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Current time:0:00Total duration:8:45

Per capita population growth and exponential growth

SYI‑1 (EU)
SYI‑1.G (LO)

Video transcript

in the previous video we started thinking about things like population growth rate and how it relates to the birth rate and the death rate within a population and we related that to some of the seemingly complex formulas that you might see on an AP Biology formula sheet now we're going to extend that conversation to discuss some of the other formulas you might see but to realize that they really are just intuition using a little bit of fancy math notation so just as a little bit of review we looked at an example where if in a population the birthrate is 60 bunnies per year we're talking about bunnies here it's a population of bunnies and the death rate is 15 bunnies per year well what's the population growth rate well in a given year you would expect 60 bunnies to be born so that would add to the population and you would expect 15 bunnies to die so that would take away from the population for a net increase of 49 bunnies per year and to put that in the language of your AP Biology formula sheet the notation they use for population growth rate they use a fancy notation so actually let me just write it over here they say n is equal to your population n is equal to population and then your population growth rate they use calculus notation so our change in population / change in time this is really talking about something in calculus known as instantaneous change but we don't have to get too bogged down with that just yet but your population growth rate which you could use this notation for is equal to your birth rate 60 bunnies per year and the notation they use for birth rate is just B they don't use the same rate notation for that I probably would have but that's fine I'm just trying to make you form familiar with what you might see and then minus the death rate minus D so this right over here is something that you would see on that formula sheet but it makes fairly intuitive sense now the next idea we're going to think about is something known as a per capita growth rate of population let me write it out in words first so here we're going to think about a per capita growth rate or population growth rate per capita population growth rate now per capita means you could view it as a on average per individual what is the average growth rate per individual what is that going to be pause this video and try to think about it well one way you could think about it it's the total population growth rate divided by the population divided by the number of people there are so it's going to be our population growth rate growth rate divided by divided by our population population now let's say that we have a population of 300 bunnies so actually let's make the math a little bit simpler let's say we have a population of 450 bunnies so what is going to be our per capita populate per capita population growth rate pause this video and try to figure that out well if we have a population of 450 bunnies 450 bunnies our population growth rate per the number of people we are a number of bunnies I should say is going to be equal to our population growth rate is 45 bunnies bunnies per year and that's going to be for every 450 bunnies 450 bunnies which will get us to 45 divided by 450 is 0.1 and then the unit's bunnies cancel with bunnies so it's 0.1 per year now why is per capita population growth rate interesting well it tells us just how likely and in most populations you need at least a male and a female in order to reproduce but there are some organisms that can just split and and reproduce asexually but it tells us on average per individual that organism per individual organism how much are they are they going to grow per year so it gives you a sense of that now connecting it to the notation that you might see on an AP Biology formula sheet it would look like this the capita population growth rate is usually denoted by the lowercase letter R and then they would say that that is going to be equal to our population growth rate which we've already seen that notation the rate of change of our population with respect to time DN DT divided by our population divided by our population now we can algebraically manipulate this a little bit to get another expression we could multiply both sides times our uppercase n times our population and we're going to get DN DT is equal to n times R or R times n let me rewrite it we could rewrite this as DN DT is equal to our per capita population growth rate times our population now this once again makes sense if you say okay this is how many people how many individuals I have and if in a given year they grow by this much on average well if you multiply the two you'll know how much the whole population has grown so if we didn't know these numbers and someone said hey watch it we could think about this let's think about now a population of a thousand bunnies so if n was equal to 1000 now let's say they're the same type of bunnies that have the same probability of reproducing and the same likelihood so we know that r is equal to 0.1 per year for this population of bunnies what is going to be our population growth rate pause this video and try to figure that out well in this situation DN DT is going to be our per capita population growth rate so it's going to be 0.1 per year times our population times 1000 bunnies bunnies I'll keep my unit's here bunnies and so this is going to be equal to 1 1000 times 1/10 is 100 where in that color it's 100 bunnies bunnies per year so hopefully you're getting an appreciation for why these types of formulas which are fairly straightforward they're using just a fancy notation are useful now this is also an interesting thing to look at because even though that this is in fancy calculus notation and they're saying that our rate of change of population is equal to R times our population this is actually a differential equation if you were to think about what this population the type of population this would describe this would actually be a population that's just that's growing exponentially so this is often known as an exponential growth equation let me write that down X potential exponential growth and in other in your math classes in your calculus classes or even in your precalculus classes you will study exponential growth in a biology class you're really just thinking about how to manipulate this a little bit but just to give you a little sense of what's going on with exponential growth if you have a population of bunnies with this type of exponential growth what is happening here this is time and this is your population so you're going to have some starting population here and it's just going to grow exponentially and the higher the R is the steeper this exponential growth curve is going to be but this describes how populations can grow if they are not constrained by the environment in any way they have just as much land as much water and as much food as they need eventually the bunnies will fill the surface of the earth and the universe now obviously we know that that is not a realistic situation that any ecosystem has some natural carrying capacity there's only so much food there's only so much land at some point there's just going to be bunnies falling from trees and and it's going to be easy for prep much easier for predators to get them and all these other things and we will discuss that in the next video how do we how do we adapt the exponential growth equation right over here to factor it a little bit more of a real-world situation where at some population you're going to be getting you're going to be hitting up against the carrying capacity of the environment
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