Logistic Growth

Summary

This video by Mr. Andersen explains the concept of logistic growth in populations, starting with exponential growth. He uses rabbits to illustrate how populations grow rapidly until they reach a natural limit, called carrying capacity. The video highlights how limited resources, like food, slow down growth as populations increase. Andersen also discusses species that grow quickly (r-selected) versus those that invest more in fewer offspring (K-selected). Using various models, he demonstrates the mathematical principles behind logistic growth, emphasizing how populations balance growth with environmental limits.

Takeaways

• 🐇 Logistic growth happens when a population reaches its carrying capacity after an initial period of exponential growth.
• 📊 N refers to the population size, and r represents the growth rate, determined by births and deaths.
• ⚖️ Exponential growth occurs when r is greater than zero, leading to rapid population increases over time.
• 🌾 Carrying capacity (K) is the maximum population size that an ecosystem can support due to limited resources, like food.
• 📉 Logistic growth follows an 'S'-shaped curve, starting with rapid growth but eventually leveling off as it approaches carrying capacity.
• 🐓 A class example using chickens in Minecraft demonstrated how populations reach carrying capacity as resources become limited.
• 📈 A NetLogo model showed how rabbit populations fluctuate over time as they consume grass, leading to crashes and stabilization.
• ➕ The exponential growth equation is dN/dt = r * N, while the logistic growth equation adds a term to account for carrying capacity.
• 🦎 r-selected species, like frogs, reproduce quickly with many offspring but often experience boom-and-bust cycles.
• 🧑‍👦 Humans, being K-selected species, reproduce slowly with significant parental investment, leading to a more stable population growth curve.

Q & A

What is logistic growth and how does it differ from exponential growth?

Logistic growth occurs when a population grows rapidly at first but eventually slows down as it reaches its carrying capacity, which is the maximum population that an environment can support. Exponential growth, on the other hand, is the rapid increase of a population without any limits, often leading to a sharp rise in numbers.

What is 'carrying capacity' (K) and how does it influence population growth?

Carrying capacity (K) refers to the maximum number of individuals that an environment can sustain due to limiting factors like resources (food, space). Once a population reaches this limit, growth slows down or stops as resources become insufficient for further growth.

How do 'r' (growth rate) and 'N' (population size) affect population growth?

The growth rate (r) and population size (N) are key factors in determining population growth. When r is greater than zero, the population grows exponentially. As N increases, the impact of the growth rate becomes more significant, but growth eventually slows as the population nears the carrying capacity (K).

What happens to population growth when r is 1, and how does this lead to exponential growth?

When r is 1, the population doubles with each generation, leading to exponential growth. For example, starting with 2 rabbits, the population will quickly increase to 4, 8, 16, 32, and so on, until it reaches a point where environmental limits stop further growth.

Why does exponential growth not occur in nature indefinitely?

Exponential growth doesn't occur indefinitely in nature because resources like food and space are limited. Eventually, populations face environmental constraints that slow down growth, leading to a leveling off at the carrying capacity.

What are 'limiting factors' and how do they affect population growth?

Limiting factors are environmental conditions that restrict population growth. These can include the availability of food, water, space, and other resources. As the population grows, these limiting factors cause the growth rate to slow down, preventing the population from exceeding the carrying capacity.

How does the 'logistic growth equation' differ from the exponential growth equation?

The logistic growth equation includes a term for carrying capacity (K), which modifies the growth rate as the population size (N) approaches K. Unlike the exponential growth equation, which only accounts for r and N, the logistic equation gradually reduces the growth rate as the population nears the environmental limit.

What is the behavior of 'r-selected' species compared to 'K-selected' species?

r-selected species reproduce quickly and in large numbers, typically in environments where resources are abundant but unstable. They experience rapid population booms followed by crashes. K-selected species, on the other hand, reproduce slowly, invest more in offspring survival, and maintain population levels near the carrying capacity.

How does the concept of 'boom and bust' cycles apply to r-selected species?

r-selected species often experience 'boom and bust' cycles, where their populations grow rapidly when resources are plentiful, but then crash when resources become scarce. This pattern repeats as their growth is not limited by parental care or gradual adaptation to the environment.

How do human population growth patterns compare to those of r-selected and K-selected species?

