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Main content
Current time:0:00Total duration:10:02

Logistic growth versus exponential growth

AP.BIO:
SYI‑1 (EU)
,
SYI‑1.G (LO)
,
SYI‑1.G.1 (EK)
,
SYI‑1.H (LO)
,
SYI‑1.H.2 (EK)

Video transcript

let's now think a little bit more about how we might model population growth and as we do so we're going to become a little bit more familiar with the types of formulas that you might see an AP Biology formula sheet so in a previous video we introduced the idea of per capita growth rate of a population and we used the letter R for that and so let's say that the per capita growth rate for a population is 0.2 that means that on average for every one individual in that population a year later it would have grown by 20% by 2/10 so for every one you would now have 1.2 of that population a year later now as we mentioned many populations that are the ones that reproduce sexually you need at least two a male and a female but there's populations of certain things that can just reproduce on their own they can just bud or they can divide if we're talking about especially unicellular organisms and from this notion we can get a related notion which is our maximum per capita growth rate of a population and let me just write that there you could view this as your per capita growth rate if the population is not limited in any way if there's ample resources water food land territory whatever that population needs to grow but that still is talking about per capita growth rate of population we're just this is the unfettered one this is the maximum and from that we can set up an exponential growth equation and we've seen this in other videos where the rate of change of our population with respect to time and is our population so DN DT is our rate of change of population with respect to time or our population growth rate right over here let me write this down population growth rate if we're dealing with a population that in no way is being limited by its ecosystem which in reality is not realistic at some point you would be well then the rate of great rate of growth of population is going to be your maximum per capita growth rate of population times your population itself and we could see set up a little table here to see how these would relate to each other so let me do that let me set up a table and so let's think about what the rate of change of population will be our population growth rate for certain populations so let's think about what it's going to be when our population is 100 when our population is 500 and when our population is 900 so given these populations what would be your population growth rate for each of them pause this video and try to answer that well when our population is 100 our population growth rate is just going to be 0.2 times that so 0.2 let me write this down this is just going to be DN DT is just going to be 0.2 our maximum per capita growth rate of population times our population times 100 which is equal to 20 so we're going to grow per year by 20 when our population is a hundred now what about when our population is 500 what is going to be our population growth rate pause the video again and try to answer that well once again we just take our maximum per capita growth rate and multiply it times our population so 0.2 times 500 our population growth rate is now 100 if we're talking about bunnies this would be and if our time is in years this would be a hundred bunnies per year or 100 individuals per year and let's think about it when our population is 900 what's our population growth rate then pause the video again alright well we're just gonna take 0.2 times 900 so it is going to be 180 individuals per year now as I just mentioned this is talking about a somewhat unrealistic situation where a population can just grow and grow and grow and never be limited in any way we know that land is limited food is limited water is real imited and so there there's this notion of a natural carrying capacity of a given population in a given environment and to describe that will use the letter K and so let's say we for the organisms that we're studying here let's say they're bunnies and in there bunnies on a relatively small island let's say that the natural carrying capacity for that island is 1,000 that the island really can't support more than 1,000 bunnies so how would we change this exponential growth equation right over here exponential to reflect that well what mathematicians and biologists have done is they've modified this they multiply this times a factor to get us what's known as logistic growth logistic growth so this is exponential growth and we're now talk about is logistic growth and what they do is they start with the exponential growth so my population growth rate you could view as your maximum per capita growth rate times your population so that's exactly what we had right over here but then they multiply that by a factor so that this thing slows down the closer and closer we get to the carrying capacity and the factor that they add is your carrying capacity - your population over over your carrying capacity now let's see if this makes intuitive sense so let's set up another table here and I'll do it with the same values so let's say we have n so our population what's going to be our population growth when our populations 100 when it's 500 and when it's 900 so I encourage you pause this video and figure out what dn/dt is at these various times well at 100 it's going to be I'll do this one I'll write it out it's going to be 0.2 times 100 times R the carrying capacity is 1,000 so it's going to be 1,000 minus 100 all of that over 1,000 so this is 900 over a thousand this is going to be 0.9 and then 0.2 times 100 is 20 so 20 times 0.9 this is going to be equal to 18 so it's a little bit lower its being slowed down a little bit but it's pretty close now let's see what happens when we get to N equals 500 pause this video and figure out what dn/dt our population growth rate would be at that time so in this case it's going to be 0.2 times 500 times 500 times this factor here which is now going to be 1,000 minus 500 that's our population now minus 500 all of that over 1,000 now what's this going to be this is 100 which we had there but it's going to be multiplied by 500 over a thousand which is 0.5 so we're only going to grow half as fast as we were in this situation because once again we don't have an infinite amount of resources here so this is going to be 100 times 0.5 which is equal to 50 and then if you look at this scenario for here when our population is 900 what is dn/dt pause the video again well it is going to be 0.2 times 900 which is 180 times this factor which is going to be 1,000 minus 900 all of that over 1,000 so now this factor is going to be 100 over a thousand which is 0.1 0.1 this part right over here is this part right over here is 180 180 times 1/10 is going to be equal to 18 so now our population growth has slowed down why is that happening here your population rate the rate of growth is growing and growing and growing because the more bunnies or whatever types of individuals you have there's just more to reproduce and there's going to keep growing exponentially but here they're getting closer and closer to the carrying capacity of whatever environment they're at and so 900 they're awfully close so now you're gonna have some bunnies that are going hungry and maybe they're on the move to reproduce as much or maybe they're getting killed or they're they're dying of this is very unpleasant thinking they're they're dying of starvation or they're not able to get water dehydration who knows what might be happening and we could also think about this visually if we were to make a quick graph right over here where if this is time and if this is population our exponential growth right here would describe something that looks like this so for exponential growth our population will grow like this the more our population is the faster it grows the more it is the faster it grows the more it is it faster it grows and will just keep going forever until it just you know there's no limit in theory obviously we know that's not realistic now with a logistic growth I'll do this in red and logistic growth in the beginning it looks a lot like exponential growth it's just a little bit slower but then as the population gets higher and higher it gets a good bit slower and it's limited by the natural carrying capacity of the environment for that population so K would be right over there it would asymptote up to it but not quite approach it and if you want to think at the limit what would happen well what happens at a population of a thousand in this circumstance well then this factor right over here just becomes zero so your population at that point just wouldn't grow any more if you were to even get there
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