbits (births) and the number of rabbits that die (deaths) contribute to the overall change in the rabbit population. These events are essential in calculating the growth rate and understanding the dynamics of population growth.

💡Per Capita

Per capita, in the context of the video, refers to the rate of change in population size relative to the current population size. Mr. Andersen emphasizes that the growth rate must be calculated per capita by dividing the number of births and deaths by the initial population size. This approach is crucial for accurately determining the impact of births and deaths on the population growth rate.

💡Spreadsheet

A spreadsheet, as mentioned in the video, is a tool used to organize, manipulate, and analyze data. Mr. Andersen uses a spreadsheet to model the exponential growth of the rabbit population, demonstrating how it can automate the calculation of population changes over time. Spreadsheets are a practical way to visualize and predict population dynamics based on the given growth rate.

💡J-Shaped Curve

The J-shaped curve is a graphical representation of exponential growth, characterized by a rapid increase in population size over time. In the video, Mr. Andersen refers to this curve when discussing the results of his spreadsheet calculations. The curve illustrates how populations can grow quickly under ideal conditions, such as abundant resources and minimal competition.

💡Hockey Stick Curve

The hockey stick curve is another term for the J-shaped curve, used to describe the pattern of exponential growth. Mr. Andersen mentions this term to describe the shape of the growth curve in his spreadsheet. The curve's name comes from its resemblance to the shape of a hockey stick, with a rapid increase in the population size occurring after an initial period of slower growth.

💡Algebraic Solution

An algebraic solution, in the context of the video, refers to a mathematical formula used to model exponential growth. Mr. Andersen provides an equation that relates the population size at a given time (N), the initial population size, the growth rate (r), and time (t). This formula allows for the prediction of population size at any point in time, demonstrating the power of algebra in modeling population dynamics.

💡Logistic Growth

Logistic growth is a model of population growth that accounts for limiting factors such as resource availability and space. In the video, Mr. Andersen briefly mentions that exponential growth cannot continue indefinitely and that logistic growth becomes relevant as populations approach carrying capacity. Logistic growth models incorporate a decreasing growth rate as the population size approaches the maximum sustainable size.

Highlights

Exponential growth is a concept that describes how populations can rapidly increase in size.

Understanding exponential growth involves grasping the mathematical principles that drive population changes.

The video uses rabbits as an example to illustrate the concept of exponential growth.

The population size is denoted by N, which changes over time.

The growth rate, denoted by r, determines how the population changes over time.

Births and deaths are the primary factors affecting the population change, calculated on a per capita basis.

An example calculation shows a birth rate of 5 and a death rate of 2 in a population of 10 rabbits, resulting in a growth rate of 0.3.

The growth rate is a factor indicating the rate at which the population increases.

In a stable ecosystem, the growth rate remains constant over time.

Despite a constant growth rate, the population increases exponentially as the base population size increases.

A spreadsheet can be used to model and predict population growth over time.

Excel is demonstrated as a tool for creating a model of exponential growth.

The spreadsheet model shows an exponential curve, also known as a J-shaped or hockey stick curve.

Changing the growth rate in the spreadsheet model affects the steepness of the exponential curve.

An algebraic solution is presented for calculating exponential growth, using the formula N(1 + r)^t.

The algebraic formula is verified with examples, showing how population size increases over time.

The video discusses the hypothetical scenario of E. coli bacteria reproduction, illustrating extreme exponential growth.

The limitations of exponential growth are acknowledged, as resources become limited and the growth rate changes.

Logistic growth, which accounts for resource limitations, is mentioned as a topic for a future video.

Transcripts

Hi. It's Mr. Andersen and in this video I'm going to talk about exponential

growth which is how populations can explode. Most students understand exponential growth,

but the math sometimes gets a little tricky. And so I am going to step you through that

in a couple of ways. And so let's start with this rabbit right here. Let's say it's part

of a population. We don't only have one rabbit, but we have a number of rabbits in our population.

We refer to that in all of the equations as N. N is going to be the population size. Now

that's going to change as we go through time. But this is going to be our original population

which is going to be N. Let's stack those rabbits up so we can count them. So our N

to start is going to be 10. So we have 10 rabbits at time 0. Now that population is

either going to increase, it's going to decrease or it's going to stay the same. And what things

are determining that? It's going to be our growth rate. And so this is the second letter

you should remember. And that's r. r is going to refer to how much it's changing over time.