Humans are more similar to K-selected species. They invest heavily in their offspring, which results in slower population growth but greater survival rates. This leads to a gradual increase in population that is more likely to stabilize near a carrying capacity, avoiding the extreme booms and busts seen in r-selected species.

Outlines

00:00

🐇 Introduction to Logistic Growth and Population Dynamics

In this section, Mr. Andersen introduces the concept of logistic growth as a follow-up to exponential growth. He explains how populations, such as rabbits, initially grow rapidly through exponential growth but eventually face limitations due to resource availability, reaching a natural carrying capacity (K). He also breaks down population size (N) and growth rate (r) using rabbits as an example to illustrate how births and deaths affect population dynamics.

05:01

📉 Limitations of Exponential Growth and Carrying Capacity

Mr. Andersen explains the limitations of exponential growth by discussing Darwin's observation on elephants. Although populations can grow quickly under ideal conditions, they eventually reach a carrying capacity due to limited resources. He illustrates this using rabbits and their consumption of grass, showing that once resources are exhausted, populations either stabilize or decline. He also shares a student project demonstrating logistic growth with chickens in a Minecraft simulation, where overcrowding naturally limits further population increase.

10:02

📊 Logistic Growth in Simulation Models

Here, Mr. Andersen introduces a NetLogo simulation called 'Rabbits, Grass, and Weeds' to model population dynamics. He demonstrates how rabbits consume grass for energy and breed, leading to exponential growth at first. However, as the grass depletes, the population eventually stabilizes, hitting a natural carrying capacity. He emphasizes how different environmental conditions (e.g., the amount of energy in the grass) can affect the population's size and growth, leading to new equilibria based on resource availability.

✏️ Mathematical Models of Population Growth

In this section, the focus shifts to the mathematical representation of population growth. Mr. Andersen explains the equation for exponential growth (dN/dt = rN) and shows how changing the variables affects population size over time. He then introduces the logistic growth equation by incorporating carrying capacity (K) into the model. By walking through several iterations, he illustrates how population growth initially mirrors exponential growth but gradually slows down as it approaches K, creating the characteristic S-shaped curve.

🐸 r-Selected and K-Selected Species

This part explores the concepts of r-selected and K-selected species. r-selected species, like frogs, reproduce rapidly and in large numbers, leading to boom-and-bust cycles in population size. In contrast, K-selected species, like chameleons, have fewer offspring but invest more energy into ensuring their survival. Mr. Andersen compares these two strategies and notes that humans, who invest heavily in their young, follow a more gradual population growth pattern with a focus on long-term sustainability.

🌳 Logistic Growth and Species Specialization

In the concluding paragraph, Mr. Andersen ties everything together by discussing logistic growth across various species. He notes that many species, including humans, follow logistic growth patterns, eventually reaching a carrying capacity determined by environmental limits. He also poses thought-provoking questions about where certain species, such as trees, fall on the spectrum between r- and K-selected strategies. The key takeaway is that logistic growth is a balancing act between rapid initial growth and long-term sustainability within an ecosystem's limits.

Keywords

💡Logistic Growth

Logistic growth refers to the type of population growth that slows as it approaches a maximum limit, known as carrying capacity. In the video, it is explained as a process that occurs after exponential growth, when a population reaches environmental limits. The rabbits’ population eventually stabilizes as they run out of grass, illustrating logistic growth.

💡Exponential Growth

Exponential growth occurs when a population increases rapidly over time without environmental constraints. The video illustrates this by showing how a small number of rabbits quickly doubles, quadruples, and grows exponentially. However, exponential growth is unsustainable in nature due to limiting factors.

💡Carrying Capacity (K)

Carrying capacity (K) is the maximum number of individuals that an environment can sustain due to limitations like food, space, and resources. The video shows that once the rabbit population exceeds the available grass, they reach a carrying capacity, and their growth slows down, demonstrating logistic growth.

💡Growth Rate (r)

The growth rate (r) defines how quickly a population changes over time. It is determined by the balance between births and deaths in the population. In the video, the growth rate is shown to influence both exponential and logistic growth, where a higher r leads to faster population increases.

