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## Differential equations

# Worked example: logistic model equations

The general logistic function is N(t)=(N₀K)/(N₀+(K-N₀)e⁻ʳᵗ). In this video, we solve a real-world word problem about logistic growth.

## Want to join the conversation?

- I wonder what is the logic to calculate to get "r", anual populatin growth rate, when N becomes 1.5N in 20 years.

I calculate in this way.

N(o)*exp(r20)= 1.5N(o),

exp(20r)= 1.5

20r = In(1.5)

r = In(1.5)/20

then i got 0.020273... I wonder why these two answer is different.

thx for ur time ^^(35 votes)- OK this confused me too, but I think I figured out where the confusion comes in.

I think that at2:55when Sal says that in 20 years, the population grows by 50%, he's not talking about the imaginary island he just made up. He's talking about unconstrained growth, growth with an infinite K, the non-Malthusian way of estimating population growth. Because remember r is the unconstrained growth constant.

In a world with no K, no cap on the population, the rate of growth would remain the same every year, which is why he can find r by just taking the 20th root of 1.5, resulting in the number 1.0205. If you multiply 1.0205 by your current population, you get the next year's population in an UNCONSTRAINED environment.

THEN Sal plugs in his r to the logistic differential equation to get the actual population growth of the island, with it's environmental constraint K. Notice at6:25that the population of the island after 20 years is almost, but not quite, 150.(9 votes)

- For everyone confused about his r, I have it figured out. The formula for Compound Annual Growth rate (CAGR) is = [(Ending value/Beginning value)^(1/# of years)] - 1. In his example the ending value would be the population after 20 years and the beginning value is the initial population. Since it has grown by 50% we know this ratio will be 1.5, He sets this to the power of (1/20) since we are concerned with the number of years being 20, and then to get 0.0205 he subtracts 1 from 1.02048 and rounds. This gives the ANNUAL growth rate (which is fine in this example because the units for t are years), but if you are concerned with instantaneous growth rate, then there is a simple conversion from your annual growth rate, A, to r (instantaneous). r = ln(1+A). Hope this helped clear some stuff up. Here is the webpage I used for the CAGR formula: http://www.investopedia.com/terms/c/cagr.asp(19 votes)
- When I solve for r using N(20)=150 I get 0.02313.(16 votes)
- This is because N(t) takes into account the population cap K, which stunts growth from the outset. Without K, a yearly growth of 2.05% would bring the population up 50% over 20 years. With K, the function actually requires a higher yearly growth rate to increase by 50% over 20 years, as you have calculated.(6 votes)

- I'm not following what Sal did at3:18when he got out the calculator and got a number for r. Did he just rearrange the whole logistic equation in his head and solve for r, or what?(10 votes)
- this is the equation he used:

future value / present value = (1+i)^n (growth rate equation google it)

i= growth rate

n=number of periods.

150/100=(1+i)^20---> i=[(1.5)^(1/20)] - 1(14 votes)

- The clarification at3:56said that r=ln(1+A). How did that come about?(8 votes)
- Can we find this limit K? Assuming that our No is 2 (lets say Adam and Eve) and that humans have been on earth for approximately 200,000 years an now the population is approximately 7 billion people. Or are we still constrained by r?(3 votes)
- If you assume that a population has grown following the Logistinc function (human population does not), you can simply solve for K, but you would need to know the value of the other variables, namely
`Nₒ`

,`r`

and a point in time`N(t)`

.

In your example you have`Nₒ=2`

, and a point in time`t=2⨉10^5`

and`N(2⨉10^5) = 7⨉10^9`

, but you still lack the value of`r`

to get the value of`K`

.

The equation solved for`K`

is:`Nₒ·e^(-rt) - N(t)`

K = ――――――――――――――――――

e^(-rt) - Nₒ(4 votes)

- Don't understand why he subtracts 1 from his calculation on the calculator to get r. Why do we subtract 1?(4 votes)
- What exactly is the significance of r in this kind of situation? What is its meaning? I don't get why it's "annual growth". Is it a preassumed term?(3 votes)
- In this example,
**r**is growth each year according to the situation set up by Sal. It is combined with t = time, in this case in years. (If time is in years, then r is the growth rate per year. Here, Sal set up a hypothetical situation where the population would grow by 50% in one generation, or about 20 years. He used that to estimate an r to use in this model.

With each kind of organism, r would be different. With bacteria, time would be hours and with mice, time might be in months. The growth rate of bacteria is fast, and the population growth rate of mice is slower than bacteria, but faster than humans by far.

In order to truly determine r for a population, you need to do population studies and counts, and also study such things as whether the population only breeds once a year, or all year long, or faster in warm months/slower in cold months, or only during the week when the Mayflies emerge, which would be true for Mayflies and maybe something else that eats Mayflies. Then you put together your model and finally can run the model and test whether the population behaves anything like your model.(3 votes)

- At3:471.02 changes to 0.020 which changes the final equation. Also, I calculated r by using the given equation with the given n(t)=150, t=20, n(0)=100, and k=1000 to get that r=0.023. What is r=ln(1+A)? I've never seen that before. Can someone explain?