And there's really only two things that are going to change that population. We're going

to have new rabbits, that's going to be births. And then we're going to have dead rabbits

and that's going to be deaths. And so those two things are going to contribute to the

change in the population. But it's per capita. In other words we have to divide by the N.

Which is going to be the original population size. And so let me make some baby rabbits.

So if I click here we've got 5 baby rabbits. And so our births would be five. And then

let's say I want to kill a couple of rabbits. Let's kill that guy. Don't worry, they're

okay. They're just virtual rabbits. And so I kill that one as well. And so we've got

births of 5. We've now got deaths of 2. And what was our N to begin with? It was 10. And

so we figure out our r value. That's going to be 5 minus 2 divided by 10, which is 3

over 10, which is going to be 0.3. And so our r value is 0.3 or our growth rate is 0.3.

What does that really mean? It's the factor at which our population is increasing. And

so if I take 10 times 0.3 I'm going to get 3. And that's how much our population increased.

And one thing you should know about that growth rate is that if the ecosystem is stable the

growth rate is essentially going to stay the same. It's not going to change over time.

And so you might think, well, if the growth rate stays the same, isn't the population

just going to increase along a consistent amount? Not really. And so let's watch what

happens. Now we're going to take 0.3 growth rate for the next generation and instead of

multiplying it times 10 we now have to multiply it times 13. And if we do that and we'll use

this equation right here, this is the change in N or the change in t. We're going to take

our growth rate with is 0.3 and now multiply it times 13. Well we don't get three anymore.

We get 3.9. And so I'm going to round. That sounds a lot like 4 rabbits. So I'm going

to add 4 rabbits. And now our population is up to 17. So even though r stayed the same

since we multiplied it times a larger value, we're going to get more rabbits in the next

generation. So let's do generation 3. We're now taking 0.3 times 17. And I get 5.1, which

is a lot like 5 rabbits. And I'm going to add those two rabbits. Or if we now multiply

that growth rate times 22, I get 6.6 which is pretty close to 7 rabbits. So we're going

to add those 7 rabbits. And so we now have got a population of 29. And so you can see

that the population is increasing. But if I were to ask you a question, I could ask

you some hard questions. The first one is not so hard. What's the population going to

be in year 5? Well to do that you take 29 times 0.3 and then we'd add that to 29. But

what if I asked you 10. Or even 30? Well this problem get's pretty hard. And so you're quickly

going to want a little bit of help. And for me when I want help the first place I go to

is to a spreadsheet. And let's go to the spreadsheet. So we're going to go to Excel. Kind of remember

those numbers there. And so let's kind of rebuild that chart. So on the left side we're

going to have 0 as our first time and 1 as our second time. If you didn't know to do

this in Excel, I can select both of those, grab this little corner here and I can increase

and it will do the counting for me. And so let's go up to population in time 30. Okay.

So we've filled that in. Now what's our original population? That's going to be 10. So I'm

got to just put that in to start with. Now what's the next population? So I'm going to

put a formula in this box. And to do that I'm just going to put an = sign. So I'm going

to put an = sign, you can see the formula right here. And so what did we do? Remember

we're going to take the original population, so let me click on that. So that's going to

be 10. And then we're going to add that, to that, we're going to add our growth rate which

was 0.3 times then we're going to click on that again. And so let's see what we get.

So we get 13. And so what we did is again, we took what was in this cell and we added

it to what was in this cell times 0.3. And so again through the magic of a spreadsheet

I can simply grab this and it's just going to use that same thing over and over. It's

going to iterate on that. And so what we're going to get is it's going to do all of the

math for us. And so what did we have in the first one? 13, 17, 22. This sounds familiar.