💡Limiting Factors

Limiting factors are environmental elements that prevent a population from growing indefinitely, such as food, water, and space. In the video, grass is a limiting factor for rabbits, as their population growth eventually stops when they consume all the available grass.

💡Boom and Bust Cycle

A boom and bust cycle refers to rapid population growth followed by a sudden decline when resources run out. In the video, this is shown in the context of frogs, whose population quickly expands but then crashes as they exhaust their resources.

💡r-Selected Species

r-selected species are organisms that reproduce quickly and in large numbers, often without much parental care. In the video, frogs are used as an example of r-selected species, which thrive in environments with abundant resources but experience population booms and crashes.

💡K-Selected Species

K-selected species are organisms that grow more slowly, produce fewer offspring, and invest more in parental care. The video contrasts r-selected species with K-selected species, like chameleons and humans, which have fewer offspring but provide more resources to ensure their survival and eventual stabilization near carrying capacity.

💡NetLogo Model

The NetLogo model is a simulation tool used in the video to visualize population dynamics, particularly the interaction between rabbits and grass. This model illustrates how populations fluctuate due to resource availability, showing how rabbits reach and maintain carrying capacity over time.

💡Population Dynamics

Population dynamics refer to the changes in population size and structure over time. In the video, the concept is discussed through different models (rabbits, grass, and chickens) to show how populations grow, fluctuate, and stabilize, depending on various factors like resources and growth rates.

Highlights

Logistic growth occurs when a population reaches a carrying capacity after exponential growth.

In population growth, N refers to the population size, and r is the growth rate determined by births and deaths.

If the growth rate (r) is greater than zero, the population experiences fast exponential growth.

Exponential growth in nature can't last forever and eventually reaches a limit due to resource constraints.

Carrying capacity (K) represents the maximum population size an environment can sustain.

The concept of carrying capacity is illustrated through the example of rabbits eating grass, with population growth slowing when resources become limited.

Students modeled logistic growth using Minecraft by showing how overcrowding limits the release of new chickens, demonstrating carrying capacity.

NetLogo simulation shows that when rabbits deplete grass, population growth decreases until reaching a balanced carrying capacity.

Exponential growth is represented mathematically by dN/dt = rN, where N is the population size, and r is the growth rate.

In logistic growth, population increase slows as it approaches carrying capacity, following the equation dN/dt = rN(K-N/K).

As population size approaches carrying capacity, the factor (K-N)/K decreases, reducing growth.

Species can be classified as r-selected or K-selected based on their growth strategy; r-selected species reproduce quickly with many offspring, while K-selected species grow slowly and invest more in fewer offspring.

Humans invest significantly in offspring, contributing to a gradual exponential increase in population followed by stabilization at carrying capacity.

The 'boom and bust' cycle is characteristic of r-selected species that experience rapid growth followed by crashes when resources are depleted.

Logistic growth is characterized by rapid initial growth that eventually slows and stabilizes as environmental limits are reached.

Transcripts

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

growth. You should probably watch my video on exponential growth if you haven't done

so first. But logistic growth is what happens after. In other words after a population eventually

maybe crashes or finds a natural kind of a carrying capacity. And so let's start by talking

about population growth in general. Imagine we have two rabbits. Our N value is 2. So

N is always going to refer to the population size. Now it's going to change over time,

but to start let's say N is equal to 2. r is going to tell us how fast that population

is going to change. We call that growth rate. It really is determined by two things. Births

and deaths. And let's say that these rabbits have two baby rabbits. And so we'd have two

births minus zero deaths. So that's going to be 2 minus 0 or 2. And then we divide it

by our original N, which was 2. So it's going to be 2 over 2 or 1. And if you have a growth

rate of one, things are going to go really, really quickly. But we could have had a growth

rate of 0.5. How could we get that? Let's say 2 rabbits are born. Minus 1 dies. So that

would be 2 minus 1 or 1. Over 2 and that's going to be 0.5. The take home message about

exponential growth remember is if r is ever greater than zero, we're going to show fast

exponential growth. And so let's say it's 1. What's going to happen? Well those four

rabbits are going to become eight. And then we're going to take 8, in this case, times

the growth rate of 1, and so we're now going to have 16. And now we're going to have 32.