N(t)=(100)(1000)/100+900e^-rt

N(t)=(1000)/1+9e^-rt

150=(1000)/1+9e^-r20

1+9e^-r20=1000/150

1+9e^-r20=20/3

9e^-r20=17/3

e^-r20=17/27

ln(e^-r20)=ln(17/27)

-20r=ln(17/27)

r=-1/20ln(17/27)

r=0.0231

Thank you(4 votes) - There is a little confusion about finding the rate "r" . Could you explain it clearly again?(2 votes)
- r is the growth per year. In other words after 1 year the population grows by a factor of r, after 2 years it grows by a factor of r^2, after 3 years by a factor of r^3, etc. We're given that the population grows by a factor of 1.5 after 20 years, so to find r Sal solved the equation r^20 = 1.5(4 votes)

## Video transcript

- So we've seen in the
last few videos if we start with a logistic differential equation where we have r which is
essentially is a constant that says how fast our we growing when we're unconstrained
by environmental limits. Then we have K which we can view as the maximum population
given our constraints. We saw that if we wanted to solve this, and we didn't want one
of the constant solutions of N of T is equal to zero, or N of T is equal to K, and we did this in the last few videos. We got the solution that N of T is equal to our initial N naught
times our maximum population. All of that over our initial population plus the difference between
our maximum population, and the initial population. So K minus N naught times
E to the negative rt. That this right over here. This logistic function. This logistic function is
a nonconstant solution, and it's the interesting one we care about if we're going to model population to the logistic differential equation. So now that we've done all
that work to come up with this, let's actually apply it. That was the whole goal, was
to model population growth. So let's come up with some assumptions. Let's first think about, well let's say that I have an island. So let's say that this is my island, and I start settling it with a 100 people. So I'm essentially saying N naught Let me do that in the N naught color. So I'm saying N naught is equal to 100. Let's say that this environment, given current technology
of farming and agriculture, and the availability of
water and whatever else, let's say it can only
support 1,000 people max. So you get the idea, so
we get K is equal to 1000. That's the limit to the population. So now what we have to think about is what is r going to be? So we have to come up
with some assumptions. So, let's say in a generation
which is about 20 years, well I'll just assume in 20 years, yeah I think it's reasonable
that the population grows by, let's say that the
population grows by 50%. In 20 years you have 50% growth. 50% increase, increase
in the actual population. So what would you have to
have your annual increase in order to after 20 years to grow by 50%. Well to think about that
I'll get out my calculator. One way to think about it, growing by 50%, that means that you are at
1.5 your original population, and if I take that to the 120th power, and we'll just do 1 divided by 20th. This essentially says how
much am I going to grow by or what is going to, this is telling me I'm going to grow by a factor
of 1.02 every year, 1.02048. So one way to think
about it is if every year I grow by 0.020 I'll just
round five then over 20 years as this compounds I
will have grown by 50%. So that would be our r. This is essentially how much
we're going to grow each year. Let me write that, growth each year. We're going to assume
our t here is in years. So we're going to assume
that our t is in years. So t is in years. So what would our logistic
function look like, given all these assumptions? We would have N of T, let me N of T, is equal to, is equal to N naught times K. That's going to be 100 times 1000. So it's going to be 100 times 1000. My initial population
times my maximum population divided by my initial population, plus the difference between
my final and initial. So that's 1000 minus 100,
so that's going to be, this right over here is going to be 900. 900 times e to the negative r. So the negative 0.0205 times t. So it will be equal to that, and to verify that this actually is, this actually does describe population in the that way we thought the logistic differential equation would let's actually plot it. So let me just pause this
video and then plot it. So there you go, I made a plot and I copy and pasted it here, and we see the behavior
that we wanted to see. We see the population right
over here at year zero, it's starting at 100. Let me do this in a color that
you're more likely to see. Population starts at
100, and we can see that, let's see after 20 years our population looks like it's almost grown to 150. So it looks like, at
least in the beginning, this term, this right
over here is dominating. We are growing by this .0205,
that would be 2.05% per year, which gets us close to 50% growth, and we see that's what
happening initially. So we go from 100 to 150
in the first 20 years, in the first generation. Then in the next generation
we should add another 75 if we weren't being kind of
constrained by the environment. So 150 plus 75 would be 225, and it looks like we got
after 20 years to about 200. So we're a little bit slower. We're a little bit slower than kind of the pure exponential growth. But the pure exponential
growth would probably have us tracking something closer to here, but still growing pretty well. But then as our population gets larger and larger and larger as we're getting closer and closer to
the maximum population, our rate of growth is approaching zero. So we constantly approach
our maximum population, but we never quite get there. It's really an asymptote. We're just approaching it as
time goes on and on and on. But you can kind of set your own threshold and say ok, when do we get to kind of 90% of maximum population? That looks like that happens, 90% of maximum population happened after 210 years on this island. So on a human scale that seems like a long time, many generations. But, I guess in a cosmic
scale it's not that long. Not even a cosmic scale, even just slightly longer
than a human scale. So, it'll happen, well this
describes what's happening. So this is a pretty interesting model, and I'd be interested
to see how it compares with actual data out there
for actual population growth. With that said, it's not this you know, everything that we've done
so far has always assumed, we're kind of assuming this
idea of a Malthusian limit. But what we've learned from human history, that this Malthusian limit
seems to keep getting pushed higher and higher based on
the improvement of technology. That we're able to grow more crops in a certain amount of area. We have better rule of law, so people don't kill each other as often. We have better control
of water and irrigation, and all of these things. So that we're able to increase the limits far beyond what we thought. I would guess if you told Thomas Malthus that in the year 2014
we have seven billion people on the earth, he would have said that's far beyond the Malthusian limit. He probably would have guessed the Malthusian limit was like a billion, or two billion given the
technology of the time, and we're already at 7 billion. As technology improves
and agriculture improves, and rule of law improves,
and everything improves, we might be able to get, who knows? There might be a time,
we might think it's crazy for their to be 20 billion
people on the planet, but given today's technology. But if technology improves
then optimistic scenarios that maybe we could keep going. That's not necessarily a good thing, but that just might be what it is.