Now it's obviously not rounded off and so this isn't the correct number of rabbits,

but we see now this exponential curve here. Or this J shaped curve. Or sometimes we call

it like a hockey stick curve, because it's quickly turning up like that. And so now we

could quickly answer those questions. At time 5 we should have around 37 rabbits. What was

the next one? I think 10. We should have around 138 rabbits. And if we go all the way down

here to time 30 we're going to have 200 and, 26199 rabbits. So that's a lot of rabbits

really really quickly. And so you can see how exponential growth takes off. But what's

fun about a spreadsheet is I could play around with it a little bit. Let's say instead of

0.3, if I change my growth rate to 0.1. So if I do that, what are we going to get? Well

if I move this all the way down again were going to get another J shaped curve. Now it's

not going to be as steep as that one was. And we didn't have as many rabbits at the

end. But we're still going to be exponential growth. Or if I were to go edit that variable

again. Let's make it 0. So let's say we take it times 0. What would we get then? Well it's

10. In other words if I go all the way down here, what are we going to get for a value?

Well, we're not increasing at all. And so it's going to be 10. It's going to stay 10

the whole time. A really hard question that I ask the students is this. Let's say we get

a negative growth rate. So let's make it -0.3. What are we going to get for a value there?

Well we get 7 here. But if I go all the way down, what do we get? Oh that's weird. We're

going to approach kind of a limit. We're going to approach 0. And that's because we're going

to take off larger amounts to begin with. And then we're going to take off less amounts

as the population gets smaller and smaller and smaller. So again that's spreadsheets.

But you don't always have a spreadsheet with you. Sometimes you just need a calculator

and a little bit of algebra. So let's go to the algebra. This is going to be the algebraic

solution to this. And so we've got an equation for exponential growth. And so change in N

over t is going to be equal to N, which is our population size, times 1 plus r, where

r is going to be the growth rate and then we're going to raise that to t, where t is

going to be equal to time. And so let's make sure that this works. Let's start right here

with 0. And so let's plug in our numbers. So we're going to put 10 in here for n. That's

that original population. 1 and then r is going to go right here. It's going to be 0.3.

And then we raise it to the 0 power. Because our time is going to be 0. Well if we simplify

that a little bit, anything raised to the 0 power is always 1. And so that's going to

be 10. And so that works out so far. But don't trust me. Let's keep going. Let's go to the

next one. Let's go to 1. So if we now put in 1 for time. It's going to be, N is still

10. This is 1 plus r again. And the r is not changing. But we're raising it to the one

power. Anything raised to the 1st power is going to be itself. So it's going to be 1.3.

And we get 13. Let's try that again with 2. So if we go to the second power again, we

plug in 2 here. It's the only thing that we're changing. So we get 1.3 to the second power.

So we're going to have to square 1.3 which is 1.69 and we get 16.9 which is a lot like

17. Or I could just throw out another time. So let's say we go time 30. So then we're

going to raise it to the 30th power. And so I get 26,199 which is going to match up exactly

with our spreadsheet. We're going to have a lot of rabbits really really quickly. And

so that's kind of an algebraic solution. It's a quick way to be given a time and then figure

out how much it's going to grow. And so a good question I could ask you is this. Let's

look at bacteria rather than rabbits. Okay? And so let's say we have one bacteria. E.

coli can reproduce in about 20 minutes. In other words 1 can make an exact copy of itself

in about 20 minutes. And so let's say that none of them die. Let's say we get rid of

the death rate. So we've got our births over N. And so how many births would we have? If

we're just dividing in half we're going to have one new E. coli. What was our original

population? It was going to be 1 as well. So now we're going to get an r of 1. And so

instead of increasing by 0.3, we're now increasing by 1 which is really increasing by 100%. So

we had 1 bacteria and now we have 100% as many bacteria or twice as many. Or 100% in

addition to that original. And so now we have 2. And so what are we going to have on the

next round? We're going to have 4. And 8. And 16. And then 32. And you can see how exponential

growth gives us a huge amount of bacteria really, really quickly. And so the question

I might ask is, is the sky the limit? And so if we're looking at exponential growth

you know, after 20 rounds like this we're going to be way up in here. I can't even read

this, 5 million, something like that. And so it does it just keep going and going and

going? No. Because what I told you is kind of a lie. r is not going to stay the same

forever. As it starts to grow they're going to run out of food, resources, space. And

so our r is going to start to change. And so then we're going to start to move into

what's logistic growth. And that's going to be a totally different set of equations and

I'm going to include that in another video. And so that's exponential growth. And I hope

that was helpful.