And now we're going to have 64. And we're going to have 128. And we're going to have

256. And pretty soon my whole screen is going to be filled up with rabbits. And I heard

my fan come on on my computer because it's really taxing to put that many rabbits on

the screen. And so does that occur in nature? No. And Darwin noted that as well. He looked

at, instead of rabbits, elephants. Elephants take a long time to reproduce. In other words

they have to be 30 before they can reproduce. And they live until they are about 90. And

what he found was that even though they might produce 3 pair of offspring during that time,

if you just let them keep breeding over and over and over again, and after five centuries,

or 500 years later, you're going to have 15,000,0000 elephants. And so exponential growth goes

really really quickly. And we know that it can't last forever. Eventually we're going

to hit a limit. And so let's look at those four rabbits again. Eventually they're going

to reach what's called K or carrying capacity. It's the maximum amount in a population that

can be supported in a general kind of, in an ecosystem or in an area. And so why is

that? Well rabbits eat things. And that is grass. And when the run out of grass they

die. Now I don't have the ratio right, you have to have way more producers than you're

going to have consumers. And so let's put our rabbits on the grass. What are they going

to do? They're going to eat the grass. And as they do so, that grass is going to go away

and they're going to have to move on to another area. And so luckily there's more grass for

them to find. But eventually they'll eat their way into a corner. So this rabbit right here

has a problem. If this grass didn't grow back, it would really be stuck. And so if we add

much more than four rabbits, we would have exceeded the carrying capacity. And in class

I ask my students to build models that showed logistic growth. And a lot of them use paper

and pencil, but one creative group used Minecraft. And what they did was constructed a little

chamber here. And they put one chicken in it. And every time the chicken would step

on one of these platforms, two doors would open up. And two more chickens would come

out. So pretty soon we had three chickens. And they'd step on platforms. And pretty soon

we had 5 chickens. And it just went really, really, really quickly. Pretty soon there

were chickens everywhere. And I ask them well it's got to show logistic growth. And these

bright students Gabe, Ethan and David said, well watch what happens. Eventually as they

stand on the platforms it becomes so crowded they can't get off anymore. And so they can't

release anymore chickens into the area and so eventually it reaches kind of a carrying

capacity of chickens in this one container. Now let me show you another model. This is

a NetLogo model called Rabbits, Grass and Weeds. So let me launch that for a second.

So in this model the rabbits are going to be these little white things with ears. And

the grass are going to be these little green squares. And so we start with 5. Now the rabbits

are going to find grass if they can. And if they can, they're going to get energy. And

if they get enough energy they can breed. But if they don't find enough grass, then

they're going to die. And so if we let it go for a second, it goes really, really quickly.

So, let me stop it there for a second. So let's watch what happened. And so this red

line right here is going to represent the number of rabbits over time. And so again

we started with 5. And pretty soon we had 350 rabbits. What happened to the grass? Well

the grass started to reproduce as well, but it crashed because the rabbits were eating

all the grass. So let's watch what happens now if we let it roll for a second. So we're

going to see a big drop in the number of rabbits. And then it's going to finally hit a limit.

So we're going to find like a perfect amount of rabbits that we have. So we're reaching

what's called carrying capacity. And you can see that if we let it go over and over and

over, the grass and the rabbits are going to go up and down, but we're going to reach

a limit. Now let's say that we gave the grass more energy. So if there's more energy in

the grass, let's increase that. We're going to see exponential growth again. And then

we're going to have even more rabbits. Or let's say we said that the grass has less

energy, as we decrease that, then we're going to have way fewer rabbits. And so that rabbit

population is going to drop. But it's going to find another carrying capacity. So what

happens as your population size increases? You run up against these limiting factors.

And those limiting factors are going to slow your growth. So let me quit that. So let's

get to the math now. So if we're looking at exponential growth, so exponential growth,

the equation is going to be this right here. And dN over dt means the change in N over

the change in time. And so it's going to be r times N, where r is the growth rate and

N is the population size. And so if we started like we did at the beginning with exponential

growth, you're going to have two rabbits. What's our change in N over change in t? It's

going to be two. How did we get that? We're multiplying r which is 1 remember. Times N

which is 2. And I'm going to get 2. What happens to that change in N? Well that's going to

give me my new population. So I'm taking 2 here plus 2 here and now I have 4. What's

my change in N over time for the next one? Again, since our r is 1. It's going to be

4 again. And so we're going to have 8. We're going to have 16. We're going to have 32.

So if we were to graph that that's going to give us that "J" shaped curve. But what I

want you to do is look at this equation right here and watch as I change it to an equation

that shows carrying capacity. And so the only thing that's going to change is what's in

the parentheses right here. And so that's going to be be the factor that's related to

carrying capacity. And so I'm going to choose an arbitrary carrying capacity. Let's say

it's 10. That the area can only support 10 rabbits. Let's watch what happens. And so

we start with two rabbits. And so I'm going to show you what's in the parentheses here.

And so we're going to have K minus N divided by K. Well what's K? What's our carrying capacity?

That's 10. Minus N. So that's minus 2. So we're going to have 8. And then what's our

carrying capacity? It's 10. So it's going to be 8 divided by 10, or it's going to be

0.8. Okay. Now we're going to figure out this. And so what's this? It's R which we said was

1. Times N, which we said was 2. Times what's in this parentheses, which is 0.8. And so

that's going to be 1.6. And so this whole equation tells us how many new rabbits we're

going to add. And we're not going to round. So I can show you what happens as we increase

this. But know this. That we're going to increase by 1.6 rabbits. So let's look at time 1. Now

we have 3.6 rabbits. Where did I get that? It's the 2 original rabbits we had plus the

1.6. Okay. What's going to be in the parentheses now? It's going to be K, which is 10 - 3.6.

So that's 6.4 divided by 10. And now it's .64. You can see that the number became a

smaller number. We're going to multiply that times 3.6. Again it's 1 times 3.6 times that

value and we get 2.3. I'm going to add that to our our original 3.6 and now we get 5.9.

What's going to be our K minus n over K? It's going to be 0.41. And so what do we get? 2.4.

And so you can see that as the population is increasing, this factor is becoming smaller

and smaller and smaller. And so now we're only adding 1.4. And now we're only adding

0.29. And now we're only adding 0.009. And so what's happening to N? You can see that

it's increasing quickly right away, but the closer we get to that carrying capacity, it's

kind of leveling off. And it's reaching that limit at that one point. So this is just mathematical.

It's not a model. But it shows that we're eventually going to approach that point. And

so again, in review. What's r? r is going to be the growth rate. What's K? K is going

to be the carrying capacity. So if our r is ever greater than 0, remember we're going

to show exponential growth. And if we ever have K, if we ever have a carrying capacity,

we're going to show logistic growth. But we also have species that specialize in these

two. We've got r selected species and K selected species. So what does that mean? An r selected

species is going to be a species that loves to grow as quickly as it can. Look how many

babies this frog is going to have. Thousands of those. And so if I were to take this frog

and introduce it into a pond where there are no frogs, we're going to have a bunch of frogs

really really quickly. We're going to have an r that's really really large. And they're

going to fill that area. They're going to run out of resources and crash. And they're

going to boom and bust, but they're going to fill it really really quickly. If we were

to look at this which is a chameleon. A chameleon is going to only have two offspring. So it's

growth rate is going to be really really slow. What is it going to do? It's going to give

them a lot of parental care. It's going to invest a lot of energy in just its few offspring.

And they're all going to survive. If we look up here with r selected species, most of those

are going to die eventually. But with this chameleon, since it's taking so much care

of them, it's going to take very good care of it, and they're all going to survive. This

is going to gradually increase and find a nice carrying capacity. So what are we? What

are humans? Well we invest a lot in our young. A lot of my students are 17 years old and

they still live at home. Their parents take care of them. So we're really investing in

them. So what kind of population curve are we going to see in humans? We're going to

see a gradual exponential increase and then a nice carrying capacity which we're eventually

going to reach. And we don't want to have this boom and bust cycle. So it's not so cut

and dry as that. There are obviously species that are somewhere in the middle. So what

about a tree do you think? A tree grows really slow but they're going to produce a lot of

offspring. And so what are they? There's probably no right answer for that. And so again, what

is logistic growth? It's growing quickly but eventually reaching a carrying capacity. And

that's based on limits in your environment. And I hope that was helpful.